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Comparison of estimation methods for the parameters of the weighted Lindley distribution
Applied Mathematics and Computation 220 (2013) 463–471
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Comparison of estimation methods for the parameters of the weighted Lindley distribution J. Mazucheli a, F. Louzada b, M.E. Ghitany c,⇑ a
Universidade Estadual de Maringá, DEs, PR, Brazil Universidade de São Paulo, ICMC, SP, Brazil c Department of Statistics & O.R., Faculty of Science, Kuwait University, Kuwait b
a r t i c l e
i n f o
a b s t r a c t The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary leastsquares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Weighted Lindley distribution Maximum likelihood Method of moments Ordinary least-squares Weighted least-squares
1. Introduction The Lindley distribution was introduced by Lindley [7], see also [8], in the context of fiducial distributions and Bayes’ theorem. Its probability density function (p.d.f.) is given by
f1 ðxjaÞ ¼
a2 ð1 þ xÞ eax ; aþ1
x > 0;
a > 0:
ð1Þ
The corresponding cumulative distribution function (c.d.f.) and hazard rate function (h.r.f.), respectively, are given by
ax eax ; F 1 ðxjaÞ ¼ 1 1 þ aþ1
x > 0;
a2 ð1 þ xÞ ; að1 þ xÞ þ 1
a > 0:
a > 0;
ð2Þ
and
h1 ðxjaÞ ¼
x > 0;
ð3Þ
Note that h1 ðxjaÞ is an increasing function in x, for all a > 0. Ghitany et al. [5] studied many statistical properties of the Lindley distribution from the reliability/survival analysis point of view. They also showed, using a real data set, that the Lindley distribution provides a better fit than the exponential distribution. Application of the Lindley distribution in the competing risks analysis and in stress-strength reliability studies are considered by Mazucheli and Achcar [9] and Al-Mutairi et al. [1], respectively. ⇑ Corresponding author. E-mail address: [email protected] (M.E. Ghitany). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.082
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Recently, Ghitany et al. [4] introduced the weighted Lindley distribution with p.d.f.
f ðxja; bÞ ¼ where CðaÞ ¼
abþ1 ða þ bÞCðbÞ
R1
xb1 ð1 þ xÞeax ;
x > 0;
a; b > 0;
ð4Þ
xa1 ex dx; a > 0, is the gamma function. The corresponding c.d.f. and h.r.f., respectively, are given by
0
Fðxja; bÞ ¼ 1
ða þ bÞCðb; axÞ þ ðaxÞb eax ða þ bÞCðbÞ
x > 0;
a; b > 0;
ð5Þ
and
hðxja; bÞ ¼
abþ1 xb1 ð1 þ xÞeax ða þ bÞCðb; axÞ þ ðaxÞb eax
x > 0;
a; b > 0;
ð6Þ
R1 where Cða; bÞ ¼ b xa1 ex dt; a > 0; b P 0, is the upper incomplete gamma function. Note that the Lindley distribution is a special case of the weighted Lindley distribution when b ¼ 1. Also, its h.r.f. hðxja; bÞ is bathtub-shaped (increasing) function in x if 0 < b < 1 (b P 1) for all a > 0. For reviews about bathtub-shaped hazard rate functions, see [11,10]. As far as the estimation of the parameters of the weighted Lindley distribution, [4] considered only the maximum likelihood estimation (MLE) method. It is of interest to compare the MLE method with other estimation methods such as the method of moments estimation (MME), ordinary least-squares estimation (OLSE) and weighted least-squares estimation (WLSE) methods. The main aim of this paper is to identify the most efficient estimators among four estimators for different shape parameters values and sample sizes. The originality of this study comes from the fact that there has been no previous work comparing all of these estimators for the two-parameter weighted Lindley distribution. Related studies for other distributions can be found in, for example, [6,12,13]. In Section 2 we discuss the four estimation methods considered in this paper. The comparison of these methods in terms of bias and mean-squared error is presented in Section 3. The four estimation methods are used in fitting two real data sets in Section 4. Some concluding remarks are presented in Section 5. 2. Methods of estimation In this section we describe the four considered estimation methods to obtain the estimates of the parameters a and b of the weighted Lindley distribution. 2.1. Maximum likelihood Let x1 ; x2 ; . . . ; xn be a random sample of size n from the weighted Lindley distribution with parameters a and b with p.d.f. (4). b MLE and b b MLE , of a and b, respectively, are given Ghitany et al. [4] showed that the maximum likelihood estimates (MLEs) a by
ab MLE ¼
b MLE ðx 1Þ þ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½b b MLE ðx 1Þ þ 4 b b MLE ð b b MLE þ 1Þ x b MLE Þ; ¼ nð b 2x
ð7Þ
where x is the sample mean, b b MLE is the solution of the non-linear equation:
n ln ðnðbÞÞ
X n 1 wðbÞ þ lnðxi Þ ¼ 0; nðbÞ þ b i¼1
ð8Þ
d and wðbÞ ¼ db ln CðbÞ is the digamma function.
2.2. Method of moments The method of moments is another technique commonly used in parameter estimation. For the weighted Lindley distribution, we have
EðXja; bÞ ¼
b ða þ b þ 1Þ ; a ða þ bÞ
VarðXja; bÞ ¼
ðb þ 1Þða þ bÞ2 a2
a2 ða þ bÞ2
:
b MME and b b MME for a and b, respectively, are obtained by solving the equations: The method of moments estimates a
J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
b MME ; b b MME Þ ¼ x; EðXj a
465
b MME ; b b MME Þ ¼ s2 ; VarðXj a
x and s2 are the sample mean and (biased) sample variance, respectively. where It follows that
ab MME ¼
b MME ðx 1Þ þ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b MME ðx 1Þ þ 4x b b MME ð b b MME þ 1Þ ½b ; 2x
ð9Þ
b MME is given by where b
b b MME ¼
bðx; s2 Þ þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ffi 2 ½bðx; s2 Þ þ 16s2 s2 þ ðx þ 1Þ2 x 3 h i ; 2s2 s2 þ ðx þ 1Þ2
ð10Þ
and
bðx; s2 Þ ¼ s4 x ðx 3 þ 2 x 2 þ x 4 s2 Þ:
ð11Þ
2.3. Ordinary and weighted least-squares Let xð1Þ 6 xð2Þ 6 6 xðnÞ be the order statistics of a random sample of size n from the weighted Lindley distribution with c.d.f. (5). It is well known that:
E F xð i Þ ¼
i iðn i þ 1Þ and Var F xðiÞ ¼ : nþ1 ðn þ 1Þ2 ðn þ 2Þ
b OLSE and b b OLSE of the parameters a and b, respectively, are obtained by minimizing the function The least square estimates a
Sða; bÞ ¼
n X FðxðiÞ ja; bÞ i¼1
2 i : nþ1
These estimates can also be obtained by solving the non-linear equations: n X FðxðiÞ ja; bÞ
i D1 ðxðiÞ ja; bÞ ¼ 0; nþ1 i¼1 n X i FðxðiÞ ja; bÞ D2 ðxðiÞ ja; bÞ ¼ 0; nþ1 i¼1
ð12Þ ð13Þ
where
D1 ðxja; bÞ ¼ ½ða þ bÞð1 þ xÞ þ 1
D2 ðxja; bÞ ¼
eax ðaxÞb ða þ bÞ2 CðbÞ
ð14Þ
;
CðbÞJðaxjbÞ Cðb; axÞJð0jbÞ CðbÞ þ ða þ bÞJð0jbÞ ðaxÞb eax ; logðaxÞ 2 ða þ bÞCðbÞ ða þ bÞCðbÞ C ðbÞ
ð15Þ
and
JðtjbÞ ¼
Z
1
logðyÞ yb1 ey dy;
t P 0:
ð16Þ
t
b WLSE and b b WLSE of the parameters a and b, respectively, are obtained by minimizing The weighted least-squares estimates a the function
Wða; bÞ ¼
n X ðn þ 1Þ2 ðn þ 2Þ i¼1
iðn i þ 1Þ
FðxðiÞ ja; bÞ
2 i : nþ1
These estimates can also be obtained by solving the non-linear equations:
1 i FðxðiÞ ja; bÞ D1 ðxðiÞ ja; bÞ ¼ 0; iðn i þ 1Þ nþ1 i¼1 n X 1 i FðxðiÞ ja; bÞ D2 ðxðiÞ ja; bÞ ¼ 0; iðn i þ 1Þ nþ1 i¼1
n X
where D1 ðxðiÞ ja; bÞ and D2 ðxðiÞ ja; bÞ are given by (14) and (15), respectively.
ð17Þ ð18Þ
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3. Simulations In this section we present results of some numerical experiments to compare the performance of the different estimators discussed in the previous section. We have taken sample sizes n ¼ 20; 30; . . . ; 100; a ¼ 0:5; 2, and b ¼ 0:5; 1; 2. For each combination ðn; a; bÞ, we have generated N ¼ 20; 000 pseudo-random samples from the weighted Lindley distribution using the acceptance–rejection algorithm proposed by Ghitany et al. [4]. The estimates were obtained using
b (: MLE, h: MME, M: OLSE and O: WLSE). Fig. 1. Bias of a
J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
467
Ox version 6.20, (see [3]), using the maximization package MaxBFGS. To assess the performance of the methods, we calculated the bias and the mean-squared error for the simulated estimates of a and b:
hÞ ¼ Biasðb
N 1X h i hÞ; ðb N i¼1
hÞ ¼ MSEðb
N 2 1X h i hÞ ; ðb N i¼1
h ¼ a; b:
Figs. 1 and 2 show, respectively, the bias of the simulated estimates of a and b. From these two figures, we observe that:
Fig. 2. Bias of b b (: MLE, h: MME, M: OLSE and O: WLSE).
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J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
1. all estimators of the parameters are positively biased, i.e. the estimators exceed the true value of the parameters, 2. the biases of all estimators of the parameters tend to zero for large n, i.e. the estimators are asymptotically unbiased for the parameters, 3. the OLS (MM) estimator has the smallest (largest) biase among the considered four estimators, 4. the LS and WLS estimators have almost identical biases.
b (: MLE, h: MME, M: OLSE and O: WLSE). Fig. 3. MSE of a
J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
469
Figs. 3, 4 show, respectively, the MSE of the simulated estimates of a and b. From these two figures, we observe that: 1. the MSE of all estimators of the parameters tend to zero for large n, i.e. all estimators are consistent for the parameters, 2. the MM estimators have the largest MSE among the considered four estimators,
Fig. 4. MSE of b b (: MLE, h: MME, M: OLSE and O: WLSE).
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J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
Table 1 Parameters estimates, K–S test, A–D test, and SSR for data set 1. Method
MLE MME OLSE WLSE
Estimated parameters
ab
b b
0.0073 0.0106 0.0049 0.0054
0.2769 0.8367 0.0702 0.0912
K–S
p-value
A–D
p-value
SSR
0.1157 0.1598 0.1338 0.1139
0.9468 0.6899 0.8627 0.9530
0.4250 1.0768 0.3674 0.3182
0.8214 0.3170 0.8788 0.9229
0.0549 0.1139 0.0553 0.0410
K–S
p-Value
A–D
p-Value
SSR
0.0723 0.0723 0.0702 0.0619
0.8620 0.8619 0.8831 0.9517
0.3325 0.3574 0.3179 0.2957
0.9112 0.8893 0.9246 0.9412
0.0494 0.0529 0.0474 0.0430
Table 2 Parameters estimates, K–S test, A–D test, and SSR for data set 2. Method
MLE MME OLSE WLSE
Estimated parameters
ab
b b
12.8227 13.2878 11.4272 11.7905
28.0882 29.1317 25.2070 25.9071
3. the ML, LS and WLS estimators have almost identical MSE, 4. All estimators have very close MSE when the parameter b > 1. 4. Examples In this section we analyze two real data sets for comparing the considered four estimation methods for the weighted Lindley distribution. Data set 1: ([14]). This data set represents the time to failure of 18 electronic devices:
5; 1; 21; 31; 46; 75; 98; 122; 145; 165; 195; 224; 245; 293; 321; 330; 350; 420: Data set 2: ([2]). This data set represents the failure stresses (in GPa) of 65 single carbon fibers of length 50 mm:
1:339; 1:434; 1:549; 1:574; 1:589; 1:613; 1:746; 1:753; 1:764; 1:807; 1:812; 1:84; 1:852; 1:852; 1:862; 1:864; 1:931; 1:952; 1:974; 2:019; 2:051; 2:055; 2:058; 2:088; 2:125; 2:162; 2:171; 2:172; 2:18; 2:194; 2:211; 2:27; 2:272; 2:28; 2:299; 2:308; 2:335; 2:349; 2:356; 2:386; 2:39; 2:41; 2:43; 2:431; 2:458; 2:471; 2:497; 2:514; 2:558; 2:577; 2:593; 2:601; 2:604; 2:62; 2:633; 2:67; 2:682; 2:699; 2:705; 2:735; 2:785; 3:02; 3:042; 3:116; 3:174: In comparing the four considered estimation methods for the weighted Lindley distribution, we adopt the following criteria: (i) highest p-value of the Kolmogorov–Smirnov test (which considers the greatest difference between the theoretical and empirical distributions), (ii) highest p-value of the Anderson–Darling test (which gives more weight to the tails of the distribution), (iii) smallest sum of squares of the residuals (SSR) between the theoretical and empirical distributions, i.e. P ^ F n ðxðiÞ Þ2 . ^ ; bÞ SSR ¼ ni¼1 ½FðxðiÞ ja Tables 1 and 2 show that the WLSE method satisfies the above adopted criteria, i.e. highest p-values of the K–S and A–D tests as well as smallest sum of squares of the residuals. Hence, for each of the given data sets, we conclude that the weighted least-squares estimation method is the most adequate among the four considered estimation methods. 5. Conclusions In this paper we compared, via intensive simulation experiments, the estimation of the parameters of the weighted Lindley distribution using four well known estimation methods, namely the maximum likelihood, method of moments, ordinary least-squares, and weighted least-squares. The simulations show that the weighted least-squares is a highly competitive method compared to the maximum likelihood method. This is also supported by the applications of two real data sets.
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Acknowledgements J. Mazucheli and F. Louzada gratefully acknowledge the financial support from the National Council for Scientific and Technological Development (CNPq) and São Paulo Research Foundation (FAPESP). References [1] D.K. Al-Mutairi, M.E. Ghitany, D. Kundu, Inferences on stress-strength reliability from Lindley distributions, Communications in Statistics – Theory and Methods 42 (2013) 1443–1463. [2] M.G. Bader, A.M. Priest, Statistical aspects of fiber and bundle strength in hybrid composites, in: T. Hayashi, K. Kawata, S. Umekawa (Eds.), Progress in Science and Engineering Composites, vol. ICCM-IV, 1982, pp. 1129–1136. [3] J.A. Doornik, Object-Oriented Matrix Programming Using Ox, 3rd ed., Timberlake Consultants Press and Oxford, London, 2007. [4] M.E. Ghitany, F. Alqallaf, D.K. Al-Mutairi, H.A. Husain, A two- parameter weighted Lindley distribution and its applications to survival data, Mathematics and Computers in Simulation 81 (2011) 1190–1201. [5] M.E. Ghitany, B. Atieh, S. Nadarajah, Lindley distribution and its application, Mathematics and Computers in Simulation 78 (4) (2008) 493–506. [6] D. Kundu, M.Z. Raqab, Generalized Rayleigh distribution: different methods of estimations, Computational Statistics and Data Analysis 49 (1) (2005) 187–200. [7] D.V. Lindley, Fiducial distributions and Bayes’ theorem, Journal of the Royal Statistical Society, B 20 (1958) 102–107. [8] D.V. Lindley, Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Inference, Cambridge University Press, New York, 1965. [9] J. Mazucheli, J.A. Achcar, The Lindley distribution applied to competing risks lifetime data, Computer Methods and Programs in Biomedicine 104 (2) (2011) 189–192. [10] S. Nadarajah, Bathtub-shaped failure rate functions, Quality and Quantity 43 (5) (2008) 855–863. [11] S. Rajarshi, M.B. Rajarshi, Bathtub distributions: A review, Communications in Statistics – Theory and Methods 17 (1988) 2597–2621. [12] A.I. Shawky, R.A. Bakoban, Exponentiated gamma distribution: different methods of estimations, Journal of Applied Mathematics, Art. ID 284296 (2012) 23 pp. [13] M. Teimouri, S.M. Hoseini, S. Nadarajah, Comparison of estimation methods for the Weibull distribution, Statistics 47 (1) (2013) 93–109. [14] F.K. Wang, A new model with bathtub-shaped failure rate using an additive Burr XII distribution, Reliability Engineering and System Safety 70 (2000) 305–312.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Comparison of estimation methods for the parameters of the weighted Lindley distribution J. Mazucheli a, F. Louzada b, M.E. Ghitany c,⇑ a
Universidade Estadual de Maringá, DEs, PR, Brazil Universidade de São Paulo, ICMC, SP, Brazil c Department of Statistics & O.R., Faculty of Science, Kuwait University, Kuwait b
a r t i c l e
i n f o
a b s t r a c t The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary leastsquares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results. Ó 2013 Elsevier Inc. All rights reserved.
Keywords: Weighted Lindley distribution Maximum likelihood Method of moments Ordinary least-squares Weighted least-squares
1. Introduction The Lindley distribution was introduced by Lindley [7], see also [8], in the context of fiducial distributions and Bayes’ theorem. Its probability density function (p.d.f.) is given by
f1 ðxjaÞ ¼
a2 ð1 þ xÞ eax ; aþ1
x > 0;
a > 0:
ð1Þ
The corresponding cumulative distribution function (c.d.f.) and hazard rate function (h.r.f.), respectively, are given by
ax eax ; F 1 ðxjaÞ ¼ 1 1 þ aþ1
x > 0;
a2 ð1 þ xÞ ; að1 þ xÞ þ 1
a > 0:
a > 0;
ð2Þ
and
h1 ðxjaÞ ¼
x > 0;
ð3Þ
Note that h1 ðxjaÞ is an increasing function in x, for all a > 0. Ghitany et al. [5] studied many statistical properties of the Lindley distribution from the reliability/survival analysis point of view. They also showed, using a real data set, that the Lindley distribution provides a better fit than the exponential distribution. Application of the Lindley distribution in the competing risks analysis and in stress-strength reliability studies are considered by Mazucheli and Achcar [9] and Al-Mutairi et al. [1], respectively. ⇑ Corresponding author. E-mail address: [email protected] (M.E. Ghitany). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.082
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J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
Recently, Ghitany et al. [4] introduced the weighted Lindley distribution with p.d.f.
f ðxja; bÞ ¼ where CðaÞ ¼
abþ1 ða þ bÞCðbÞ
R1
xb1 ð1 þ xÞeax ;
x > 0;
a; b > 0;
ð4Þ
xa1 ex dx; a > 0, is the gamma function. The corresponding c.d.f. and h.r.f., respectively, are given by
0
Fðxja; bÞ ¼ 1
ða þ bÞCðb; axÞ þ ðaxÞb eax ða þ bÞCðbÞ
x > 0;
a; b > 0;
ð5Þ
and
hðxja; bÞ ¼
abþ1 xb1 ð1 þ xÞeax ða þ bÞCðb; axÞ þ ðaxÞb eax
x > 0;
a; b > 0;
ð6Þ
R1 where Cða; bÞ ¼ b xa1 ex dt; a > 0; b P 0, is the upper incomplete gamma function. Note that the Lindley distribution is a special case of the weighted Lindley distribution when b ¼ 1. Also, its h.r.f. hðxja; bÞ is bathtub-shaped (increasing) function in x if 0 < b < 1 (b P 1) for all a > 0. For reviews about bathtub-shaped hazard rate functions, see [11,10]. As far as the estimation of the parameters of the weighted Lindley distribution, [4] considered only the maximum likelihood estimation (MLE) method. It is of interest to compare the MLE method with other estimation methods such as the method of moments estimation (MME), ordinary least-squares estimation (OLSE) and weighted least-squares estimation (WLSE) methods. The main aim of this paper is to identify the most efficient estimators among four estimators for different shape parameters values and sample sizes. The originality of this study comes from the fact that there has been no previous work comparing all of these estimators for the two-parameter weighted Lindley distribution. Related studies for other distributions can be found in, for example, [6,12,13]. In Section 2 we discuss the four estimation methods considered in this paper. The comparison of these methods in terms of bias and mean-squared error is presented in Section 3. The four estimation methods are used in fitting two real data sets in Section 4. Some concluding remarks are presented in Section 5. 2. Methods of estimation In this section we describe the four considered estimation methods to obtain the estimates of the parameters a and b of the weighted Lindley distribution. 2.1. Maximum likelihood Let x1 ; x2 ; . . . ; xn be a random sample of size n from the weighted Lindley distribution with parameters a and b with p.d.f. (4). b MLE and b b MLE , of a and b, respectively, are given Ghitany et al. [4] showed that the maximum likelihood estimates (MLEs) a by
ab MLE ¼
b MLE ðx 1Þ þ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½b b MLE ðx 1Þ þ 4 b b MLE ð b b MLE þ 1Þ x b MLE Þ; ¼ nð b 2x
ð7Þ
where x is the sample mean, b b MLE is the solution of the non-linear equation:
n ln ðnðbÞÞ
X n 1 wðbÞ þ lnðxi Þ ¼ 0; nðbÞ þ b i¼1
ð8Þ
d and wðbÞ ¼ db ln CðbÞ is the digamma function.
2.2. Method of moments The method of moments is another technique commonly used in parameter estimation. For the weighted Lindley distribution, we have
EðXja; bÞ ¼
b ða þ b þ 1Þ ; a ða þ bÞ
VarðXja; bÞ ¼
ðb þ 1Þða þ bÞ2 a2
a2 ða þ bÞ2
:
b MME and b b MME for a and b, respectively, are obtained by solving the equations: The method of moments estimates a
J. Mazucheli et al. / Applied Mathematics and Computation 220 (2013) 463–471
b MME ; b b MME Þ ¼ x; EðXj a
465
b MME ; b b MME Þ ¼ s2 ; VarðXj a
x and s2 are the sample mean and (biased) sample variance, respectively. where It follows that
ab MME ¼
b MME ðx 1Þ þ b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b MME ðx 1Þ þ 4x b b MME ð b b MME þ 1Þ ½b ; 2x
ð9Þ
b MME is given by where b
b b MME ¼
bðx; s2 Þ þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ffi 2 ½bðx; s2 Þ þ 16s2 s2 þ ðx þ 1Þ2 x 3 h i ; 2s2 s2 þ ðx þ 1Þ2
ð10Þ
and
bðx; s2 Þ ¼ s4 x ðx 3 þ 2 x 2 þ x 4 s2 Þ:
ð11Þ
2.3. Ordinary and weighted least-squares Let xð1Þ 6 xð2Þ 6 6 xðnÞ be the order statistics of a random sample of size n from the weighted Lindley distribution with c.d.f. (5). It is well known that:
E F xð i Þ ¼
i iðn i þ 1Þ and Var F xðiÞ ¼ : nþ1 ðn þ 1Þ2 ðn þ 2Þ
b OLSE and b b OLSE of the parameters a and b, respectively, are obtained by minimizing the function The least square estimates a
Sða; bÞ ¼
n X FðxðiÞ ja; bÞ i¼1
2 i : nþ1
These estimates can also be obtained by solving the non-linear equations: n X FðxðiÞ ja; bÞ
i D1 ðxðiÞ ja; bÞ ¼ 0; nþ1 i¼1 n X i FðxðiÞ ja; bÞ D2 ðxðiÞ ja; bÞ ¼ 0; nþ1 i¼1
ð12Þ ð13Þ
where
D1 ðxja; bÞ ¼ ½ða þ bÞð1 þ xÞ þ 1
D2 ðxja; bÞ ¼
eax ðaxÞb ða þ bÞ2 CðbÞ
ð14Þ
;
CðbÞJðaxjbÞ Cðb; axÞJð0jbÞ CðbÞ þ ða þ bÞJð0jbÞ ðaxÞb eax ; logðaxÞ 2 ða þ bÞCðbÞ ða þ bÞCðbÞ C ðbÞ
ð15Þ
and
JðtjbÞ ¼
Z
1
logðyÞ yb1 ey dy;
t P 0:
ð16Þ
t
b WLSE and b b WLSE of the parameters a and b, respectively, are obtained by minimizing The weighted least-squares estimates a the function
Wða; bÞ ¼
n X ðn þ 1Þ2 ðn þ 2Þ i¼1
iðn i þ 1Þ
FðxðiÞ ja; bÞ
2 i : nþ1
These estimates can also be obtained by solving the non-linear equations:
1 i FðxðiÞ ja; bÞ D1 ðxðiÞ ja; bÞ ¼ 0; iðn i þ 1Þ nþ1 i¼1 n X 1 i FðxðiÞ ja; bÞ D2 ðxðiÞ ja; bÞ ¼ 0; iðn i þ 1Þ nþ1 i¼1
n X
where D1 ðxðiÞ ja; bÞ and D2 ðxðiÞ ja; bÞ are given by (14) and (15), respectively.
ð17Þ ð18Þ
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3. Simulations In this section we present results of some numerical experiments to compare the performance of the different estimators discussed in the previous section. We have taken sample sizes n ¼ 20; 30; . . . ; 100; a ¼ 0:5; 2, and b ¼ 0:5; 1; 2. For each combination ðn; a; bÞ, we have generated N ¼ 20; 000 pseudo-random samples from the weighted Lindley distribution using the acceptance–rejection algorithm proposed by Ghitany et al. [4]. The estimates were obtained using
b (: MLE, h: MME, M: OLSE and O: WLSE). Fig. 1. Bias of a
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Ox version 6.20, (see [3]), using the maximization package MaxBFGS. To assess the performance of the methods, we calculated the bias and the mean-squared error for the simulated estimates of a and b:
hÞ ¼ Biasðb
N 1X h i hÞ; ðb N i¼1
hÞ ¼ MSEðb
N 2 1X h i hÞ ; ðb N i¼1
h ¼ a; b:
Figs. 1 and 2 show, respectively, the bias of the simulated estimates of a and b. From these two figures, we observe that:
Fig. 2. Bias of b b (: MLE, h: MME, M: OLSE and O: WLSE).
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1. all estimators of the parameters are positively biased, i.e. the estimators exceed the true value of the parameters, 2. the biases of all estimators of the parameters tend to zero for large n, i.e. the estimators are asymptotically unbiased for the parameters, 3. the OLS (MM) estimator has the smallest (largest) biase among the considered four estimators, 4. the LS and WLS estimators have almost identical biases.
b (: MLE, h: MME, M: OLSE and O: WLSE). Fig. 3. MSE of a
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Figs. 3, 4 show, respectively, the MSE of the simulated estimates of a and b. From these two figures, we observe that: 1. the MSE of all estimators of the parameters tend to zero for large n, i.e. all estimators are consistent for the parameters, 2. the MM estimators have the largest MSE among the considered four estimators,
Fig. 4. MSE of b b (: MLE, h: MME, M: OLSE and O: WLSE).
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Table 1 Parameters estimates, K–S test, A–D test, and SSR for data set 1. Method
MLE MME OLSE WLSE
Estimated parameters
ab
b b
0.0073 0.0106 0.0049 0.0054
0.2769 0.8367 0.0702 0.0912
K–S
p-value
A–D
p-value
SSR
0.1157 0.1598 0.1338 0.1139
0.9468 0.6899 0.8627 0.9530
0.4250 1.0768 0.3674 0.3182
0.8214 0.3170 0.8788 0.9229
0.0549 0.1139 0.0553 0.0410
K–S
p-Value
A–D
p-Value
SSR
0.0723 0.0723 0.0702 0.0619
0.8620 0.8619 0.8831 0.9517
0.3325 0.3574 0.3179 0.2957
0.9112 0.8893 0.9246 0.9412
0.0494 0.0529 0.0474 0.0430
Table 2 Parameters estimates, K–S test, A–D test, and SSR for data set 2. Method
MLE MME OLSE WLSE
Estimated parameters
ab
b b
12.8227 13.2878 11.4272 11.7905
28.0882 29.1317 25.2070 25.9071
3. the ML, LS and WLS estimators have almost identical MSE, 4. All estimators have very close MSE when the parameter b > 1. 4. Examples In this section we analyze two real data sets for comparing the considered four estimation methods for the weighted Lindley distribution. Data set 1: ([14]). This data set represents the time to failure of 18 electronic devices:
5; 1; 21; 31; 46; 75; 98; 122; 145; 165; 195; 224; 245; 293; 321; 330; 350; 420: Data set 2: ([2]). This data set represents the failure stresses (in GPa) of 65 single carbon fibers of length 50 mm:
1:339; 1:434; 1:549; 1:574; 1:589; 1:613; 1:746; 1:753; 1:764; 1:807; 1:812; 1:84; 1:852; 1:852; 1:862; 1:864; 1:931; 1:952; 1:974; 2:019; 2:051; 2:055; 2:058; 2:088; 2:125; 2:162; 2:171; 2:172; 2:18; 2:194; 2:211; 2:27; 2:272; 2:28; 2:299; 2:308; 2:335; 2:349; 2:356; 2:386; 2:39; 2:41; 2:43; 2:431; 2:458; 2:471; 2:497; 2:514; 2:558; 2:577; 2:593; 2:601; 2:604; 2:62; 2:633; 2:67; 2:682; 2:699; 2:705; 2:735; 2:785; 3:02; 3:042; 3:116; 3:174: In comparing the four considered estimation methods for the weighted Lindley distribution, we adopt the following criteria: (i) highest p-value of the Kolmogorov–Smirnov test (which considers the greatest difference between the theoretical and empirical distributions), (ii) highest p-value of the Anderson–Darling test (which gives more weight to the tails of the distribution), (iii) smallest sum of squares of the residuals (SSR) between the theoretical and empirical distributions, i.e. P ^ F n ðxðiÞ Þ2 . ^ ; bÞ SSR ¼ ni¼1 ½FðxðiÞ ja Tables 1 and 2 show that the WLSE method satisfies the above adopted criteria, i.e. highest p-values of the K–S and A–D tests as well as smallest sum of squares of the residuals. Hence, for each of the given data sets, we conclude that the weighted least-squares estimation method is the most adequate among the four considered estimation methods. 5. Conclusions In this paper we compared, via intensive simulation experiments, the estimation of the parameters of the weighted Lindley distribution using four well known estimation methods, namely the maximum likelihood, method of moments, ordinary least-squares, and weighted least-squares. The simulations show that the weighted least-squares is a highly competitive method compared to the maximum likelihood method. This is also supported by the applications of two real data sets.
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