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Non-asymptotic fuzzy estimators based on confidence intervals
Information Sciences xxx (2014) xxx–xxx
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Non-asymptotic fuzzy estimators based on confidence intervals D.S. Sfiris 1, B.K. Papadopoulos ⇑ Department of Civil Engineering, Section of Mathematics, Democritus University of Thrace, 12 Vas. Sofias Str, 67100 Xanthi, Greece
a r t i c l e
i n f o
Article history: Received 20 December 2010 Received in revised form 31 August 2013 Accepted 28 March 2014 Available online xxxx 2010 MSC: 03E72 62A86 62F25
a b s t r a c t In this paper, we study and generalize the method of fuzzy estimation which is based on confidence intervals. We provide definitions and the corresponding formulas. The method we propose allows new additional ways in constructing fuzzy estimators. In particular, instead of constructing fuzzy estimators as the exact set of confidence intervals of the parameters to be estimated, we generalize them, in such a way that we eliminate discontinuities, and ensure compact support preserving the statistically derived triangular shape of the estimators. Our approach is particularly useful in critical situations, where subtle fuzzy comparisons between almost equal statistical quantities have to be made. An example of this application is provided. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Fuzzy set theory Fuzzy statistical analysis Tolerance and confidence region
1. Introduction Statistical estimation is one of the main techniques used in inferential statistics. A statistical model may be perceived as a particular type of description of a probability distribution. Such models contain parameters whose values are estimated from samples, a required task to proceed in statistical data analysis [1]. With the growth of fuzzy methodologies, the integration of statistics with fuzzy set theory has attracted considerable attention [6,12,9,22,19,20,3]. Kruse [12] proposed the use of fuzzy linguistic values for estimating the parameters of statistical distributions in a fuzzy way. Gil et al. [9] introduced the notion of fuzzy Bayes’ estimator for real valued parameters. Yao and Hwang [22] investigated point estimators of fuzzy random variables for vague data. Wagner [19] described how to construct a fuzzy number on the basis of confidence intervals from a given sample but only for a determined set of a-cuts. Wu [20] proposed the concept of fuzzy estimators constructed from a family of closed intervals under the consideration of fuzzy random variables. Using a different approach, Buckley [3] proposed a fuzzy estimation method in order to construct fuzzy numbers for parameters in probability density (mass) functions that have been estimated from random samples, using all confidence intervals. Various applications of this approach to fuzzy estimation can be found in the literature (see [16,10,17,11]). Falsafain et al. [7] studied the construction of explicit and unique membership functions for such fuzzy estimators and recently Falsafain and Taheri [8] presented an improvement for non-symetric distributions. In this paper we generalize the fuzzy estimation approach established by Buckley [3]. ⇑ Corresponding author. 1
E-mail addresses: dsfi[email protected] (D.S. Sfiris), [email protected] (B.K. Papadopoulos). Principal corresponding author.
http://dx.doi.org/10.1016/j.ins.2014.03.131 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
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Our motivation in based in the need to compare two almost equal statistical quantities. In such critical situations, the shape of the fuzzy numbers that are being produced is very important. While in the literature [3,7,8], fuzzy estimation being an extension of confidence interval estimation makes an assumption that the shape of fuzzy estimators is bound to the exact set of confidence intervals, we show that this is not the only possible way to construct fuzzy estimators. We provide a generalization which not only includes previous approaches as a subcase, but also permits the construction of many families of such fuzzy estimators which we call them non-asymptotic fuzzy estimators (NAFEs). The new generalized definition guarantees that NAFE are proper fuzzy numbers, preserving the statistically derived triangular shape and are appropriate for subtle fuzzy comparisons. An extra benefit of our proposed method is that one can now choose how close to be to the original point estimation, that is how strict is the fuzzy estimation. We show that the comparison of fuzzy estimators is greatly affected by their shapes, so a criterion is required to decide which shape is appropriate and we introduce the false alarm rate index. As an experiment, we tested the behavior and effectiveness of seven families of NAFE. The results have shown that by implementing NAFE on almost equal statistical quantities, we can significantly reduce the false alarm rate, i.e. we can correctly find which option is slightly better than the other when using either point or confidence interval estimation or fuzzy estimation. Thus our own non-asymptotic fuzzy estimators clearly address the issue of comparing almost equal quantities. An application of NAFE in the sensitive area of financial decisions is presented. For more applications of NAFE see [4,18]. The remainder of this paper is organized as follows: In Section 2.1 we introduce notions and definitions from Fuzzy Set theory. In Section 2.2 we review the literature and particularly Buckley’s method of fuzzy estimation. In Section 3 we present the new method of non-asymptotic fuzzy estimators, in Section 3.2 we develop the corresponding membership function of non-asymptotic fuzzy estimators and in Section 3.3 we propose membership functions for some common statistical models. In the next Sections 4, 5 we discuss a fuzzy comparison method and provide the falst alarm criterion for the selection of the appropriate shape of the estimators. Section 6 is an application. Our conclusions follow in Section 7. 2. Fuzzy estimation based on confidence intervals 2.1. Fuzzy sets notions and definitions Before presenting the fuzzy estimation method, we introduce the relevant mathematical notations and the basic notions from the fuzzy set theory. ~ : X ! ½0; 1 is called a fuzzy set or a fuzzy subset of X, where AðxÞ e Let X be a universal set. Every function of the form A is e We place a tilde over a fuzzy set symbol so as to distinguish it interpreted as the membership degree of x in the fuzzy set A. e from classical set. Classical sets are also called crisp sets and are special cases of fuzzy sets where AðxÞ is only zero or one. e are defined by the sets The a-cuts of a fuzzy set A
n o ~ A ¼ x 2 R : AðxÞ P a ; a 2 ð0; 1
a~
ð1Þ
~ is the closure in the topology of X of the union of all the a-cuts [5], that is while its support 0 A
n o ~ ~ ¼ x : AðxÞ >0 : A ¼ [a2ð0;1 a A
0~
ð2Þ
e We say that A e is a fuzzy number if the following conditions It is known that a-cuts uniquely determine the fuzzy set A. hold: e is normal, that is there exists x 2 R such that A ~ ðxÞ ¼ 1, 1. A e is a convex fuzzy set, that is for every t 2 ½0; 1 and x1 ; x2 2 R, we have 2. A
n o ~ ðð1 t Þx1 þ tx2 Þ P min Aðx ~ 1 Þ; Aðx ~ 2Þ ; A e is upper semi-continuous on R, i.e. 8x0 2 R and 8e > 0 there exists a neighborhood V ðx0 Þ such that 3. A uðxÞ 6 uðx0 Þ þ e; 8x 2 V ðx0 Þ, e Eq. (2), is compact. 4. the support of A, e can be written as intervals of the form We also mention that the a-cuts of a fuzzy number A a~
A ¼ ½A1 ðaÞ; A2 ðaÞ
ð3Þ
where A1 ðaÞ; A2 ðaÞ can be regarded as functions on ½0; 1 [15]. Then, 1. A1 ðaÞ is non-decreasing and left continuous, 2. A2 ðaÞ is non-increasing and left continuous, 3. A1 ðaÞ 6 A2 ðaÞ.
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Conversely, according to Wu and Zhang [21], for any functions A1 ðaÞ and A2 ðaÞ defined on ð0; 1 which satisfy 1–3 in the ~ such that Eq. (3) holds for any a 2 ð0; 1. If A ~ is a normal fuzzy set, a A ~ are closed interabove, there exists a unique fuzzy set A ~ 0 A, ~ is bounded, then A ~ is also a fuzzy number. vals for every a 2 ð0; 1 and the support of A; The graph of a fuzzy number may have various shapes. A triangular fuzzy number is defined by a triplet ða1 ; a2 ; a3 Þ where the vertex of the triangle is at x ¼ a2 and its base is the interval ½a1 ; a2 . A triangular shaped fuzzy number is partially derived by a1 < a2 < a3 because its sides are continuous curves, not straight line segments. We also refer to complete fuzzy numbers which implies that their support is closed and bounded (compact). For the realization of arithmetic operations we use the following a-cuts operations, based on the extension principle as ~ and B ~ be two fuzzy numbers and [23]. Let A a~
~ ¼ ½B1 ðaÞ; B2 ðaÞ A ¼ ½A1 ðaÞ; A2 ðaÞ; a B
be their a-cuts. The fuzzy addition and fuzzy subtraction are defined as a
h
i ~B ~ ¼ ½A1 ðaÞ þ B1 ðaÞ; A2 ðaÞ þ B2 ðaÞ; A
a
h
i ~B ~ ¼ ½A1 ðaÞ B2 ðaÞ; A2 ðaÞ B1 ðaÞ A
respectively for every a 2 ½0; 1. The multiplication of a fuzzy number with a scalar may be defined as a
h
i ~ ¼ ka A e ¼ ½k; k½A1 ðaÞ; A2 ðaÞ ¼ ½kA1 ðaÞ; kA2 ðaÞ; ifk P 0 kA ½kA1 ðaÞ; kA2 ðaÞ; ifk < 0
where the scalar k is identified as the closed interval ½k; k. ~B ~øB ~ and B, ~ and division A ~ of fuzzy numbers A ~ based on Zadeh’s extension principle [13, p.44-49], are The product A defined as a
h
i ~B ~ ¼ ½min ðA1 a B1 a ; A1 a B2 a ; A2 a B1 a ; A2 a B2 a Þ; max ðA1 a B1 a ; A1 a B2 a ; A2 a B1 a ; A2 a B2 a Þ A
a
h
i ~øB ~ ¼ ½min ðA1 a =B1 a ; A1 a =B2 a ; A2 a =B1 a ; A2 a =B2 a Þ; max ðA1 a =B1 a ; A1 a =B2 a ; A2 a =B1 a ; A2 a =B2 a Þ A
where A1 a ¼ A1 ðaÞ; A2 a ¼ A2 ðaÞ B1 a ¼ B1 ðaÞ; B2 a ¼ B2 ðaÞ for short and 0 R
aB ~
for the division.
2.2. Buckley’s method of fuzzy estimation In this section, we recall the fuzzy estimation method of [3] and present an example to be used later for comparison with our new method. Let X be a random variable with probability density function f ðx; hÞ for single parameter h. Assume that h is unknown and it must be estimated from a random sample X 1 ; X 2 ; . . . ; X n . Denote ð1 bÞ100% confidence intervals for h by ½h1 ðbÞ; h2 ðbÞ for all 0:01 6 b < 1. Starting at 0.01 is arbitrary and you could begin at 0.001 or 0.005, etc. Add to this the interval ½h ; h for the 0% confidence interval for h. Then place these confidence intervals, one on top of the other, to produce a triangular shaped fuzzy number ~ h whose a-cuts are the following confidence intervals a~
h ¼ ½h1 ðaÞ; h2 ðaÞ; 0:01 6 a 6 1:
ð4Þ
h in order to make it a complete fuzzy number, one needs to drop the graph of ~ h straight down to To finish the ‘‘bottom’’ of ~ complete its a-cuts, a~
h ¼ ½h1 ð0:01Þ; h2 ð0:01Þ; 0 6 a < 0:01:
ð5Þ
An example to estimate the mean of the normal distribution follows. Numerical Example 1. Let x1 ; x2 ; . . . ; xn be a sample of n ¼ 100 observations drawn from a normal population. Let us consider that the observations have an unknown mean value l and a known variance r2 ¼ 16. Suppose the sample mean value turns out to be x ¼ 40. Then, according to the method described, the fuzzy estimator for the mean l is a fuzzy number l~ whose a-cuts are defined by the ð1 bÞ100% confidence intervals of l for all amax 6 b 6 1, where amax corresponds to the ~ before we drop its membership function straight down. Thus, if amax ¼ 0:1 (90% confidence interval), then largest a-cut of l based on Eq. (4) and Eq. (5) we obtain
8h 4 4 i > ffi ; 40 þ U1 1 a2 pffiffiffiffiffi ffi ; a 2 ½0:1; 1 < 40 U1 1 a2 pffiffiffiffiffi 100 100 a~ i l¼ h 1 1 > 4 ffi 4 ffi : 40 U 1 0:1 pffiffiffiffiffi pffiffiffiffiffi ; 40 þ U 1 0:1 ; a 2 ½0; 0:1Þ 2 2 100 100
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where U1 denotes the reverse cumulative distribution function of the standard normal distribution. Fig. 1 illustrates the shape of the fuzzy estimator obtained for this example. Notice that the support of a fuzzy estimator as above, asymptotically extends from 1 to þ1 and its a-cuts become exactly the confidence intervals of the estimated parameter. Hence we may say that the above estimators are asymptotic in nature. 3. Non-asymptotic fuzzy estimators 3.1. The proposed definition We now present how to generalize the fuzzy estimators, showing new ways to construct them. Let X 1 ; X 2 ; . . . ; X n be a random sample of size n from a distribution with unknown parameter h and let ½h1 ðaÞ; h2 ðaÞ denote the ð1 aÞ100% confidence intervals for h. Using any monotonic increasing, continuous and onto function
hðaÞ : ð0; 1 !
hc 2
i ; 0:5 ; c 2 ð0; 1Þ
ð6Þ
then a family of fuzzy estimators h~c is derived by the following superimposed intervals a~ hc
¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ; a 2 ð0; 1:
ð7Þ
We call them non-asymptotic fuzzy estimators (NAFE) and we will prove in Theorem 1 that they are proper fuzzy numbers with compact support. Theorem 1. A non-asymptotic fuzzy estimator is a complete triangular shaped fuzzy number whose a-cuts are the ð1 2hðaÞÞ100%; a 2 ð0; 1, confidence intervals for h and its compact support is exactly the ð1 cÞ100% confidence interval for h. Proof. By definition, hðaÞ is a monotonic increasing, continuous and onto function. We conclude that h1 ð2hðaÞÞ is monotonic non-decreasing and continuous as a composite function and h2 ð2hðaÞÞ is monotonic non-increasing and continuous, hence, ð1 2hðaÞÞ100% confidence intervals are nested, therefore h1 ð2hðaÞÞ 6 h2 ð2hðaÞÞ. From Theorem 2.1 of Wu and Zhang [21] (see Section 2) and based on the previous conditions, there exists a unique fuzzy set that can be constructed from the set of ð1 2hðaÞÞ100% confidence intervals for h. To prove that this fuzzy set is a fuzzy number, it is sufficient to prove that all the a-cuts are contained in a closed and bounded interval. This interval is 0 h~c ¼ ½h1 ð2hð0ÞÞ; h2 ð2hð0ÞÞ ¼ ½h1 ðcÞ; h2 ðcÞ, which is exactly the ð1 cÞ100% confidence interval for h. h According to definition, choosing a proper monotonic increasing, continuous and onto function (6) allows us to construct fuzzy estimators of a particular NAFE family associated with it. The following seven functions are proper candidates, but not the only ones: 1. hðaÞ ¼ 2. hðaÞ ¼ 3. hðaÞ ¼ 4. hðaÞ ¼ 5. hðaÞ ¼ 6. hðaÞ ¼ 7. hðaÞ ¼
1 2 1 2 1 2 1 2 1 2 1 2
2c a þ 2c (Linear NAFE), 2c a2 þ 2c (Square NAFE), 2c a2 þ 3c (Cube NAFE), pffiffiffi 2c a þ 2c (Square Root NAFE), 2c a1=3 (Cube Root NAFE), 2c a1=3 (Power of 1/4 NAFE),
a 2
c
2
;a P c . (Buckley’s fuzzy estimators). ;a < c
Fig. 1. Fuzzy estimator for the mean
l of normal distribution of Example 1.
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Each function determines the shape of the membership function of the fuzzy estimators, where the support of each estimator is the ð1 cÞ100% confidence interval. For example one can choose c ¼ 0:1 (90% confidence interval), c ¼ 0:05 (95% confidence interval) or any other value. Thus we generalize Buckley’s fuzzy estimators which are the case seven above. 3.2. Membership function of non-asymptotic fuzzy estimators Constructing a fuzzy estimator based on its a-cuts definition does not directly relate any point x with a membership value. [7] proved explicit and unique membership functions for Buckley’s fuzzy estimators. In Theorem 2, we prove the general membership function for all non-asymptotic fuzzy estimators. Theorem 2. Suppose that X 1 ; X 2 ; . . . ; X n is a random sample of size n from a distribution with unknown parameter h. If ½h1 ðaÞ; h2 ðaÞ denote the ð1 aÞ100% confidence intervals for h, then a non-asymptotic fuzzy estimator of h has the following membership function
n o h~c ðxÞ ¼ min ðpÞ1 ðxÞ; ðqÞ1 ðxÞ; 1 ; where p ¼ h1 2h; q ¼ h2 2h denote compositions of functions and hðaÞ : ð0; 1 ! continuous and onto function.
c 2
; 0:5 ; c 2 ð0; 1Þ is monotonic increasing,
Proof. Since, by definition, the a-cuts of a non-asymptotic fuzzy estimator are a h~c ¼ ½ðh1 2hÞðaÞ; ðh2 2hÞðaÞ or ¼ ½pðaÞ; qðaÞ, it holds that, if x 2a h~c , then pðaÞ 6 x 6 qðaÞ. Hence, from pðaÞ 6 x we obtain a 6 ðpÞ1 ðxÞ, and from n o x 6 qðaÞ we obtain a 6 ðqÞ1 ðxÞ. Moreover, a 6 1, therefore a ¼ min ðpÞ1 ðxÞ; ðqÞ1 ðxÞ; 1 ¼ h~c ðxÞ. h
a h~ c
Theorems 1 and 2 hold for all probability distributions. In the following section, we propose specific membership functions for the parameters of some widespread statistical models. 3.3. Using non-asymptotic fuzzy estimators to estimate the parameters of the normal and exponential distributions Let us consider the linear function
hðaÞ ¼
1 c c a þ ; a 2 ð0; 1; c 2 ð0; 1Þ: 2 2 2
ð8Þ
Proposition 1 (Non-asymptotic fuzzy mean of normal distribution – large samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is large enough, then
8
c 2 > pxffiffi U rx ; x prffiffin U1 1 2c 6 x 6 x > 1 c 1c > = n <
c l~ c ðxÞ ¼ 2 U xx pffiffi prffiffi 1 1 2c > 1c r= n 1c ; x 6 x 6 x þ n U > > : 0; otherwise
ð9Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for l and the acuts of this fuzzy number are the closed intervals a
r r l~c ¼ x zhðaÞ pffiffiffi ; x þ zhðaÞ pffiffiffi ; a 2 ð0; 1 n
where zhðaÞ ¼ U1 ð1 hðaÞÞ; hðaÞ ¼ distribution.
n
1 2
ð10Þ
2c a þ 2c, and U denotes the cumulative distribution function of the standard normal
Proof. We first show that there exists a unique fuzzy number whose a-cuts are given in Eq. (10); then beginning from these a-cuts, we prove the membership function of this fuzzy number. Let qðaÞ ¼ x zhðaÞ prffiffin and rðaÞ ¼ x þ zhðaÞ prffiffin be functions on ð0; 1. We prove that qðaÞ 6 r ðaÞ. This is equivalent to
r r 1 x zhðaÞ pffiffiffi < x þ zhðaÞ pffiffiffi () zhðaÞ > 0 () U1 ð1 hðaÞÞ > 0 () hðaÞ < 2 n n which is always true.
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Now, let a1 P a2 . We prove that qðaÞ is non-decreasing. Indeed,
r
r
qða1 Þ P qða2 Þ () x zhða1 Þ pffiffiffi P x zhða2 Þ pffiffiffi () n n zhða1 Þ 6 zhða2 Þ () U1 ð1 hða1 ÞÞ 6 U1 ð1 hða2 ÞÞ () hða1 Þ P hða2 Þ which is always true, since hðaÞ is non-decreasing. Similarly, we can prove that r ðaÞ is non-increasing. Finally, we notice that qðaÞ and rðaÞ are continuous as a composition of continuous functions. Consequently, according to Theorem 2.1 [21] (see Section 2), there exists a unique fuzzy number l~c : X ! ½0; 1 determined by Eq. (10). Lastly, we only need to construct the membership function of this fuzzy number to close the proof (see Appendix A). h Remark 1. If the sample size is small and it has been drawn from a normally distributed population with unknown r, then r in Eq. (9) and (10) is replaced by s and zhðaÞ is replaced by thðaÞ , where t hðaÞ ¼ T 1 ð1 hðaÞÞ and T denotes the cumulative distribution function of the t-student distribution with m ¼ n 1 degrees of freedom. The proof can be done in an analogous way to the proof of Proposition 1. Numerical Example 2. We consider the same example presented in Section 2.2, this time to construct a non-asymptotic fuzzy estimator for l. To obtain the same support for the fuzzy estimator, we used 90% confidence interval for l, that is ~ c is c ¼ 0:1. Deploying Eq. (9), the membership function of l
l~ 0:1
8
2 4 ffi 1 > pffiffiffiffiffiffi 0:1 ; 40 pffiffiffiffiffi U 4=x40 U ð0:95Þ 6 x 6 40 > 0:9 0:9 > 100 100 <
¼ 2 4 ffi 1 pffiffiffiffiffiffi 0:1 ; 40 6 x 6 40 þ pffiffiffiffiffi U 4=40x U ð0:95Þ > 0:9 0:9 100 100 > > : 0; otherwise
~ c (Eq. (10)) are The a-cuts of l a
4
4
100
100
l~c ¼ 40 z0:9a2þ0:1 pffiffiffiffiffiffiffiffiffi ; 40 þ z0:9a2þ0:1 pffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1
Fig. 2 illustrates the shape of the fuzzy estimator obtained for this example. Proposition 2 (Non-asymptotic fuzzy variance of normal distribution – small samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is small, then
8
2 n1Þs2 ðn1Þs2 2c > > 12 c F ðn1x Þs ; ð1 2c 6 x 6 > 1 c M F ð 2 Þ > <
2 2 2 2 ~ ð n1 Þs ð n1 Þs ð n1 c r c ð xÞ ¼ 2 F 1c ; M 6 x 6 F 1 Þsc > 1c x > ð2Þ > > : 0; otherwise
ð11Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for r2 and the acuts of this fuzzy number are the closed intervals
1 0.8 0.6 0.4 0.2
37.45
40
42.55
x
Fig. 2. Non-asymptotic fuzzy mean estimator of Example 2.
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a ~2 rc
¼
" # ðn 1Þs2 ðn 1Þs2 ; a 2 ð0; 1 ; 2 2
ð12Þ
vn1;hðaÞ vn1;1hðaÞ
where v2n1;: is the value of the chi-squared distribution with k ¼ n 1 degrees of freedom, v2n1;hðaÞ ¼ F 1 ð1 hðaÞÞ; hðaÞ ¼ 12 2c a þ 2c, and F denotes the cumulative distribution function of the chi-squared distribution. M is the median of the distribution. Proof. We proceed to this proof based on Theorems 1 and 2. The ð1 aÞ100% confidence intervals for
r2
1a
r2 are
" # ðn 1Þs2 ðn 1Þs2 ¼ ; 2 ; 2
vn1;a=2 vn1;1a=2
therefore, according to Theorem 1, the non-asymptotic fuzzy estimator for r2 is a fuzzy number the a-cuts of which are the closed intervals
"
a ~2 rc
ðn 1Þs2 ðn 1Þs2 ¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ ¼ ; 2 2
#
vn1;hðaÞ vn1;1hðaÞ
~ 2c ¼ ½pðaÞ; qðaÞ. After calculating p1 ðxÞ and ðqÞ1 ðxÞ, We now consider pðaÞ ¼ h1 ð2hðaÞÞ and qðaÞ ¼ h2 ð2hðaÞÞ, so that a r we obtain
p1 ðxÞ ¼
2c 2 ðn 1Þs2 F 1c 1c x
and
ðqÞ1 ðxÞ ¼
2 ðn 1Þs2 c F 1c x 1c
and therefore, based on Theorem 2,
n
o
r~ 2c ðxÞ ¼ min p1 ðxÞ; ðqÞ1 ðxÞ; 1
that is equal to the proposed membership function.
h
Proposition 3 (Non-asymptotic fuzzy variance of normal distribution - large samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is large enough, then
8
qffiffiffiffiffiffiffi
2c n1 s2 s2 pffiffiffiffiffi > 12 c U 1 ; 6 x 6 s2 > c 2 2 1 c x > 1þU1 ð12Þ n1 > < q ffiffiffiffiffiffi ffi
r~ 2c ðxÞ ¼ 2c 2 U n1 1 s2 ; s2 6 x 6 s2 pffiffiffiffiffi > c 2 x 2 > 1c 1c 1U1 ð12Þ n1 > > : 0; otherwise
ð13Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for r2 and the acuts of this fuzzy number are the closed intervals
"
a ~2 rc
# s2 s2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1 ¼ 1 þ zhðaÞ 2=n 1 1 zhðaÞ 2=n 1
where z2hðaÞ ¼ U1 ð1 hðaÞÞ , hðaÞ ¼ distribution.
1 2
ð14Þ
2c a þ 2c, and U denotes the cumulative distribution function of the standard normal
Proof. The proof is analogous to those provided for Propositions 1 and 2. h Proposition 4 (Non-asymptotic fuzzy mean of exponential distribution). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. Then
1 a~ k
c
ðxÞ ¼
8 ^ ^k 1 ðxÞ; F 12nkc 6 x 6 2n > c 2 > M F 2nx^kÞ ð 2Þ > 2 1c 1c ð < ^
^
1 2nk k 6 x 6 12n2 c ; c 2 F 2nx^ > kÞ1c M F ð 2 Þ > 1c ð > : 0; otherwise
ð15Þ
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for l and the acuts of this fuzzy number are the closed intervals
" 1 1 ¼ a~ ^k kc
2n
1 ; v22n;hðaÞ ^k
#
2n
v22n;1hðaÞ
; a 2 ð0; 1
ð16Þ
x is the maximum likelihood estimate, v22n;: is the value of the chi-squared distribution with k ¼ 2n degrees of freedom, k ¼ 1= where ^ 1 2 v2n;hðaÞ ¼ F ð1 hðaÞÞ, hðaÞ ¼ 12 2c a þ 2c, and F denotes the cumulative distribution function of the chi-squared distribution. M is the median of the distribution. Proof. The proof is analogous to those provided for Propositions 1 and 2. h
4. Fuzzy hypothesis testing Buckley [3] and Taheri and Arefi [17] described how fuzzy hypothesis testing is rendered into the comparison of two fuzzy estimators. Here we discuss the comparison of non-asymptotic fuzzy estimators. Let X 1 ; X 2 ; . . . ; X n ; Y 1 ; Y 2 ; . . . ; Y n be two random samples from a variable X with distinct probability density functions ~c ; q ~ c be two non-asymptotic fuzzy estimators constructed based fx ðx; hÞ and fy ðx; qÞ of unknown parameters h and q. Let also h on sample observations x1 ; x2 ; . . . ; xn and y1 ; y2 ; . . . ; yn . To evaluate whether the second fuzzy estimator is greater than the first, we use the following fuzzy decision rule: h1;c if AR =AT P /, 1. H0 : ~ h2;c is greater than ~ 2. H1 : ~ h1;c if AR =AT < /, h2;c is approximately ~
where a~ hc
¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ; a P 0
a~
qc ¼ ½q1 ð2hðaÞÞ; q2 ð2hðaÞÞ; a P 0
are the a-cuts of the fuzzy estimators from Eq. (1), 0~
~ c ¼ ½q1 ðcÞ; q2 ðcÞ hc ¼ ½h1 ðcÞ; h2 ðcÞ;0 q
are the supports of the fuzzy estimators,
Z
q2 ðcÞ
q~ c ðxÞdx
q1 ðcÞ
~ c, is the total area under the fuzzy number q
AR ¼ AT
Z s
q~ c ðxÞdx
Z
h2 ðcÞ
~hc ðxÞdx
s
q1 ðcÞ
~ c but to the right of ~ hc , and is the area under the graph of q
n
o
~ ~ 1 ~ c ð xÞ s ¼ ~h1 c ðas Þ ¼ q c ðas Þ; as ¼ sup min hc ðxÞ; q x2X
~ c ðxÞ has the highest membership value. / is a constant control parameter expressing hc ðxÞ; q is point where the intersection of ~ the desired ratio of the second fuzzy number exceeding the first. ~ c ðxÞ are known, one can find an analytical expression for as , i.e. considhc ðxÞ; q Note that when the membership functions ~ ering the non-asymptotic fuzzy mean estimators for normal distribution (Proposition 1), we obtain r
as ¼
1ffi pffiffiffi x n1 2
r1 ffi pffiffiffi n1
2ffi þ prffiffiffi x n2 1 2ffi þ prffiffiffi n2
:
Numerical Example 3. Let X 1 ; X 2 ; . . . ; X n and Y 1 ; Y 2 ; . . . ; Y n be two normally distributed random samples of size n ¼ 1000 from N l1 ; r21 and N l2 ; r22 respectively. Given observed values x1 ; x2 ; . . . ; xn ; y1 ; y2 ; . . . ; yn , the sample means turned out to be x1 ¼ 5:0586; x2 ¼ 5:1456 and the sample standard deviations s1 ¼ 2:4711; s2 ¼ 6:3365. We are interested in testing the fuzzy null hypothesis H0 : l1 is greater than l2 .
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For this example, we constructed seven families of NAFE (linear, square, cube, square root, cube root, Power of 1/4 and Buckley’s type) with a support of ð1 cÞ100% ¼ 90% confidence interval. The general form of their a-cuts is
a~
2:4711
2:4711
l1;0:1 ¼ 5:0586 zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 5:0586 þ zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1
a~
1000
1000
6:3365
6:3365
l2;0:1 ¼ 5:1456 zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 5:1456 þ zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1 1000
1000
and their supports are 0~
l1;c ¼ ½4:924; 5:182;0 l~ 2;c ¼ ½4:811; 5:470
We compare the fuzzy estimators by applying the methodology described in this section. Fig. 3 illustrates the areas under two non-aysmptotic fuzzy estimators needed for the calculation of AR as AR ¼ AT E1 E2 . The result of the comparison for all NAFE families for / ¼ 0:4 is shown in Table 1, Column 6. As we see, the shape of the fuzzy estimators used may totally change the final decision. Using Cube, Square NAFE or Buckley’s estimation, the suggestion is to reject the hypothesis H0 . On the contrary, using Linear, Sqrt, Cube root or Power of 1/4 NAFE the suggested decision is to accept H0 . The question which shape is appropriate is discussed in the next section. 5. The false alarm criterion We now provide a criterion for evaluating our choices with regards to the shape of the fuzzy estimators. We call it the false alarm rate index. It will be explained by an example. Assume that we have 100 pairs of randomly generated data samples from two normal populations, where l1 ¼ 5; l2 ¼ 5:05 and standard deviations r1 ¼ r2 ¼ 2:5. Each sample has size 1000. Seven shapes of NAFE (linear, square, cube, square root, cube root, Power of 1/4 and Buckley’s type) are considered with a support of ð1 cÞ100% ¼ 90% confidence interval (Fig. 4). The goal is to confirm the null hypothesis H0 : l2 greater than l1 . The methodology described in Section 4 was applied allowing us to count the number of cases where the fuzzy comparison approach suggested to reject H0 when x1 < x2 (Nr). Let Np be the number of cases, where x1 < x2 . The false alarm rate criterion is defined as
FAindex ¼
Nr : Np
The appropriate shape has the lowest false alarm value. 1 0.8 0.6 0.4
AR
0.2
E1
4.924
4.811
E2 5.182
5.470
x
Fig. 3. Fuzzy comparison of two linear NAFE of Example 3.
Table 1 Comparison results. NAFE family
AT
E1 þ E2
AR
AR =AT
Decision
Cube Square Buckley’s Linear Sqrt Cube root Power of 1/4
0.4436 0.3795 0.3031 0.2635 0.1637 0.1196 0.0946
0.3202 0.2649 0.1936 0.1547 0.0692 0.0373 0.0215
0.1234 0.1146 0.1095 0.1087 0.0964 0.0823 0.0732
0.2782 0.3019 0.3613 0.4127 0.5775 0.6881 0.7729
Reject H0 Reject H0 Reject H0 Accept H0 Accept H0 Accept H0 Accept H0
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
The false alarm rates for 10 different / values are given in Table 2 of Appendix B. Linear, Square root and Power of 1/4 NAFE produced a lower false alarms rate than Buckley’s fuzzy estimators, while the Square and Cube fuzzy estimators produced a higher false alarms rate. This is so regardless of any particular c value, showing the stability of the methodology. 6. A critical situation in a financial decision The problem discussed below has also been our motivation to study various shapes of fuzzy estimators. Suppose that an investor has two almost similar choices. According to decision theory, the final choice can be determined by calculating the 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 x1
x2
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x1
x2
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x1
x2
x
x1
x2
x
x1
x2
x
x1
x2
x
1 0.8 0.6 0.4 0.2
x1
x2
x
Fig. 4. Shapes of various fuzzy estimators.
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
11
expected utility of each choice and then choosing the one with the higher expected utility. An example of such a utility function is the exponential utility function given by
UðxÞ ¼ 1 expkx ; k > 0
ð17Þ
satisfying
U 0 ðxÞ ¼ kexpkx > 0; U Prime ðxÞ ¼ k2 expkx < 0; k¼
U 00 ðxÞ U 0 ðxÞ
where k is the degree of risk-sensitivity [2]. The non-asymptotic fuzzy estimators will decide the best investment choice. Rather than directly maximizing the expected utility, it is possible, under certain assumptions, to reach to the same solution following the Mean–Variance approach [14]. Thus, if the choices have normally returns, then the expected utility can be expressed as a function of the mean and variance k2 2
E½UðxÞ ¼ 1 expklþ 2 r
ð18Þ
All we have to do is to estimate the l and r and then using Eq. (18) we can obtain an estimation for E[U (x)]. Finally, using the a-cuts notation and operations (see Section 2), Eq. (18) is fuzzified as follows: 2
a
k2 2 k2 2 k2 2 2 a k2 a E~c ½UðxÞ ¼ 1 expk l~ c þ 2 r~2 c ¼ 1 expk½l1 ðaÞ;l2 ðaÞþ 2 ½r1 ðaÞ;r2 ðaÞ ¼ 1 exp kl1 ðaÞþ 2 r1 ðaÞ;kl2 ðaÞþ 2 r2 ðaÞ
a
Ec ½U ðxÞ ¼ ½LðaÞ; RðaÞ; a > 0
or
k xzhðbÞ psffi k2
LðaÞ ¼ 1 exp
n
2
ð19Þ
s2
1þzhðbÞ
pffiffiffiffiffi 2
n1
Table 2 Mean values and variances. Investment choices
Mean
Choice 1 Choice 2
5.8 7.5
l
Variance
r2
3.460 8.062
Table 3 Expected utilities. Investment choices
Expected utilities
Choice 1 Choice 2
0.795 0.850
Fig. 5. Fuzzy expected utilities.
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
Fig. 6. Fuzzy comparison. Table 4 Fuzzy comparison areas.
k xþzhðbÞ psffi k2
RðaÞ ¼ 1 exp
n
2
AT
AR
AR =AT
0.1646
0.1503
0.9131
pffiffiffiffiffi 2
s2
1zhðbÞ
n1
The investor has now two almost similar fuzzy estimators to compare for his final decision. The appropriate shape of the fuzzy estimators makes the difference and will be decided using the false alarm criterion of Section 5. Numerical Example 4. Suppose an investor has two alternatives with normally distributed historical returns X 1 ; X 2 ; . . . ; X n and Y 1 ; Y 2 ; . . . ; Y n ðn ¼ 100Þ. Table 2 illustrates the crisp mean values and variances for the two choices. Let the degree of risk aversion for the investor be constant k ¼ 0:3. We are going to use the model of Eq. (18) k2 2 E½UðxÞ ¼ 1 expklþ 2 r to estimate the corresponding expected utilities. Table 3 shows the computed expected utilities. By moving into the fuzzy domain, we construct from the historical data the two fuzzy estimators (fuzzy mean (Eq. (10)) and fuzzy variance (Eq. (14)) of each investor’s choice, in order to compute the fuzzy expected utilities (Fig. 5). In our example, the fuzzy expected utilities have been computed using Eq. (19). We are now able to proceed to the fuzzy comparison itself. By overlapping the fuzzy expected utilities, this comparison clearly reveals that the second choice is preferable (Fig. 6). Numerically, the same fuzzy comparison is in Table 4. We proceed to a fuzzy hypothesis testing (Section 4) with a desirable control parameter / ¼ 0:4. AT is the total area under the second fuzzy estimator. AR is the area under the second fuzzy estimator that exceeds the first fuzzy estimator (black colour in Fig. 6). The ratio AR =AT ¼ 0:9131 being greater than / ¼ 0:4 validates our hypothesis that the second choice is preferable. 7. Conclusion As evidenced by recent research, the use of fuzzy numbers based on confidence intervals improves point and confidence interval estimation. In this paper, we provided a generalization of the methodology, showing how to construct a family of suitable fuzzy estimators, called non-asymptotic fuzzy estimators (NAFE). In particular, along with the generalized definition which allows us to construct NAFE, we proved that such fuzzy estimators are fuzzy numbers with compact support. We also provided the general membership function for all non-asymptotic fuzzy estimators and proposed specific membership functions for the parameters of some widespread statistical models. Furthermore, we dealed with the issue of fuzzy comparison by using non-asymptotic fuzzy estimators as a vehicle for a fuzzy hypothesis testing. The use of such non-asymptotic fuzzy estimators are important in critical situations, where almost equal statistical quantities are to be compared. Appendix A. Continuation of the proof of Theorem 2 Let qðaÞ ¼ h1 ð2hðaÞÞ ¼ x zhðaÞ prffiffin and r ðaÞ ¼ h2 ð2hðaÞÞ ¼ x þ zhðaÞ prffiffin be functions on ½0; 1. We find q1 ðxÞ and r1 ðxÞ as below.
x x x x r x zhðaÞ pffiffiffi ¼ x () zhðaÞ ¼ pffiffiffi () U1 ð1 hðaÞÞ ¼ pffiffiffi () n r= n r= n x x x x 1c c pffiffiffi () pffiffiffi () aþ ¼1U 1 hðaÞ ¼ U 2 2 r= n r= n
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx Table 5 False alarm rate results. Estimation method
/¼0
/ ¼ 0:1
/ ¼ 0:2
/ ¼ 0:3
/ ¼ 0:4
/ ¼ 0:5
Point Estim. Cube NAFE Square NAFE Buckely’s Estim. Linear NAFE Square root NAFE Cube root NAFE Power of 1/4 NAFE
0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0 7.61 7.01 5.07 4.48 2.69 1.79 1.34
0 17.46 15.22 12.69 10.75 6.87 3.88 3.28
0 27.01 22.99 19.7 17.16 11.49 7.76 6.27
0 37.61 32.69 27.31 23.73 15.22 11.49 9.1
0 48.96 41.64 37.16 30.6 19.85 15.22 11.79
/ ¼ 0:6
/ ¼ 0:7
/ ¼ 0:8
/ ¼ 0:9
/¼1
Point Estim. Cube NAFE Square NAFE Buckley’s Estim. Linear NAFE Square root NAFE Cube root NAFE Power of 1/4 NAFE
0 60.6 55.1 55.1 40.2 26.9 19.7 15.7
0 71.6 66.3 66.3 50.2 33.9 25.1 20.8
0 80.9 75.8 75.8 63.1 43.3 33.0 27.3
0 88.1 85.2 85.2 75.8 58.7 46.3 37.2
0 95.5 95.5 95.5 95.5 95.5 95.5 95.5
a¼
2 x x c pffiffiffi U ¼ q1 ðxÞ 1c 1c r= n
ð20Þ
Since 0 6 a 6 1, Eq. (20) is satisfied when
x x 2 x x c 1 2c pffiffiffi pffiffiffi 6 U 6 1 () 6 U () 1c 2 1c 2 r= n r= n
x x 1 c r c 6 pffiffiffi 6 U1 1 U1 () 0 6 x x 6 pffiffiffi U1 1 () 2 2 2 n r= n 06
r c x pffiffiffi U1 1 6 x 6 x 2 n
ð21Þ
Similarly, we find
r1 ðxÞ ¼
2c 2 x x pffiffiffi þ U 1c 1c r= n
ð22Þ
satisfied when
r c x 6 x 6 x þ pffiffiffi U1 1 2 n
ð23Þ
By combining Eqs. (20)–(23) based on Theorem 1 we obtain
8
2 > U xpxffiffi c ; x prffiffin U1 1 2c 6 x 6 x > > < 1c r= n 1c c l~ c ðxÞ ¼ 2 U xx pffiffi prffiffi 1 1 2c > 1c r= n 1c ; x 6 x 6 x þ n U > > : 0; otherwise Appendix B See Table 5. References [1] A.H. Ang, W.H. Tang, Probability Concepts in Engineering: Emphasis on Applications in Civil & Environmental Engineering, 2nd ed., John Wiley & Sons, USA, 2006. [2] C. Barz, Risk-averse Capacity Control in Revenue Management, Springer, Berlin, 2007. [3] J.J. Buckley, Fuzzy statistics: hypothesis testing, Soft Comput. 9 (7) (2005) 512–518. [4] K.A. Chrysafis, B.K. Papadopoulos, On theoretical pricing of options with fuzzy estimators, J. Comput. Appl. Math. 223 (2) (2006) 552–566. [5] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, in: Metric Space of Fuzzy Sets: Theory and Application, World Scientific, Singapore, 1994. [6] D. Dubois, H. Prade, Fuzzy sets and probability: misunderstandings, bridges and gaps, in: Proceedings of 2nd IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’93), vol. 2, 1993, pp. 1059–1068. [7] A. Falsafain, S.M. Taheri, M. Mashinchi, Fuzzy estimation of parameters in statistical models, Int. J. Math. Sci. (2) (2008) , 79–8.
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[8] A. Falsafain, S.M. Taheri, On Buckley’s approach to fuzzy estimation, Soft Comput. 15 (2011) 345–349. [9] M.A. Gil, N. Corral, P. Gil, The fuzzy decision problem: an approach to the point estimation problem with fuzzy information, Eur. J. Oper. Res. 22 (1) (1985) 26–34. [10] B. Hsu, M. Shu, Fuzzy inference to assess manufacturing process capability with imprecise data, Eur. J. Oper. Res. 186 (2) (2008) 652–670. [11] I. Kaya, C. Kahraman, Fuzzy process capability analyses with fuzzy normal distribution, Expert Syst. Appl. 37 (7) (2010) 5390–5403. [12] R. Kruse, Statistical estimation with linguistic data, Inf. Sci. 33 (3) (1984) 197–207. [13] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, PTR, Upper Saddle River, NJ, 1995. [14] H. Levy, H.M. Markowitz, Approximating expected utility by a function of mean and variance, Am. Econ. Rev. 69 (3) (1979) 308–317. [15] H.T. Nguyen, A note on the extension principle for fuzzy sets, J. Math. Anal. Appl. 64 (1978) 369–380. [16] A. Parchami, M. Mashinchi, Fuzzy estimation for process capability indices, Inf. Sci. 177 (6) (2007) 1452–1462. [17] S.M. Taheri, M. Arefi, Testing fuzzy hypotheses based on fuzzy test statistic, Soft Comput. 13 (6) (2008) 617–625. [18] L. Tsironis, D.S. Sfiris, B.K. Papadopoulos, Fuzzy performance evaluation of workflow stochastic petri nets by means of block reduction, IEEE Trans. Syst. Man Cybernet. – Part A: Syst. Humans 40 (2) (2010) 352–362. [19] U. Wagner, Statistical based fuzzy sets, in: Fuzzy Information Processing Society, Proceedings, NAFIPS, Annual Meeting of the North American IEEE, 2002, pp. 163–166. [20] H.C. Wu, The fuzzy estimators of fuzzy parameters based on fuzzy random variables, Eur. J. Oper. Res. 146 (1) (2003) 101–114. [21] C. Wu, B. Zhang, Embedding problem of noncompact fuzzy number space E (II), Fuzzy Sets Syst. 110 (1999) 135–142. [22] J.S. Yao, C.M. Hwang, Point estimation for the n sizes of random sample with one vague data, Fuzzy Sets Syst. 80 (2) (1996) 205–215. [23] M. Ming, M. Friedman, A. Kandel, A new fuzzy arithmetic, Fuzzy Sets Syst. 108 (1999) 83–90.
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Non-asymptotic fuzzy estimators based on confidence intervals D.S. Sfiris 1, B.K. Papadopoulos ⇑ Department of Civil Engineering, Section of Mathematics, Democritus University of Thrace, 12 Vas. Sofias Str, 67100 Xanthi, Greece
a r t i c l e
i n f o
Article history: Received 20 December 2010 Received in revised form 31 August 2013 Accepted 28 March 2014 Available online xxxx 2010 MSC: 03E72 62A86 62F25
a b s t r a c t In this paper, we study and generalize the method of fuzzy estimation which is based on confidence intervals. We provide definitions and the corresponding formulas. The method we propose allows new additional ways in constructing fuzzy estimators. In particular, instead of constructing fuzzy estimators as the exact set of confidence intervals of the parameters to be estimated, we generalize them, in such a way that we eliminate discontinuities, and ensure compact support preserving the statistically derived triangular shape of the estimators. Our approach is particularly useful in critical situations, where subtle fuzzy comparisons between almost equal statistical quantities have to be made. An example of this application is provided. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Fuzzy set theory Fuzzy statistical analysis Tolerance and confidence region
1. Introduction Statistical estimation is one of the main techniques used in inferential statistics. A statistical model may be perceived as a particular type of description of a probability distribution. Such models contain parameters whose values are estimated from samples, a required task to proceed in statistical data analysis [1]. With the growth of fuzzy methodologies, the integration of statistics with fuzzy set theory has attracted considerable attention [6,12,9,22,19,20,3]. Kruse [12] proposed the use of fuzzy linguistic values for estimating the parameters of statistical distributions in a fuzzy way. Gil et al. [9] introduced the notion of fuzzy Bayes’ estimator for real valued parameters. Yao and Hwang [22] investigated point estimators of fuzzy random variables for vague data. Wagner [19] described how to construct a fuzzy number on the basis of confidence intervals from a given sample but only for a determined set of a-cuts. Wu [20] proposed the concept of fuzzy estimators constructed from a family of closed intervals under the consideration of fuzzy random variables. Using a different approach, Buckley [3] proposed a fuzzy estimation method in order to construct fuzzy numbers for parameters in probability density (mass) functions that have been estimated from random samples, using all confidence intervals. Various applications of this approach to fuzzy estimation can be found in the literature (see [16,10,17,11]). Falsafain et al. [7] studied the construction of explicit and unique membership functions for such fuzzy estimators and recently Falsafain and Taheri [8] presented an improvement for non-symetric distributions. In this paper we generalize the fuzzy estimation approach established by Buckley [3]. ⇑ Corresponding author. 1
E-mail addresses: dsfi[email protected] (D.S. Sfiris), [email protected] (B.K. Papadopoulos). Principal corresponding author.
http://dx.doi.org/10.1016/j.ins.2014.03.131 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
Our motivation in based in the need to compare two almost equal statistical quantities. In such critical situations, the shape of the fuzzy numbers that are being produced is very important. While in the literature [3,7,8], fuzzy estimation being an extension of confidence interval estimation makes an assumption that the shape of fuzzy estimators is bound to the exact set of confidence intervals, we show that this is not the only possible way to construct fuzzy estimators. We provide a generalization which not only includes previous approaches as a subcase, but also permits the construction of many families of such fuzzy estimators which we call them non-asymptotic fuzzy estimators (NAFEs). The new generalized definition guarantees that NAFE are proper fuzzy numbers, preserving the statistically derived triangular shape and are appropriate for subtle fuzzy comparisons. An extra benefit of our proposed method is that one can now choose how close to be to the original point estimation, that is how strict is the fuzzy estimation. We show that the comparison of fuzzy estimators is greatly affected by their shapes, so a criterion is required to decide which shape is appropriate and we introduce the false alarm rate index. As an experiment, we tested the behavior and effectiveness of seven families of NAFE. The results have shown that by implementing NAFE on almost equal statistical quantities, we can significantly reduce the false alarm rate, i.e. we can correctly find which option is slightly better than the other when using either point or confidence interval estimation or fuzzy estimation. Thus our own non-asymptotic fuzzy estimators clearly address the issue of comparing almost equal quantities. An application of NAFE in the sensitive area of financial decisions is presented. For more applications of NAFE see [4,18]. The remainder of this paper is organized as follows: In Section 2.1 we introduce notions and definitions from Fuzzy Set theory. In Section 2.2 we review the literature and particularly Buckley’s method of fuzzy estimation. In Section 3 we present the new method of non-asymptotic fuzzy estimators, in Section 3.2 we develop the corresponding membership function of non-asymptotic fuzzy estimators and in Section 3.3 we propose membership functions for some common statistical models. In the next Sections 4, 5 we discuss a fuzzy comparison method and provide the falst alarm criterion for the selection of the appropriate shape of the estimators. Section 6 is an application. Our conclusions follow in Section 7. 2. Fuzzy estimation based on confidence intervals 2.1. Fuzzy sets notions and definitions Before presenting the fuzzy estimation method, we introduce the relevant mathematical notations and the basic notions from the fuzzy set theory. ~ : X ! ½0; 1 is called a fuzzy set or a fuzzy subset of X, where AðxÞ e Let X be a universal set. Every function of the form A is e We place a tilde over a fuzzy set symbol so as to distinguish it interpreted as the membership degree of x in the fuzzy set A. e from classical set. Classical sets are also called crisp sets and are special cases of fuzzy sets where AðxÞ is only zero or one. e are defined by the sets The a-cuts of a fuzzy set A
n o ~ A ¼ x 2 R : AðxÞ P a ; a 2 ð0; 1
a~
ð1Þ
~ is the closure in the topology of X of the union of all the a-cuts [5], that is while its support 0 A
n o ~ ~ ¼ x : AðxÞ >0 : A ¼ [a2ð0;1 a A
0~
ð2Þ
e We say that A e is a fuzzy number if the following conditions It is known that a-cuts uniquely determine the fuzzy set A. hold: e is normal, that is there exists x 2 R such that A ~ ðxÞ ¼ 1, 1. A e is a convex fuzzy set, that is for every t 2 ½0; 1 and x1 ; x2 2 R, we have 2. A
n o ~ ðð1 t Þx1 þ tx2 Þ P min Aðx ~ 1 Þ; Aðx ~ 2Þ ; A e is upper semi-continuous on R, i.e. 8x0 2 R and 8e > 0 there exists a neighborhood V ðx0 Þ such that 3. A uðxÞ 6 uðx0 Þ þ e; 8x 2 V ðx0 Þ, e Eq. (2), is compact. 4. the support of A, e can be written as intervals of the form We also mention that the a-cuts of a fuzzy number A a~
A ¼ ½A1 ðaÞ; A2 ðaÞ
ð3Þ
where A1 ðaÞ; A2 ðaÞ can be regarded as functions on ½0; 1 [15]. Then, 1. A1 ðaÞ is non-decreasing and left continuous, 2. A2 ðaÞ is non-increasing and left continuous, 3. A1 ðaÞ 6 A2 ðaÞ.
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Conversely, according to Wu and Zhang [21], for any functions A1 ðaÞ and A2 ðaÞ defined on ð0; 1 which satisfy 1–3 in the ~ such that Eq. (3) holds for any a 2 ð0; 1. If A ~ is a normal fuzzy set, a A ~ are closed interabove, there exists a unique fuzzy set A ~ 0 A, ~ is bounded, then A ~ is also a fuzzy number. vals for every a 2 ð0; 1 and the support of A; The graph of a fuzzy number may have various shapes. A triangular fuzzy number is defined by a triplet ða1 ; a2 ; a3 Þ where the vertex of the triangle is at x ¼ a2 and its base is the interval ½a1 ; a2 . A triangular shaped fuzzy number is partially derived by a1 < a2 < a3 because its sides are continuous curves, not straight line segments. We also refer to complete fuzzy numbers which implies that their support is closed and bounded (compact). For the realization of arithmetic operations we use the following a-cuts operations, based on the extension principle as ~ and B ~ be two fuzzy numbers and [23]. Let A a~
~ ¼ ½B1 ðaÞ; B2 ðaÞ A ¼ ½A1 ðaÞ; A2 ðaÞ; a B
be their a-cuts. The fuzzy addition and fuzzy subtraction are defined as a
h
i ~B ~ ¼ ½A1 ðaÞ þ B1 ðaÞ; A2 ðaÞ þ B2 ðaÞ; A
a
h
i ~B ~ ¼ ½A1 ðaÞ B2 ðaÞ; A2 ðaÞ B1 ðaÞ A
respectively for every a 2 ½0; 1. The multiplication of a fuzzy number with a scalar may be defined as a
h
i ~ ¼ ka A e ¼ ½k; k½A1 ðaÞ; A2 ðaÞ ¼ ½kA1 ðaÞ; kA2 ðaÞ; ifk P 0 kA ½kA1 ðaÞ; kA2 ðaÞ; ifk < 0
where the scalar k is identified as the closed interval ½k; k. ~B ~øB ~ and B, ~ and division A ~ of fuzzy numbers A ~ based on Zadeh’s extension principle [13, p.44-49], are The product A defined as a
h
i ~B ~ ¼ ½min ðA1 a B1 a ; A1 a B2 a ; A2 a B1 a ; A2 a B2 a Þ; max ðA1 a B1 a ; A1 a B2 a ; A2 a B1 a ; A2 a B2 a Þ A
a
h
i ~øB ~ ¼ ½min ðA1 a =B1 a ; A1 a =B2 a ; A2 a =B1 a ; A2 a =B2 a Þ; max ðA1 a =B1 a ; A1 a =B2 a ; A2 a =B1 a ; A2 a =B2 a Þ A
where A1 a ¼ A1 ðaÞ; A2 a ¼ A2 ðaÞ B1 a ¼ B1 ðaÞ; B2 a ¼ B2 ðaÞ for short and 0 R
aB ~
for the division.
2.2. Buckley’s method of fuzzy estimation In this section, we recall the fuzzy estimation method of [3] and present an example to be used later for comparison with our new method. Let X be a random variable with probability density function f ðx; hÞ for single parameter h. Assume that h is unknown and it must be estimated from a random sample X 1 ; X 2 ; . . . ; X n . Denote ð1 bÞ100% confidence intervals for h by ½h1 ðbÞ; h2 ðbÞ for all 0:01 6 b < 1. Starting at 0.01 is arbitrary and you could begin at 0.001 or 0.005, etc. Add to this the interval ½h ; h for the 0% confidence interval for h. Then place these confidence intervals, one on top of the other, to produce a triangular shaped fuzzy number ~ h whose a-cuts are the following confidence intervals a~
h ¼ ½h1 ðaÞ; h2 ðaÞ; 0:01 6 a 6 1:
ð4Þ
h in order to make it a complete fuzzy number, one needs to drop the graph of ~ h straight down to To finish the ‘‘bottom’’ of ~ complete its a-cuts, a~
h ¼ ½h1 ð0:01Þ; h2 ð0:01Þ; 0 6 a < 0:01:
ð5Þ
An example to estimate the mean of the normal distribution follows. Numerical Example 1. Let x1 ; x2 ; . . . ; xn be a sample of n ¼ 100 observations drawn from a normal population. Let us consider that the observations have an unknown mean value l and a known variance r2 ¼ 16. Suppose the sample mean value turns out to be x ¼ 40. Then, according to the method described, the fuzzy estimator for the mean l is a fuzzy number l~ whose a-cuts are defined by the ð1 bÞ100% confidence intervals of l for all amax 6 b 6 1, where amax corresponds to the ~ before we drop its membership function straight down. Thus, if amax ¼ 0:1 (90% confidence interval), then largest a-cut of l based on Eq. (4) and Eq. (5) we obtain
8h 4 4 i > ffi ; 40 þ U1 1 a2 pffiffiffiffiffi ffi ; a 2 ½0:1; 1 < 40 U1 1 a2 pffiffiffiffiffi 100 100 a~ i l¼ h 1 1 > 4 ffi 4 ffi : 40 U 1 0:1 pffiffiffiffiffi pffiffiffiffiffi ; 40 þ U 1 0:1 ; a 2 ½0; 0:1Þ 2 2 100 100
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where U1 denotes the reverse cumulative distribution function of the standard normal distribution. Fig. 1 illustrates the shape of the fuzzy estimator obtained for this example. Notice that the support of a fuzzy estimator as above, asymptotically extends from 1 to þ1 and its a-cuts become exactly the confidence intervals of the estimated parameter. Hence we may say that the above estimators are asymptotic in nature. 3. Non-asymptotic fuzzy estimators 3.1. The proposed definition We now present how to generalize the fuzzy estimators, showing new ways to construct them. Let X 1 ; X 2 ; . . . ; X n be a random sample of size n from a distribution with unknown parameter h and let ½h1 ðaÞ; h2 ðaÞ denote the ð1 aÞ100% confidence intervals for h. Using any monotonic increasing, continuous and onto function
hðaÞ : ð0; 1 !
hc 2
i ; 0:5 ; c 2 ð0; 1Þ
ð6Þ
then a family of fuzzy estimators h~c is derived by the following superimposed intervals a~ hc
¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ; a 2 ð0; 1:
ð7Þ
We call them non-asymptotic fuzzy estimators (NAFE) and we will prove in Theorem 1 that they are proper fuzzy numbers with compact support. Theorem 1. A non-asymptotic fuzzy estimator is a complete triangular shaped fuzzy number whose a-cuts are the ð1 2hðaÞÞ100%; a 2 ð0; 1, confidence intervals for h and its compact support is exactly the ð1 cÞ100% confidence interval for h. Proof. By definition, hðaÞ is a monotonic increasing, continuous and onto function. We conclude that h1 ð2hðaÞÞ is monotonic non-decreasing and continuous as a composite function and h2 ð2hðaÞÞ is monotonic non-increasing and continuous, hence, ð1 2hðaÞÞ100% confidence intervals are nested, therefore h1 ð2hðaÞÞ 6 h2 ð2hðaÞÞ. From Theorem 2.1 of Wu and Zhang [21] (see Section 2) and based on the previous conditions, there exists a unique fuzzy set that can be constructed from the set of ð1 2hðaÞÞ100% confidence intervals for h. To prove that this fuzzy set is a fuzzy number, it is sufficient to prove that all the a-cuts are contained in a closed and bounded interval. This interval is 0 h~c ¼ ½h1 ð2hð0ÞÞ; h2 ð2hð0ÞÞ ¼ ½h1 ðcÞ; h2 ðcÞ, which is exactly the ð1 cÞ100% confidence interval for h. h According to definition, choosing a proper monotonic increasing, continuous and onto function (6) allows us to construct fuzzy estimators of a particular NAFE family associated with it. The following seven functions are proper candidates, but not the only ones: 1. hðaÞ ¼ 2. hðaÞ ¼ 3. hðaÞ ¼ 4. hðaÞ ¼ 5. hðaÞ ¼ 6. hðaÞ ¼ 7. hðaÞ ¼
1 2 1 2 1 2 1 2 1 2 1 2
2c a þ 2c (Linear NAFE), 2c a2 þ 2c (Square NAFE), 2c a2 þ 3c (Cube NAFE), pffiffiffi 2c a þ 2c (Square Root NAFE), 2c a1=3 (Cube Root NAFE), 2c a1=3 (Power of 1/4 NAFE),
a 2
c
2
;a P c . (Buckley’s fuzzy estimators). ;a < c
Fig. 1. Fuzzy estimator for the mean
l of normal distribution of Example 1.
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Each function determines the shape of the membership function of the fuzzy estimators, where the support of each estimator is the ð1 cÞ100% confidence interval. For example one can choose c ¼ 0:1 (90% confidence interval), c ¼ 0:05 (95% confidence interval) or any other value. Thus we generalize Buckley’s fuzzy estimators which are the case seven above. 3.2. Membership function of non-asymptotic fuzzy estimators Constructing a fuzzy estimator based on its a-cuts definition does not directly relate any point x with a membership value. [7] proved explicit and unique membership functions for Buckley’s fuzzy estimators. In Theorem 2, we prove the general membership function for all non-asymptotic fuzzy estimators. Theorem 2. Suppose that X 1 ; X 2 ; . . . ; X n is a random sample of size n from a distribution with unknown parameter h. If ½h1 ðaÞ; h2 ðaÞ denote the ð1 aÞ100% confidence intervals for h, then a non-asymptotic fuzzy estimator of h has the following membership function
n o h~c ðxÞ ¼ min ðpÞ1 ðxÞ; ðqÞ1 ðxÞ; 1 ; where p ¼ h1 2h; q ¼ h2 2h denote compositions of functions and hðaÞ : ð0; 1 ! continuous and onto function.
c 2
; 0:5 ; c 2 ð0; 1Þ is monotonic increasing,
Proof. Since, by definition, the a-cuts of a non-asymptotic fuzzy estimator are a h~c ¼ ½ðh1 2hÞðaÞ; ðh2 2hÞðaÞ or ¼ ½pðaÞ; qðaÞ, it holds that, if x 2a h~c , then pðaÞ 6 x 6 qðaÞ. Hence, from pðaÞ 6 x we obtain a 6 ðpÞ1 ðxÞ, and from n o x 6 qðaÞ we obtain a 6 ðqÞ1 ðxÞ. Moreover, a 6 1, therefore a ¼ min ðpÞ1 ðxÞ; ðqÞ1 ðxÞ; 1 ¼ h~c ðxÞ. h
a h~ c
Theorems 1 and 2 hold for all probability distributions. In the following section, we propose specific membership functions for the parameters of some widespread statistical models. 3.3. Using non-asymptotic fuzzy estimators to estimate the parameters of the normal and exponential distributions Let us consider the linear function
hðaÞ ¼
1 c c a þ ; a 2 ð0; 1; c 2 ð0; 1Þ: 2 2 2
ð8Þ
Proposition 1 (Non-asymptotic fuzzy mean of normal distribution – large samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is large enough, then
8
c 2 > pxffiffi U rx ; x prffiffin U1 1 2c 6 x 6 x > 1 c 1c > = n <
c l~ c ðxÞ ¼ 2 U xx pffiffi prffiffi 1 1 2c > 1c r= n 1c ; x 6 x 6 x þ n U > > : 0; otherwise
ð9Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for l and the acuts of this fuzzy number are the closed intervals a
r r l~c ¼ x zhðaÞ pffiffiffi ; x þ zhðaÞ pffiffiffi ; a 2 ð0; 1 n
where zhðaÞ ¼ U1 ð1 hðaÞÞ; hðaÞ ¼ distribution.
n
1 2
ð10Þ
2c a þ 2c, and U denotes the cumulative distribution function of the standard normal
Proof. We first show that there exists a unique fuzzy number whose a-cuts are given in Eq. (10); then beginning from these a-cuts, we prove the membership function of this fuzzy number. Let qðaÞ ¼ x zhðaÞ prffiffin and rðaÞ ¼ x þ zhðaÞ prffiffin be functions on ð0; 1. We prove that qðaÞ 6 r ðaÞ. This is equivalent to
r r 1 x zhðaÞ pffiffiffi < x þ zhðaÞ pffiffiffi () zhðaÞ > 0 () U1 ð1 hðaÞÞ > 0 () hðaÞ < 2 n n which is always true.
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Now, let a1 P a2 . We prove that qðaÞ is non-decreasing. Indeed,
r
r
qða1 Þ P qða2 Þ () x zhða1 Þ pffiffiffi P x zhða2 Þ pffiffiffi () n n zhða1 Þ 6 zhða2 Þ () U1 ð1 hða1 ÞÞ 6 U1 ð1 hða2 ÞÞ () hða1 Þ P hða2 Þ which is always true, since hðaÞ is non-decreasing. Similarly, we can prove that r ðaÞ is non-increasing. Finally, we notice that qðaÞ and rðaÞ are continuous as a composition of continuous functions. Consequently, according to Theorem 2.1 [21] (see Section 2), there exists a unique fuzzy number l~c : X ! ½0; 1 determined by Eq. (10). Lastly, we only need to construct the membership function of this fuzzy number to close the proof (see Appendix A). h Remark 1. If the sample size is small and it has been drawn from a normally distributed population with unknown r, then r in Eq. (9) and (10) is replaced by s and zhðaÞ is replaced by thðaÞ , where t hðaÞ ¼ T 1 ð1 hðaÞÞ and T denotes the cumulative distribution function of the t-student distribution with m ¼ n 1 degrees of freedom. The proof can be done in an analogous way to the proof of Proposition 1. Numerical Example 2. We consider the same example presented in Section 2.2, this time to construct a non-asymptotic fuzzy estimator for l. To obtain the same support for the fuzzy estimator, we used 90% confidence interval for l, that is ~ c is c ¼ 0:1. Deploying Eq. (9), the membership function of l
l~ 0:1
8
2 4 ffi 1 > pffiffiffiffiffiffi 0:1 ; 40 pffiffiffiffiffi U 4=x40 U ð0:95Þ 6 x 6 40 > 0:9 0:9 > 100 100 <
¼ 2 4 ffi 1 pffiffiffiffiffiffi 0:1 ; 40 6 x 6 40 þ pffiffiffiffiffi U 4=40x U ð0:95Þ > 0:9 0:9 100 100 > > : 0; otherwise
~ c (Eq. (10)) are The a-cuts of l a
4
4
100
100
l~c ¼ 40 z0:9a2þ0:1 pffiffiffiffiffiffiffiffiffi ; 40 þ z0:9a2þ0:1 pffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1
Fig. 2 illustrates the shape of the fuzzy estimator obtained for this example. Proposition 2 (Non-asymptotic fuzzy variance of normal distribution – small samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is small, then
8
2 n1Þs2 ðn1Þs2 2c > > 12 c F ðn1x Þs ; ð1 2c 6 x 6 > 1 c M F ð 2 Þ > <
2 2 2 2 ~ ð n1 Þs ð n1 Þs ð n1 c r c ð xÞ ¼ 2 F 1c ; M 6 x 6 F 1 Þsc > 1c x > ð2Þ > > : 0; otherwise
ð11Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for r2 and the acuts of this fuzzy number are the closed intervals
1 0.8 0.6 0.4 0.2
37.45
40
42.55
x
Fig. 2. Non-asymptotic fuzzy mean estimator of Example 2.
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a ~2 rc
¼
" # ðn 1Þs2 ðn 1Þs2 ; a 2 ð0; 1 ; 2 2
ð12Þ
vn1;hðaÞ vn1;1hðaÞ
where v2n1;: is the value of the chi-squared distribution with k ¼ n 1 degrees of freedom, v2n1;hðaÞ ¼ F 1 ð1 hðaÞÞ; hðaÞ ¼ 12 2c a þ 2c, and F denotes the cumulative distribution function of the chi-squared distribution. M is the median of the distribution. Proof. We proceed to this proof based on Theorems 1 and 2. The ð1 aÞ100% confidence intervals for
r2
1a
r2 are
" # ðn 1Þs2 ðn 1Þs2 ¼ ; 2 ; 2
vn1;a=2 vn1;1a=2
therefore, according to Theorem 1, the non-asymptotic fuzzy estimator for r2 is a fuzzy number the a-cuts of which are the closed intervals
"
a ~2 rc
ðn 1Þs2 ðn 1Þs2 ¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ ¼ ; 2 2
#
vn1;hðaÞ vn1;1hðaÞ
~ 2c ¼ ½pðaÞ; qðaÞ. After calculating p1 ðxÞ and ðqÞ1 ðxÞ, We now consider pðaÞ ¼ h1 ð2hðaÞÞ and qðaÞ ¼ h2 ð2hðaÞÞ, so that a r we obtain
p1 ðxÞ ¼
2c 2 ðn 1Þs2 F 1c 1c x
and
ðqÞ1 ðxÞ ¼
2 ðn 1Þs2 c F 1c x 1c
and therefore, based on Theorem 2,
n
o
r~ 2c ðxÞ ¼ min p1 ðxÞ; ðqÞ1 ðxÞ; 1
that is equal to the proposed membership function.
h
Proposition 3 (Non-asymptotic fuzzy variance of normal distribution - large samples). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. If the sample size is large enough, then
8
qffiffiffiffiffiffiffi
2c n1 s2 s2 pffiffiffiffiffi > 12 c U 1 ; 6 x 6 s2 > c 2 2 1 c x > 1þU1 ð12Þ n1 > < q ffiffiffiffiffiffi ffi
r~ 2c ðxÞ ¼ 2c 2 U n1 1 s2 ; s2 6 x 6 s2 pffiffiffiffiffi > c 2 x 2 > 1c 1c 1U1 ð12Þ n1 > > : 0; otherwise
ð13Þ
is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for r2 and the acuts of this fuzzy number are the closed intervals
"
a ~2 rc
# s2 s2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1 ¼ 1 þ zhðaÞ 2=n 1 1 zhðaÞ 2=n 1
where z2hðaÞ ¼ U1 ð1 hðaÞÞ , hðaÞ ¼ distribution.
1 2
ð14Þ
2c a þ 2c, and U denotes the cumulative distribution function of the standard normal
Proof. The proof is analogous to those provided for Propositions 1 and 2. h Proposition 4 (Non-asymptotic fuzzy mean of exponential distribution). Let X 1 ; X 2 ; . . . ; X n be a random sample and let x1 ; . . . ; xn be sample values assumed by the sample. Let also c 2 ð0; 1Þ. Then
1 a~ k
c
ðxÞ ¼
8 ^ ^k 1 ðxÞ; F 12nkc 6 x 6 2n > c 2 > M F 2nx^kÞ ð 2Þ > 2 1c 1c ð < ^
^
1 2nk k 6 x 6 12n2 c ; c 2 F 2nx^ > kÞ1c M F ð 2 Þ > 1c ð > : 0; otherwise
ð15Þ
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is the membership function of a fuzzy number, the support of which is exactly the ð1 cÞ100% confidence interval for l and the acuts of this fuzzy number are the closed intervals
" 1 1 ¼ a~ ^k kc
2n
1 ; v22n;hðaÞ ^k
#
2n
v22n;1hðaÞ
; a 2 ð0; 1
ð16Þ
x is the maximum likelihood estimate, v22n;: is the value of the chi-squared distribution with k ¼ 2n degrees of freedom, k ¼ 1= where ^ 1 2 v2n;hðaÞ ¼ F ð1 hðaÞÞ, hðaÞ ¼ 12 2c a þ 2c, and F denotes the cumulative distribution function of the chi-squared distribution. M is the median of the distribution. Proof. The proof is analogous to those provided for Propositions 1 and 2. h
4. Fuzzy hypothesis testing Buckley [3] and Taheri and Arefi [17] described how fuzzy hypothesis testing is rendered into the comparison of two fuzzy estimators. Here we discuss the comparison of non-asymptotic fuzzy estimators. Let X 1 ; X 2 ; . . . ; X n ; Y 1 ; Y 2 ; . . . ; Y n be two random samples from a variable X with distinct probability density functions ~c ; q ~ c be two non-asymptotic fuzzy estimators constructed based fx ðx; hÞ and fy ðx; qÞ of unknown parameters h and q. Let also h on sample observations x1 ; x2 ; . . . ; xn and y1 ; y2 ; . . . ; yn . To evaluate whether the second fuzzy estimator is greater than the first, we use the following fuzzy decision rule: h1;c if AR =AT P /, 1. H0 : ~ h2;c is greater than ~ 2. H1 : ~ h1;c if AR =AT < /, h2;c is approximately ~
where a~ hc
¼ ½h1 ð2hðaÞÞ; h2 ð2hðaÞÞ; a P 0
a~
qc ¼ ½q1 ð2hðaÞÞ; q2 ð2hðaÞÞ; a P 0
are the a-cuts of the fuzzy estimators from Eq. (1), 0~
~ c ¼ ½q1 ðcÞ; q2 ðcÞ hc ¼ ½h1 ðcÞ; h2 ðcÞ;0 q
are the supports of the fuzzy estimators,
Z
q2 ðcÞ
q~ c ðxÞdx
q1 ðcÞ
~ c, is the total area under the fuzzy number q
AR ¼ AT
Z s
q~ c ðxÞdx
Z
h2 ðcÞ
~hc ðxÞdx
s
q1 ðcÞ
~ c but to the right of ~ hc , and is the area under the graph of q
n
o
~ ~ 1 ~ c ð xÞ s ¼ ~h1 c ðas Þ ¼ q c ðas Þ; as ¼ sup min hc ðxÞ; q x2X
~ c ðxÞ has the highest membership value. / is a constant control parameter expressing hc ðxÞ; q is point where the intersection of ~ the desired ratio of the second fuzzy number exceeding the first. ~ c ðxÞ are known, one can find an analytical expression for as , i.e. considhc ðxÞ; q Note that when the membership functions ~ ering the non-asymptotic fuzzy mean estimators for normal distribution (Proposition 1), we obtain r
as ¼
1ffi pffiffiffi x n1 2
r1 ffi pffiffiffi n1
2ffi þ prffiffiffi x n2 1 2ffi þ prffiffiffi n2
:
Numerical Example 3. Let X 1 ; X 2 ; . . . ; X n and Y 1 ; Y 2 ; . . . ; Y n be two normally distributed random samples of size n ¼ 1000 from N l1 ; r21 and N l2 ; r22 respectively. Given observed values x1 ; x2 ; . . . ; xn ; y1 ; y2 ; . . . ; yn , the sample means turned out to be x1 ¼ 5:0586; x2 ¼ 5:1456 and the sample standard deviations s1 ¼ 2:4711; s2 ¼ 6:3365. We are interested in testing the fuzzy null hypothesis H0 : l1 is greater than l2 .
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For this example, we constructed seven families of NAFE (linear, square, cube, square root, cube root, Power of 1/4 and Buckley’s type) with a support of ð1 cÞ100% ¼ 90% confidence interval. The general form of their a-cuts is
a~
2:4711
2:4711
l1;0:1 ¼ 5:0586 zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 5:0586 þ zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1
a~
1000
1000
6:3365
6:3365
l2;0:1 ¼ 5:1456 zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; 5:1456 þ zhðaÞ pffiffiffiffiffiffiffiffiffiffiffiffi ; a 2 ð0; 1 1000
1000
and their supports are 0~
l1;c ¼ ½4:924; 5:182;0 l~ 2;c ¼ ½4:811; 5:470
We compare the fuzzy estimators by applying the methodology described in this section. Fig. 3 illustrates the areas under two non-aysmptotic fuzzy estimators needed for the calculation of AR as AR ¼ AT E1 E2 . The result of the comparison for all NAFE families for / ¼ 0:4 is shown in Table 1, Column 6. As we see, the shape of the fuzzy estimators used may totally change the final decision. Using Cube, Square NAFE or Buckley’s estimation, the suggestion is to reject the hypothesis H0 . On the contrary, using Linear, Sqrt, Cube root or Power of 1/4 NAFE the suggested decision is to accept H0 . The question which shape is appropriate is discussed in the next section. 5. The false alarm criterion We now provide a criterion for evaluating our choices with regards to the shape of the fuzzy estimators. We call it the false alarm rate index. It will be explained by an example. Assume that we have 100 pairs of randomly generated data samples from two normal populations, where l1 ¼ 5; l2 ¼ 5:05 and standard deviations r1 ¼ r2 ¼ 2:5. Each sample has size 1000. Seven shapes of NAFE (linear, square, cube, square root, cube root, Power of 1/4 and Buckley’s type) are considered with a support of ð1 cÞ100% ¼ 90% confidence interval (Fig. 4). The goal is to confirm the null hypothesis H0 : l2 greater than l1 . The methodology described in Section 4 was applied allowing us to count the number of cases where the fuzzy comparison approach suggested to reject H0 when x1 < x2 (Nr). Let Np be the number of cases, where x1 < x2 . The false alarm rate criterion is defined as
FAindex ¼
Nr : Np
The appropriate shape has the lowest false alarm value. 1 0.8 0.6 0.4
AR
0.2
E1
4.924
4.811
E2 5.182
5.470
x
Fig. 3. Fuzzy comparison of two linear NAFE of Example 3.
Table 1 Comparison results. NAFE family
AT
E1 þ E2
AR
AR =AT
Decision
Cube Square Buckley’s Linear Sqrt Cube root Power of 1/4
0.4436 0.3795 0.3031 0.2635 0.1637 0.1196 0.0946
0.3202 0.2649 0.1936 0.1547 0.0692 0.0373 0.0215
0.1234 0.1146 0.1095 0.1087 0.0964 0.0823 0.0732
0.2782 0.3019 0.3613 0.4127 0.5775 0.6881 0.7729
Reject H0 Reject H0 Reject H0 Accept H0 Accept H0 Accept H0 Accept H0
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
The false alarm rates for 10 different / values are given in Table 2 of Appendix B. Linear, Square root and Power of 1/4 NAFE produced a lower false alarms rate than Buckley’s fuzzy estimators, while the Square and Cube fuzzy estimators produced a higher false alarms rate. This is so regardless of any particular c value, showing the stability of the methodology. 6. A critical situation in a financial decision The problem discussed below has also been our motivation to study various shapes of fuzzy estimators. Suppose that an investor has two almost similar choices. According to decision theory, the final choice can be determined by calculating the 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 x1
x2
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x1
x2
x
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x1
x2
x
x1
x2
x
x1
x2
x
x1
x2
x
1 0.8 0.6 0.4 0.2
x1
x2
x
Fig. 4. Shapes of various fuzzy estimators.
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
11
expected utility of each choice and then choosing the one with the higher expected utility. An example of such a utility function is the exponential utility function given by
UðxÞ ¼ 1 expkx ; k > 0
ð17Þ
satisfying
U 0 ðxÞ ¼ kexpkx > 0; U Prime ðxÞ ¼ k2 expkx < 0; k¼
U 00 ðxÞ U 0 ðxÞ
where k is the degree of risk-sensitivity [2]. The non-asymptotic fuzzy estimators will decide the best investment choice. Rather than directly maximizing the expected utility, it is possible, under certain assumptions, to reach to the same solution following the Mean–Variance approach [14]. Thus, if the choices have normally returns, then the expected utility can be expressed as a function of the mean and variance k2 2
E½UðxÞ ¼ 1 expklþ 2 r
ð18Þ
All we have to do is to estimate the l and r and then using Eq. (18) we can obtain an estimation for E[U (x)]. Finally, using the a-cuts notation and operations (see Section 2), Eq. (18) is fuzzified as follows: 2
a
k2 2 k2 2 k2 2 2 a k2 a E~c ½UðxÞ ¼ 1 expk l~ c þ 2 r~2 c ¼ 1 expk½l1 ðaÞ;l2 ðaÞþ 2 ½r1 ðaÞ;r2 ðaÞ ¼ 1 exp kl1 ðaÞþ 2 r1 ðaÞ;kl2 ðaÞþ 2 r2 ðaÞ
a
Ec ½U ðxÞ ¼ ½LðaÞ; RðaÞ; a > 0
or
k xzhðbÞ psffi k2
LðaÞ ¼ 1 exp
n
2
ð19Þ
s2
1þzhðbÞ
pffiffiffiffiffi 2
n1
Table 2 Mean values and variances. Investment choices
Mean
Choice 1 Choice 2
5.8 7.5
l
Variance
r2
3.460 8.062
Table 3 Expected utilities. Investment choices
Expected utilities
Choice 1 Choice 2
0.795 0.850
Fig. 5. Fuzzy expected utilities.
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx
Fig. 6. Fuzzy comparison. Table 4 Fuzzy comparison areas.
k xþzhðbÞ psffi k2
RðaÞ ¼ 1 exp
n
2
AT
AR
AR =AT
0.1646
0.1503
0.9131
pffiffiffiffiffi 2
s2
1zhðbÞ
n1
The investor has now two almost similar fuzzy estimators to compare for his final decision. The appropriate shape of the fuzzy estimators makes the difference and will be decided using the false alarm criterion of Section 5. Numerical Example 4. Suppose an investor has two alternatives with normally distributed historical returns X 1 ; X 2 ; . . . ; X n and Y 1 ; Y 2 ; . . . ; Y n ðn ¼ 100Þ. Table 2 illustrates the crisp mean values and variances for the two choices. Let the degree of risk aversion for the investor be constant k ¼ 0:3. We are going to use the model of Eq. (18) k2 2 E½UðxÞ ¼ 1 expklþ 2 r to estimate the corresponding expected utilities. Table 3 shows the computed expected utilities. By moving into the fuzzy domain, we construct from the historical data the two fuzzy estimators (fuzzy mean (Eq. (10)) and fuzzy variance (Eq. (14)) of each investor’s choice, in order to compute the fuzzy expected utilities (Fig. 5). In our example, the fuzzy expected utilities have been computed using Eq. (19). We are now able to proceed to the fuzzy comparison itself. By overlapping the fuzzy expected utilities, this comparison clearly reveals that the second choice is preferable (Fig. 6). Numerically, the same fuzzy comparison is in Table 4. We proceed to a fuzzy hypothesis testing (Section 4) with a desirable control parameter / ¼ 0:4. AT is the total area under the second fuzzy estimator. AR is the area under the second fuzzy estimator that exceeds the first fuzzy estimator (black colour in Fig. 6). The ratio AR =AT ¼ 0:9131 being greater than / ¼ 0:4 validates our hypothesis that the second choice is preferable. 7. Conclusion As evidenced by recent research, the use of fuzzy numbers based on confidence intervals improves point and confidence interval estimation. In this paper, we provided a generalization of the methodology, showing how to construct a family of suitable fuzzy estimators, called non-asymptotic fuzzy estimators (NAFE). In particular, along with the generalized definition which allows us to construct NAFE, we proved that such fuzzy estimators are fuzzy numbers with compact support. We also provided the general membership function for all non-asymptotic fuzzy estimators and proposed specific membership functions for the parameters of some widespread statistical models. Furthermore, we dealed with the issue of fuzzy comparison by using non-asymptotic fuzzy estimators as a vehicle for a fuzzy hypothesis testing. The use of such non-asymptotic fuzzy estimators are important in critical situations, where almost equal statistical quantities are to be compared. Appendix A. Continuation of the proof of Theorem 2 Let qðaÞ ¼ h1 ð2hðaÞÞ ¼ x zhðaÞ prffiffin and r ðaÞ ¼ h2 ð2hðaÞÞ ¼ x þ zhðaÞ prffiffin be functions on ½0; 1. We find q1 ðxÞ and r1 ðxÞ as below.
x x x x r x zhðaÞ pffiffiffi ¼ x () zhðaÞ ¼ pffiffiffi () U1 ð1 hðaÞÞ ¼ pffiffiffi () n r= n r= n x x x x 1c c pffiffiffi () pffiffiffi () aþ ¼1U 1 hðaÞ ¼ U 2 2 r= n r= n
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D.S. Sfiris, B.K. Papadopoulos / Information Sciences xxx (2014) xxx–xxx Table 5 False alarm rate results. Estimation method
/¼0
/ ¼ 0:1
/ ¼ 0:2
/ ¼ 0:3
/ ¼ 0:4
/ ¼ 0:5
Point Estim. Cube NAFE Square NAFE Buckely’s Estim. Linear NAFE Square root NAFE Cube root NAFE Power of 1/4 NAFE
0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0 7.61 7.01 5.07 4.48 2.69 1.79 1.34
0 17.46 15.22 12.69 10.75 6.87 3.88 3.28
0 27.01 22.99 19.7 17.16 11.49 7.76 6.27
0 37.61 32.69 27.31 23.73 15.22 11.49 9.1
0 48.96 41.64 37.16 30.6 19.85 15.22 11.79
/ ¼ 0:6
/ ¼ 0:7
/ ¼ 0:8
/ ¼ 0:9
/¼1
Point Estim. Cube NAFE Square NAFE Buckley’s Estim. Linear NAFE Square root NAFE Cube root NAFE Power of 1/4 NAFE
0 60.6 55.1 55.1 40.2 26.9 19.7 15.7
0 71.6 66.3 66.3 50.2 33.9 25.1 20.8
0 80.9 75.8 75.8 63.1 43.3 33.0 27.3
0 88.1 85.2 85.2 75.8 58.7 46.3 37.2
0 95.5 95.5 95.5 95.5 95.5 95.5 95.5
a¼
2 x x c pffiffiffi U ¼ q1 ðxÞ 1c 1c r= n
ð20Þ
Since 0 6 a 6 1, Eq. (20) is satisfied when
x x 2 x x c 1 2c pffiffiffi pffiffiffi 6 U 6 1 () 6 U () 1c 2 1c 2 r= n r= n
x x 1 c r c 6 pffiffiffi 6 U1 1 U1 () 0 6 x x 6 pffiffiffi U1 1 () 2 2 2 n r= n 06
r c x pffiffiffi U1 1 6 x 6 x 2 n
ð21Þ
Similarly, we find
r1 ðxÞ ¼
2c 2 x x pffiffiffi þ U 1c 1c r= n
ð22Þ
satisfied when
r c x 6 x 6 x þ pffiffiffi U1 1 2 n
ð23Þ
By combining Eqs. (20)–(23) based on Theorem 1 we obtain
8
2 > U xpxffiffi c ; x prffiffin U1 1 2c 6 x 6 x > > < 1c r= n 1c c l~ c ðxÞ ¼ 2 U xx pffiffi prffiffi 1 1 2c > 1c r= n 1c ; x 6 x 6 x þ n U > > : 0; otherwise Appendix B See Table 5. References [1] A.H. Ang, W.H. Tang, Probability Concepts in Engineering: Emphasis on Applications in Civil & Environmental Engineering, 2nd ed., John Wiley & Sons, USA, 2006. [2] C. Barz, Risk-averse Capacity Control in Revenue Management, Springer, Berlin, 2007. [3] J.J. Buckley, Fuzzy statistics: hypothesis testing, Soft Comput. 9 (7) (2005) 512–518. [4] K.A. Chrysafis, B.K. Papadopoulos, On theoretical pricing of options with fuzzy estimators, J. Comput. Appl. Math. 223 (2) (2006) 552–566. [5] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, in: Metric Space of Fuzzy Sets: Theory and Application, World Scientific, Singapore, 1994. [6] D. Dubois, H. Prade, Fuzzy sets and probability: misunderstandings, bridges and gaps, in: Proceedings of 2nd IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’93), vol. 2, 1993, pp. 1059–1068. [7] A. Falsafain, S.M. Taheri, M. Mashinchi, Fuzzy estimation of parameters in statistical models, Int. J. Math. Sci. (2) (2008) , 79–8.
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