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Inequalities involving expectations to characterize distributions
Statistics and Probability Letters 83 (2013) 2113–2118
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Inequalities involving expectations to characterize distributions Subarna Bhattacharjee a , Asok K. Nanda b,∗ , Satya Kr. Misra c a
Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India
b
Department of Mathematics and Statistics, IISER Kolkata, Mohanpur Campus, Mohanpur-741252, West Bengal, India
c
School of Applied Sciences, KIIT University, Bhubaneswar-751024, Odisha, India
article
abstract
info
Article history: Received 8 January 2013 Received in revised form 20 May 2013 Accepted 20 May 2013 Available online 27 May 2013
The characterization of distributions is well known in the field of Statistics and Reliability. This paper characterizes a few distributions with the help of failure rate, mean residual, log-odds rate, and aging intensity functions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Gini’s mean difference Logistic distribution Log-odds rate
1. Introduction We mention a few functions to be used in the sequel from the vast literature of reliability theory. These functions characterize the aging phenomenon of any living unit or a system of components. Suppose that X is a continuous random variable with probability density function (pdf) f (·), cumulative distributive function (cdf) F (·) and survival function (sf) F¯ (·) ≡ 1 − F (·). The failure rate function of X , denoted by r (·), is defined as the f (x) ratio of the pdf to the sf, i.e., r (x) = F¯ (x) , where defined. The mean residual life function, denoted by m(·) of X , is defined as m(x) = E (X − x | X > x) =
∞ x
F¯ (u)du
F¯ (x)
. The aging intensity function, which analyzes the aging property of a system
quantitatively (cf. Jiang et al., 2003), is defined as xr (x) L(x) = x , r (u)du 0
=
where defined,
−xf (x) . ¯F (x) ln F¯ (x)
The log-odds rate (LOR) of X is defined as LORX (x) =
=
∗
d dx
LOX (x), f (x)
F (x) F¯ (x)
Corresponding author. Tel.: +91 9474722103. E-mail addresses: [email protected], [email protected], [email protected] (A.K. Nanda).
0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.05.022
(1.1)
2114
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118 F (·)
where LOX (·) = ln F¯ (·) is the log-odds function. Also, by changing the variable Y = ln(X ), the log-odds rate in terms of ln(x) is obtained as g (y) , LORY (y) = ¯ (y) G(y) G ey f (ey )
, (1.2) F (ey ) F¯ (ey ) where the random variable Y has the pdf and the cdf denoted by g (·) and G(·) respectively. Henceforth, we denote LORX (x) by LOR(x) and LORY (y) by LOR(y) when there is no ambiguity. The odds ratio has a large number of applications in different fields viz. reliability, large sample theory, discriminant analysis and many others. The usefulness of the log-odds rate function in comparing the reliability of two systems is discussed in Navarro et al. (2008). They have characterized different probability distributions based on the relationships among conditional moment, failure rate and log-odds rate. It has been used to characterize probability distributions by Sunoj et al. (2007), where the authors also mention the different usage of the log-odds rate in reliability and repairable systems. Characterization of distribution by the log-odds rate is also discussed in Wang et al. (2003). They have noted that the increasing log-odds ratio is less stringent than increasing failure rate, and is therefore potentially of broader applicability. Brown et al. (2012) have used the log-odds ratio to construct large sample Wilson-type confidence intervals. It has been numerically demonstrated by Platt (1998) that the asymptotic bias of the maximum modified profile likelihood estimator of a common odds ratio is negligible for odds ratio less than 5. While estimating discriminant coefficients, Sheena and Gupta (2004) obtained the estimators in terms of gradient of log-odds. Fisher’s linear discriminant function can be viewed as a posterior log-odds that a subject belongs to one population versus the other given the data vector, see, for instance, Haff (1986). He has also shown that the vector of discriminant coefficients is the gradient of the posterior log-odds. Zimmer et al. (1998) and Wang et al. (2003) showed that F has constant LOR in x (resp. ln x) if and only if F has logistic (resp. log–logistic) distribution with respective cdf given by 1 F1 (x) = , x ∈ R, µ ∈ R, s > 0, x−µ 1 + exp − s =
and
xα
F2 (x) =
, x > 0, α > 0. 1 + xα The characterization of probability distributions arising in reliability theory is done in Kagan et al. (1973), Kotz (1974), Galambos (1975a,b), Klebanov (1978), Nanda (2010) among others. The characterization of distributions through truncation is done in Laurent (1974). Looking into the importance of the log-odds ratio, here we characterize a few well-known statistical distributions through r (·), m(·), L(·) and LOR(·). 2. Main results In this section, we characterize a few probability distributions viz., exponential, Weibull, logistic and log–logistic distributions. We start this section by stating one known result from Makino (1984) and Nanda (2010). Theorem 2.1. For any nonnegative random variable X ,
E
1
1
≥
r (X )
E (r (X ))
,
and
E
1 m(X )
≥
1 E (m(X ))
.
The equality holds if and only if X is exponentially distributed. This motivates us to prove the following theorem. Before that we give a lemma from Nanda et al. (2007) to be used in sequel. Lemma 2.1. For a nonnegative random variable X , L(x) = c, for x ≥ 0, c being a constant, if and only if X follows two-parameter Weibull distribution with shape parameter c. Theorem 2.2. For any nonnegative random variable X ,
E
1 L(X )
≥
1 E (L(X ))
.
The equality holds if and only if X follows two-parameter Weibull family of distributions.
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
2115
Proof. From the Cauchy–Schwartz inequality, we have
E
1
E (L(X )) ≥ 1.
L( X )
(2.3)
The equality in (2.3) holds if and only if there exists a constant A such that, for all x > 0,
−F (x) ln(F (x)) =A xf (x) F (x) ln(F (x)) −xf (x)
which is equivalent to the fact that L(x) = constant. Hence, the result follows from Lemma 2.1. As stated in Nanda (2010), the exponential distribution can be characterized by E
m(X ) r (X )
. We observe a similar result as
under. Theorem 2.3. For any nonnegative random variable X ,
E
L(X )
≥
r (X )
E
1 r (X ) L(X )
.
The equality holds if and only if X is exponentially distributed. Proof. Using Cauchy–Schwartz inequality, it follows that
E
r (X )
E
L( X )
L(X )
≥ 1.
r (X )
(2.4)
The equality in (2.4) holds if and only if there exists a constant B such that, for all x ≥ 0, x
r (t )dt = Bx.
(2.5)
0
Differentiating (2.5) with respect to x, we get r (x) = B, for all x > 0, which holds if and only if X has an exponential distribution. Nanda (2010) characterizes exponential and Rayleigh distributions through E (Xr (X )) and E
r (X ) X
respectively. This
motivates us to think of an interesting result which would characterize two-parameter Weibull distribution, given in the next theorem. Theorem 2.4. For any nonnegative random variable X ,
2
E (XL(X )f (X )) ≥ 4 E (Xf (X ))
.
The equality holds if and only if X follows two-parameter Weibull distribution. Proof. By the Cauchy–Schwartz inequality, we have
∞
−f (x)F¯ (x) ln(F¯ (x)) dx
0
∞ 0
2 ∞ −x2 f 3 (x) 2 dx ≥ xf (x)dx . F¯ (x) ln(F¯ (x)) 0
On using F¯ (0) = 1 and limx→∞ F¯ (x) = 0, we get ∞
1 −f (x)F¯ (x) ln(F¯ (x)) dx = . 4
0
It is easy to note that ∞
0
−x2 f 3 (x) dx = E (XL(X )f (X )) F¯ (x) ln(F¯ (x))
and
0
2
∞
xf (x)dx 2
2 = E (Xf (X )) .
(2.6)
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S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
Hence, from (2.6), we get 1 4
E (XL(X )f (X )) ≥ E (Xf (X ))
2
.
The equality in (2.6) holds if and only if there exists a constant C such that
−f (x)F¯ (x) ln(F¯ (x)) = C
−x2 f 3 (x) , F¯ (x) ln(F¯ (x))
which is equivalent to
x2 f 2 (x)
1
2 = , C F¯ 2 (x) ln(F¯ (x))
i.e., L(x) = constant. Thus, the result follows from Lemma 2.1.
The next remark about E (Xf (X )) may be noted for independent interest. Remark 2.1. For a nonnegative random variable X having exponential distribution with F¯X (t ) = exp(−λt ), for λ > 0, t ≥ 0, E (X ) = 1/λ, whereas E (Xf (X )) = 1/4. Similarly, for a nonnegative random variable X having two-parameter Weibull α distribution with F¯X (t ) = exp(− tβ ), for α, β > 0, t ≥ 0, we have, E (X ) = β 1/α Γ ( 1+α ), but E (Xf (X )) = α/4. It tells that α E (Xf (X )) is easier to compute for some distributions. The next theorem characterizes logistic distribution with the help of LOR function. Theorem 2.5. For any random variable X ,
E
1
≥
LOR(X )
1 E (LOR(X ))
.
The equality holds if and only if X follows logistic distribution. Proof. From the Cauchy–Schwartz inequality, we have
E
1 LOR(X )
E (LOR(X )) ≥ 1.
The equality in (2.7) holds if and only if there exists a constant C0 > 0 such that, for all x, f (x) F (x)F¯ (x)
= C0 ,
which is equivalent to the fact that
1 F ( x)
+
1 1 − F ( x)
dF (x) = C0 dx
which gives F (x) 1 − F (x)
= A 0 e C0 x ,
where A0 is the constant of integration. Thus, for K0 = 1/A0 , and for all x, F (x) =
e C0 x K0 + e C 0 x
,
This completes the proof.
The next corollary characterizes log–logistic distribution with the help of the LOR function.
(2.7)
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
2117
Corollary 2.1. For any nonnegative random variable Y , we have
E
1
1
≥
LOR(Y )
E (LOR(Y ))
.
The equality holds if and only if X follows log–logistic distribution, where Y = ln X .
The Italian statistician Corrado Gini suggested the quantity ∆1 , known as Gini’s mean difference, based on the expected, absolute difference between pair of units as a measure of dispersion. The definition of ∆1 is as under. Definition 2.1. If X is a continuous random variable with pdf f (·), then for two arbitrary values x, y of X , Gini’s mean difference is defined as
∆1 =
∞
∞
|x − y|f (x)f (y)dxdy −∞
−∞
A well known measure of statistical dispersion, called Gini’s coefficient of concentration, defined as G=
∆1 2E (X )
,
is often used by the scientists in measuring social inequality in terms of concentration of money, resources, etc. Next, we state a lemma to be used in the upcoming theorems. Lemma 2.2. If X is a continuous random variable with distribution function F (·), then
∆1 = 2
∞
F (x) 1 − F (x) dx. −∞
The next corollary specifies ∆1 for a nonnegative random variable. Corollary 2.2. For a piece-wise differentiable distribution function F (·) of a nonnegative random variable X having mean E (X ) < ∞, we have
∆1 = 2
∞
F (x) 1 − F (x) dx, 0
and G=
∞
1 E (X )
F (x) 1 − F (x) dx. 0
Theorem 2.6. For any random variable X ,
2
E LOR(X ) ≥
,
∆1
where ∆1 is the Gini’s Mean difference of X . The equality holds if and only if X follows logistic distribution. Proof. By Cauchy–Schwartz inequality, we have
∞
f 2 (x)
−∞
F¯ (x)F (x)
∞
dx
¯ F (x)F (x)dx ≥ 1.
−∞
Note that
∞ −∞
f 2 ( x) F¯ (x)F (x)
dx = E LOR(X )
and, from Lemma 2.2,
∞
F (x)F¯ (x)dx = −∞
Thus, (2.8) reduces to
E LOR(X ) ≥
2
∆1
.
∆1 2
.
(2.8)
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S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
The equality holds if and only if there exists a constant C ∗ such that
f 2 ( x) F¯ (x)F (x)
= C ∗ F (x)F¯ (x).
This gives LORX (x) = C ∗ , which holds if and only if X follows logistic distribution.
Theorem 2.7. For any nonnegative random variable X ,
2 2 E (X ) ,
E X 2 LOR(X ) ≥
∆1
where ∆1 is the Gini’s Mean difference of X . The equality holds if and only if X follows log–logistic distribution. Proof. Let us assume that the pdf and the cdf of X be f (·), and F (·) respectively. By Cauchy–Schwartz inequality, we have
∞
y2 f 2 (y) F¯ (y)F (y)
0
∞
F (y)F¯ (y)dy
dy
∞
2 yf (y)dy
≥
.
(2.9)
0
0
Since, ∞
E X LOR(X ) =
2
y2 f (y) F¯ (y)F (y)
0
dy,
we have, from Corollary 2.2, that ∞
F (y)F¯ (y)dy =
∆1
0
2
where ∆1 is the Gini’s Mean difference of the random variable X . Thus, (2.9) reduces to
E X 2 LOR(X ) ≥
2 2 E (X ) .
∆1
The equality holds if and only if there exists a constant C1∗ such that
y2 f 2 (y) F¯ (y)F (y)
= C1∗ F (y)F¯ (y).
This gives LORX (y) = C1∗ , which holds if and only if X follows log–logistic distribution.
References Brown, B.M., Suesse, T., Yap, V.B., 2012. Wilson confidence intervals for the two-sample log-odds ratio in stratified 2×2 contingency tables. Communications in Statistics-Theory and Methods 41 (18), 3355–3370. Galambos, J., 1975a. Characterizations of probability distributions by properties of order statistics I. In: Patil, G.P., Kotz, S., Ord, J.K. (Eds.), Statistical Distributions in Scientific Work, Vol. 3. pp. 71–88. Galambos, J., 1975b. Characterizations of probability distributions by properties of order statistics II. In: Patil, G.P., Kotz, S., Ord, J.K. (Eds.), Statistical Distributions in Scientific Work, Vol. 3. pp. 89–101. Haff, L.R., 1986. On linear log-odds and estimation of discriminant coefficients. Communications in Statistics-Theory and Methods 15 (7), 2131–2144. Jiang, R., Ji, P., Xiao, X., 2003. Aging property of univariate failure rate models. Reliability Engineering and System Safety 39, 113–116. Kagan, A.M., Linnik, Y.V., Rao, C.R., 1973. Characterization Problems in Mathematical Statistics. Wiley, New York. Klebanov, L.B., 1978. Some problems of characterizations of distributions arising in reliability theory. Theory of Probability and its Applications 23, 798–801. Kotz, S., 1974. Characterization of statistical distributions: a supplement to recent surveys. International Statistical Review 42, 39–65. Laurent, A.G., 1974. On characterization of some distributions by truncation properties. Journal of the American Statistical Association 69, 823–827. Makino, A.T., 1984. Mean hazard rate and its applications to the normal approximation of the Weibull distribution. Naval Research Logistics Quarterly 31, 1–8. Nanda, A.K., 2010. Characterizations of distributions through failure rate and mean residual life functions. Statistics and Probability Letters 80, 752–755. Nanda, A.K., Bhattacharjee, S., Alam, S.S., 2007. Properties of ageing intensity function. Statistics and Probability Letters 77, 365–373. Navarro, J., Ruiz, J.M., del Aguila, Y., 2008. Characterizations and ordering properties based on log-odds functions. Statistics 42 (4), 313–328. Platt, R.W., 1998. Estimation using the modified profile likelihood in log odds ratio regression analysis. Communications in Statistics—Simulation and Computation 27 (4), 905–915. Sheena, Y., Gupta, A.K., 2004. New estimators of discriminant coefficients as the gradient of log-odds. Annals of the Institute of Statistical Mathematics 56 (4), 757–770. Sunoj, S.M., Sankaran, P.G., Maya, S.S., 2007. Characterizations of distributions using log-odds rate. Statistics 41, 443–451. Wang, Yao, Hossain, A.M., Zimmer, W.J., 2003. Monotone log-odds rate distributions in reliability analysis. Communications in Statistics-Theory and Methods 32, 2227–2244. Zimmer, W.J., Wang, Y., Pathak, P.K., 1998. Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30, 376–385.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Inequalities involving expectations to characterize distributions Subarna Bhattacharjee a , Asok K. Nanda b,∗ , Satya Kr. Misra c a
Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India
b
Department of Mathematics and Statistics, IISER Kolkata, Mohanpur Campus, Mohanpur-741252, West Bengal, India
c
School of Applied Sciences, KIIT University, Bhubaneswar-751024, Odisha, India
article
abstract
info
Article history: Received 8 January 2013 Received in revised form 20 May 2013 Accepted 20 May 2013 Available online 27 May 2013
The characterization of distributions is well known in the field of Statistics and Reliability. This paper characterizes a few distributions with the help of failure rate, mean residual, log-odds rate, and aging intensity functions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Gini’s mean difference Logistic distribution Log-odds rate
1. Introduction We mention a few functions to be used in the sequel from the vast literature of reliability theory. These functions characterize the aging phenomenon of any living unit or a system of components. Suppose that X is a continuous random variable with probability density function (pdf) f (·), cumulative distributive function (cdf) F (·) and survival function (sf) F¯ (·) ≡ 1 − F (·). The failure rate function of X , denoted by r (·), is defined as the f (x) ratio of the pdf to the sf, i.e., r (x) = F¯ (x) , where defined. The mean residual life function, denoted by m(·) of X , is defined as m(x) = E (X − x | X > x) =
∞ x
F¯ (u)du
F¯ (x)
. The aging intensity function, which analyzes the aging property of a system
quantitatively (cf. Jiang et al., 2003), is defined as xr (x) L(x) = x , r (u)du 0
=
where defined,
−xf (x) . ¯F (x) ln F¯ (x)
The log-odds rate (LOR) of X is defined as LORX (x) =
=
∗
d dx
LOX (x), f (x)
F (x) F¯ (x)
Corresponding author. Tel.: +91 9474722103. E-mail addresses: [email protected], [email protected], [email protected] (A.K. Nanda).
0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.05.022
(1.1)
2114
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118 F (·)
where LOX (·) = ln F¯ (·) is the log-odds function. Also, by changing the variable Y = ln(X ), the log-odds rate in terms of ln(x) is obtained as g (y) , LORY (y) = ¯ (y) G(y) G ey f (ey )
, (1.2) F (ey ) F¯ (ey ) where the random variable Y has the pdf and the cdf denoted by g (·) and G(·) respectively. Henceforth, we denote LORX (x) by LOR(x) and LORY (y) by LOR(y) when there is no ambiguity. The odds ratio has a large number of applications in different fields viz. reliability, large sample theory, discriminant analysis and many others. The usefulness of the log-odds rate function in comparing the reliability of two systems is discussed in Navarro et al. (2008). They have characterized different probability distributions based on the relationships among conditional moment, failure rate and log-odds rate. It has been used to characterize probability distributions by Sunoj et al. (2007), where the authors also mention the different usage of the log-odds rate in reliability and repairable systems. Characterization of distribution by the log-odds rate is also discussed in Wang et al. (2003). They have noted that the increasing log-odds ratio is less stringent than increasing failure rate, and is therefore potentially of broader applicability. Brown et al. (2012) have used the log-odds ratio to construct large sample Wilson-type confidence intervals. It has been numerically demonstrated by Platt (1998) that the asymptotic bias of the maximum modified profile likelihood estimator of a common odds ratio is negligible for odds ratio less than 5. While estimating discriminant coefficients, Sheena and Gupta (2004) obtained the estimators in terms of gradient of log-odds. Fisher’s linear discriminant function can be viewed as a posterior log-odds that a subject belongs to one population versus the other given the data vector, see, for instance, Haff (1986). He has also shown that the vector of discriminant coefficients is the gradient of the posterior log-odds. Zimmer et al. (1998) and Wang et al. (2003) showed that F has constant LOR in x (resp. ln x) if and only if F has logistic (resp. log–logistic) distribution with respective cdf given by 1 F1 (x) = , x ∈ R, µ ∈ R, s > 0, x−µ 1 + exp − s =
and
xα
F2 (x) =
, x > 0, α > 0. 1 + xα The characterization of probability distributions arising in reliability theory is done in Kagan et al. (1973), Kotz (1974), Galambos (1975a,b), Klebanov (1978), Nanda (2010) among others. The characterization of distributions through truncation is done in Laurent (1974). Looking into the importance of the log-odds ratio, here we characterize a few well-known statistical distributions through r (·), m(·), L(·) and LOR(·). 2. Main results In this section, we characterize a few probability distributions viz., exponential, Weibull, logistic and log–logistic distributions. We start this section by stating one known result from Makino (1984) and Nanda (2010). Theorem 2.1. For any nonnegative random variable X ,
E
1
1
≥
r (X )
E (r (X ))
,
and
E
1 m(X )
≥
1 E (m(X ))
.
The equality holds if and only if X is exponentially distributed. This motivates us to prove the following theorem. Before that we give a lemma from Nanda et al. (2007) to be used in sequel. Lemma 2.1. For a nonnegative random variable X , L(x) = c, for x ≥ 0, c being a constant, if and only if X follows two-parameter Weibull distribution with shape parameter c. Theorem 2.2. For any nonnegative random variable X ,
E
1 L(X )
≥
1 E (L(X ))
.
The equality holds if and only if X follows two-parameter Weibull family of distributions.
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
2115
Proof. From the Cauchy–Schwartz inequality, we have
E
1
E (L(X )) ≥ 1.
L( X )
(2.3)
The equality in (2.3) holds if and only if there exists a constant A such that, for all x > 0,
−F (x) ln(F (x)) =A xf (x) F (x) ln(F (x)) −xf (x)
which is equivalent to the fact that L(x) = constant. Hence, the result follows from Lemma 2.1. As stated in Nanda (2010), the exponential distribution can be characterized by E
m(X ) r (X )
. We observe a similar result as
under. Theorem 2.3. For any nonnegative random variable X ,
E
L(X )
≥
r (X )
E
1 r (X ) L(X )
.
The equality holds if and only if X is exponentially distributed. Proof. Using Cauchy–Schwartz inequality, it follows that
E
r (X )
E
L( X )
L(X )
≥ 1.
r (X )
(2.4)
The equality in (2.4) holds if and only if there exists a constant B such that, for all x ≥ 0, x
r (t )dt = Bx.
(2.5)
0
Differentiating (2.5) with respect to x, we get r (x) = B, for all x > 0, which holds if and only if X has an exponential distribution. Nanda (2010) characterizes exponential and Rayleigh distributions through E (Xr (X )) and E
r (X ) X
respectively. This
motivates us to think of an interesting result which would characterize two-parameter Weibull distribution, given in the next theorem. Theorem 2.4. For any nonnegative random variable X ,
2
E (XL(X )f (X )) ≥ 4 E (Xf (X ))
.
The equality holds if and only if X follows two-parameter Weibull distribution. Proof. By the Cauchy–Schwartz inequality, we have
∞
−f (x)F¯ (x) ln(F¯ (x)) dx
0
∞ 0
2 ∞ −x2 f 3 (x) 2 dx ≥ xf (x)dx . F¯ (x) ln(F¯ (x)) 0
On using F¯ (0) = 1 and limx→∞ F¯ (x) = 0, we get ∞
1 −f (x)F¯ (x) ln(F¯ (x)) dx = . 4
0
It is easy to note that ∞
0
−x2 f 3 (x) dx = E (XL(X )f (X )) F¯ (x) ln(F¯ (x))
and
0
2
∞
xf (x)dx 2
2 = E (Xf (X )) .
(2.6)
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S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
Hence, from (2.6), we get 1 4
E (XL(X )f (X )) ≥ E (Xf (X ))
2
.
The equality in (2.6) holds if and only if there exists a constant C such that
−f (x)F¯ (x) ln(F¯ (x)) = C
−x2 f 3 (x) , F¯ (x) ln(F¯ (x))
which is equivalent to
x2 f 2 (x)
1
2 = , C F¯ 2 (x) ln(F¯ (x))
i.e., L(x) = constant. Thus, the result follows from Lemma 2.1.
The next remark about E (Xf (X )) may be noted for independent interest. Remark 2.1. For a nonnegative random variable X having exponential distribution with F¯X (t ) = exp(−λt ), for λ > 0, t ≥ 0, E (X ) = 1/λ, whereas E (Xf (X )) = 1/4. Similarly, for a nonnegative random variable X having two-parameter Weibull α distribution with F¯X (t ) = exp(− tβ ), for α, β > 0, t ≥ 0, we have, E (X ) = β 1/α Γ ( 1+α ), but E (Xf (X )) = α/4. It tells that α E (Xf (X )) is easier to compute for some distributions. The next theorem characterizes logistic distribution with the help of LOR function. Theorem 2.5. For any random variable X ,
E
1
≥
LOR(X )
1 E (LOR(X ))
.
The equality holds if and only if X follows logistic distribution. Proof. From the Cauchy–Schwartz inequality, we have
E
1 LOR(X )
E (LOR(X )) ≥ 1.
The equality in (2.7) holds if and only if there exists a constant C0 > 0 such that, for all x, f (x) F (x)F¯ (x)
= C0 ,
which is equivalent to the fact that
1 F ( x)
+
1 1 − F ( x)
dF (x) = C0 dx
which gives F (x) 1 − F (x)
= A 0 e C0 x ,
where A0 is the constant of integration. Thus, for K0 = 1/A0 , and for all x, F (x) =
e C0 x K0 + e C 0 x
,
This completes the proof.
The next corollary characterizes log–logistic distribution with the help of the LOR function.
(2.7)
S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
2117
Corollary 2.1. For any nonnegative random variable Y , we have
E
1
1
≥
LOR(Y )
E (LOR(Y ))
.
The equality holds if and only if X follows log–logistic distribution, where Y = ln X .
The Italian statistician Corrado Gini suggested the quantity ∆1 , known as Gini’s mean difference, based on the expected, absolute difference between pair of units as a measure of dispersion. The definition of ∆1 is as under. Definition 2.1. If X is a continuous random variable with pdf f (·), then for two arbitrary values x, y of X , Gini’s mean difference is defined as
∆1 =
∞
∞
|x − y|f (x)f (y)dxdy −∞
−∞
A well known measure of statistical dispersion, called Gini’s coefficient of concentration, defined as G=
∆1 2E (X )
,
is often used by the scientists in measuring social inequality in terms of concentration of money, resources, etc. Next, we state a lemma to be used in the upcoming theorems. Lemma 2.2. If X is a continuous random variable with distribution function F (·), then
∆1 = 2
∞
F (x) 1 − F (x) dx. −∞
The next corollary specifies ∆1 for a nonnegative random variable. Corollary 2.2. For a piece-wise differentiable distribution function F (·) of a nonnegative random variable X having mean E (X ) < ∞, we have
∆1 = 2
∞
F (x) 1 − F (x) dx, 0
and G=
∞
1 E (X )
F (x) 1 − F (x) dx. 0
Theorem 2.6. For any random variable X ,
2
E LOR(X ) ≥
,
∆1
where ∆1 is the Gini’s Mean difference of X . The equality holds if and only if X follows logistic distribution. Proof. By Cauchy–Schwartz inequality, we have
∞
f 2 (x)
−∞
F¯ (x)F (x)
∞
dx
¯ F (x)F (x)dx ≥ 1.
−∞
Note that
∞ −∞
f 2 ( x) F¯ (x)F (x)
dx = E LOR(X )
and, from Lemma 2.2,
∞
F (x)F¯ (x)dx = −∞
Thus, (2.8) reduces to
E LOR(X ) ≥
2
∆1
.
∆1 2
.
(2.8)
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S. Bhattacharjee et al. / Statistics and Probability Letters 83 (2013) 2113–2118
The equality holds if and only if there exists a constant C ∗ such that
f 2 ( x) F¯ (x)F (x)
= C ∗ F (x)F¯ (x).
This gives LORX (x) = C ∗ , which holds if and only if X follows logistic distribution.
Theorem 2.7. For any nonnegative random variable X ,
2 2 E (X ) ,
E X 2 LOR(X ) ≥
∆1
where ∆1 is the Gini’s Mean difference of X . The equality holds if and only if X follows log–logistic distribution. Proof. Let us assume that the pdf and the cdf of X be f (·), and F (·) respectively. By Cauchy–Schwartz inequality, we have
∞
y2 f 2 (y) F¯ (y)F (y)
0
∞
F (y)F¯ (y)dy
dy
∞
2 yf (y)dy
≥
.
(2.9)
0
0
Since, ∞
E X LOR(X ) =
2
y2 f (y) F¯ (y)F (y)
0
dy,
we have, from Corollary 2.2, that ∞
F (y)F¯ (y)dy =
∆1
0
2
where ∆1 is the Gini’s Mean difference of the random variable X . Thus, (2.9) reduces to
E X 2 LOR(X ) ≥
2 2 E (X ) .
∆1
The equality holds if and only if there exists a constant C1∗ such that
y2 f 2 (y) F¯ (y)F (y)
= C1∗ F (y)F¯ (y).
This gives LORX (y) = C1∗ , which holds if and only if X follows log–logistic distribution.
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