Discrete strong unimodality of order statistics

Statistics and Probability Letters 103 (2015) 176–185

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Discrete strong unimodality of order statistics Mahdi Alimohammadi a , Mohammad Hossein Alamatsaz a , Erhard Cramer b,∗ a

Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran

b

Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany

article

info

Article history: Received 19 March 2015 Received in revised form 20 April 2015 Accepted 20 April 2015 Available online 30 April 2015

abstract We show that discrete logconcavity of probability mass functions implies logconcavity of their distribution and survival functions. Applications are obtained and discrete strong unimodality of order statistics (OSs) is established. Illustrative examples are provided and finally discrete logconvexity of OSs is discussed. © 2015 Elsevier B.V. All rights reserved.

Keywords: Discrete order statistics Logconcavity Unimodality Total positivity IFR (DFR) DRHR (IRHR)

1. Introduction The notions of unimodality and strong unimodality exist for both discrete and continuous distributions, each one having its own interpretation. A cumulative distribution function F is said to be unimodal if there exists a value x = a such that F (x) is convex for x < a and concave for x > a. Since the property of unimodality is not preserved under convolutions, Ibragimov (1956) introduced and characterized strongly unimodal distributions. F is said to be strongly unimodal if its convolution with any unimodal distribution is unimodal. Clearly, any strongly unimodal distribution is unimodal but the converse is not necessarily true. He proved that this is equivalent to F having a logconcave density f . According to the above definition, the only unimodal discrete distributions are the degenerate ones. However, a discrete distribution with probability mass function {pi }, on the lattice of integers, is admitted to be unimodal about a, if



pi ≥ pi−1 , pi ≥ pi+1 ,

i≤a i ≥ a.

{pi } is said to be strongly unimodal, if the convolution of {pi } with any discrete unimodal distribution {qi } is unimodal. As in the continuous case, Keilson and Gerber (1971) showed that a necessary and sufficient condition for {pi } to be strongly unimodal is that {pi } is a logconcave sequence, i.e., p2i ≥ pi−1 pi+1

∀i.

(1)

An excellent source for the details of these concepts is provided by Dharmadhikari and Joag-Dev (1988) (for convolutions of discrete logconcave random variables, see also Johnson and Goldschmidt, 2006).

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (M. Alimohammadi), [email protected] (M.H. Alamatsaz), [email protected] (E. Cramer). http://dx.doi.org/10.1016/j.spl.2015.04.019 0167-7152/© 2015 Elsevier B.V. All rights reserved.

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Because of the importance of ordered random variables (e.g., order statistics (OSs)) in many branches of statistical theory and applications, unimodality and strong unimodality of such random variables have been extensively studied in the literature in the continuous case (see, for example Alam, 1972; Huang and Ghosh, 1982; Basak and Basak, 2002; Aliev, 2003; Cramer et al., 2004; Cramer, 2004). The results of Chen et al. (2009) and Alimohammadi and Alamatsaz (2011) concerning generalized order statistics (for the definition see Kamps, 1995) unified most of these results. The recent work of Alimohammadi et al. (2014) almost exhausted these results in the continuous case. Here, we were motivated to investigate strong unimodality of OSs in the discrete case. In this article, we obtain some basic and useful results and show that, similar to the continuous case, discrete logconcavity of a probability mass function (pmf) implies logconcavity of its cumulative distribution (cdf) and survival functions (sf). As an application, we shall show that discrete logconcave distributions are IFR (increasing failure rate) and DRHR (decreasing reverse hazard rate) and discrete logconvex ones are DFR (decreasing failure rate) while they cannot be IRHR (increasing reverse hazard rate). Then, using these fundamental results, we shall establish discrete strong unimodality of OSs by means of the notion of total positivity and a lemma due to Misra and van der Meulen (2003). Some illustrative examples are also provided in this regard. Finally, we obtain some results about discrete logconvexity of OSs. Throughout the paper, increasing and decreasing mean non-decreasing and non-increasing, respectively. Further, ratios are supposed to be well defined whenever they are used. 2. Some preliminaries results Let X be a discrete random variable with support SX = {L, . . . , U } ⊆ Z = {0, ±1, ±2, . . .}, pmf pi = P (X = i), cdf Fi = P (X ≤ i), and sf F¯i = P (X > i), i ∈ SX . In this section, we review some useful related results. We first need the following comprehensive fundamental lemma which is an interesting result on its own. Some parts of the lemma are known in the literature but they are recalled and proved for completeness. First, we note that it is not difficult to see that if {pi } is logconvex, i.e., p2i ≤ pi−1 pi+1 , ∀i, then U = ∞. Furthermore, in Theorem 2.3, we shall show that if {pi } is logconvex, then pi /F¯i−1 is decreasing which in turn implies that {pi } is decreasing (and thus unimodal). Therefore, L must be finite because of the summability of {pi }. Thus, in the rest of the article, we assume that the support of a logconvex pmf {pi } is SX = {L, . . . , ∞} with L > −∞. Lemma 2.1. Let {pi } be a logconcave (logconvex) pmf and i, j ∈ SX . Then, we have: (i) pi pj ≥ (≤)pi−1 pj+1 , for j ≥ i; (ii) Fi pj ≥ (≤)Fi−1 pj+1 , for j ≥ i − 1; (iii) pi F¯j−1 ≥ (≤)pi−1 F¯j , for j ≥ i − 1. For logconvexity, the extra condition of i − 1 ∈ SX is needed on the boundary of the support. Proof. Assume that {pi } is logconcave. Then, we have: (i) The assertion is obvious by noticing that logconcavity of the sequence {pi } implies that {pi /pi−1 } is decreasing. (ii) From (i), for j ≥ i we have i 

pk pj ≥

k=−∞

i 

pk−1 pj+1 H⇒ Fi pj ≥ Fi−1 pj+1 ,

j ≥ i.

(2)

k=−∞

For i = j, (2) yields that Fj+1 p j +1

=1+

Fj p j +1

≥

Fj−1 pj

+1=

Fj pj

.

So, (2) is also valid for j ≥ i − 1. Thus, (ii) is true. (iii) Again, from (i), for j ≥ i we have ∞ 

pi pk ≥

∞ 

k=j

pi−1 pk+1 H⇒ pi F¯j−1 ≥ pi−1 F¯j ,

j ≥ i.

(3)

k=j

Now, letting j = i in (3) we obtain F¯i−2 pi−1

=1+

F¯i−1 pi−1

≥

F¯i pi

+1=

F¯i−1 pi

.

Thus, (3) follows for every j ≥ i − 1. The results for logconvexity are proved similarly.



To establish our main results, we first present a useful theorem concerning the inherence of discrete logconcavity from a pmf to its cdf and sf.

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Theorem 2.2. Let {pi } be a logconcave pmf. Then, Fi−1 /Fi and F¯i−1 /F¯i are increasing in i, or, equivalently, Fi2 ≥ Fi−1 Fi+1

F¯i2 ≥ F¯i−1 F¯i+1 .

and

Furthermore, for any real x in [i, i + 1) if we define x− and x+ such that i − 1 ≤ x− < i ,

i + 1 ≤ x+ < i + 2,

∀i,

then, logconcavity of {pi } implies that F 2 (x) ≥ F (x− )F (x+ ) and

F¯ 2 (x) ≥ F¯ (x− )F¯ (x+ ).

Proof. Put j = i in Lemma 2.1(ii) and (iii). Then, we have Fi Fi−1

=1+

pi

≥

Fi−1

pi+1 Fi

+1=

Fi+1 Fi

and F¯i−1 F¯i

=1+

pi F¯i

≥

pi−1

+1=

F¯i−1

Thus, we have the results.

F¯i−2 F¯i−1

.

(4)



Note that, the converse of Theorem 2.2 may not be true in general. As a counterexample, consider the pmf with p0 = 5/8, p1 = 2/8 and p2 = 1/8 which is not logconcave but both F and F¯ are logconcave. The analogue of Theorem 2.2 for logconvex pmfs will be presented after Theorem 2.3 but with a major difference. Now, we discuss some applications in reliability theory. It is well known that the monotonicity of a failure rate of a life distribution plays a very important role in modeling failure time data. Therefore, identification and properties of IFR and DFR distributions have been extensively studied in the literature for the continuous case (standard references on this topic are Barlow and Proschan, 1975; Marshall and Olkin, 2007). However, for the discrete case, due to the complexity of the failure rate, determination of IFR and DFR properties are not straightforward. We denote hazard and reverse hazard rate functions by hi = pi /F¯i−1 and ri = pi /Fi , respectively. Theorem 2.3. Let X be a discrete integer-valued random variable with pmf {pi }. We have: (i) if {pi } is logconcave (logconvex), then it is IFR (DFR); (ii) if {pi } is logconcave, then it is DRHR. Proof. Let j = i − 1 in Lemma 2.1(iii) and (ii). Then, we obtain hi =

pi F¯i−1

≥ (≤)

pi−1 F¯i−2

= hi−1

and ri−1 =

pi−1 Fi−1

≥

pi Fi

= ri .

Thus, the desired results follow.



Remark 2.4. An (1995, Section 3) obtained Lemma 2.1(i) and Theorem 2.3(i) for the IFR case. However, independently of An (1995), later Gupta et al. (1997) proved Theorem 2.3(i) using another approach (see also Johnson et al., 2005, p. 519). Here, one may think that the method used in the proof of Theorem 2.3 may be utilized to obtain the same result for the IRHR case. However, it is worth mentioning that if logconvexity of {pi } would imply IRHR, then pi /Fi and thus {pi } would be increasing which contradicts U = ∞, mentioned before. Indeed, if {pi } is logconvex, then ∆Fi = Fi − Fi−1 = pi is decreasing, i.e., F is concave. Furthermore, an increasing concave function is logconcave. Thus, we have proved the following where the later claim is proved similar to (4) in the reverse direction. Theorem 2.5. If {pi } is logconvex, then F is logconcave and F¯ is logconvex. 3. Main results Let X1 , . . . , Xn be a random sample of size n from a discrete distribution F . We denote the cdf, sf, and pmf of its OSs X1:n ≤ · · · ≤ Xn:n by Fir :n , F¯ir :n , and pri :n , i ∈ SX , 1 ≤ r ≤ n, respectively. OSs have been extensively investigated in the

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179

literature. For the discrete case, we refer to, e.g., Balakrishnan (1986), Nagaraja (1992), David and Nagaraja (2003), Dembińska (2008), and Arnold et al. (2008, Chapter 3) (see also, Balakrishnan and Cramer, 2014, Section 2.8). There are some expressions for the pmf of the rth OS as follows (Arnold et al., 2008, p. 42): pri :n = Fir :n − Fir−:n1

=

n    n

j

j =r

=r

n  r

Fi

j n −j

Fi F¯i

− Fij−1 F¯in−−1j



yr −1 (1 − y)n−r dy.

(5)

Fi−1

Alam (1972) showed that for an absolutely continuous distribution with a density function f , the condition of convexity of 1/f is sufficient to ensure unimodality of its OSs. In the discrete case, he only investigated unimodality of the smallest and largest OSs. In the continuous case, Barlow and Proschan (1966, Theorem 7.2) and Huang and Ghosh (1982) showed that OSs from a strongly unimodal distribution are strongly unimodal. We first recall that

¯ Definition 3.1. Let X and Y be two random variables with density functions (or pmf) f and g, cdf’s F and G, and sf’s F¯ and G, respectively. We say that X is smaller than Y in the: ¯ (x) for all x; (a) usual stochastic order (denoted by F ≤st G) if F¯ (x) ≤ G (b) likelihood ratio order (denoted by F ≤lr G) if g (x)/f (x) is increasing in x. It is well known that F ≤lr G H⇒ F ≤st G. For a comprehensive discussion on stochastic orders, we refer the reader to Shaked and Shanthikumar (2007). The lemma below, due to Misra and van der Meulen (2003), is often used in establishing the monotonicity of a fraction in which the numerator and denominator are integrals or summations. Lemma 3.2. Assume that Θ is a subset of the real line R, and let U be a nonnegative random variable having a cdf belonging to the family P = {H (·|θ ), θ ∈ Θ } which is such that, for θ1 , θ2 ∈ Θ , H (·|θ1 ) ≤st (≥st )H (·|θ2 ),

whenever θ1 < θ2 .

Let ψ(u, θ ) be a real valued function defined on R × Θ , which is measurable in u for each θ such that Eθ [ψ(U , θ )] exists. Then, Eθ [ψ(U , θ )] is (i) increasing in θ , if ψ(u, θ ) is increasing in θ and increasing (decreasing) in u; (ii) decreasing in θ , if ψ(u, θ ) is decreasing in θ and decreasing (increasing) in u. The following definition introduces the useful concept of total positivity (see Karlin, 1968). Definition 3.3. Let X and Y be subsets of the real line R. A kernel k : X × Y → R is said to be totally positive of order 2

(TP2 ) if

k(x1 , y1 )k(x2 , y2 ) − k(x1 , y2 )k(x2 , y1 ) ≥ 0, for all x1 ≤ x2 in X and y1 ≤ y2 in Y. Now, we are ready to provide our main results. Theorem 3.4. If {pi } is strongly unimodal, then its corresponding smallest and largest OSs are strongly unimodal. Proof. According to (1), we have to show that p1i :n /p1i−:n1 is decreasing in i. Now, since pi1:n = F¯i1−:n1 − F¯i1:n = F¯in−1 − F¯in = pi

n−1 

j n −j −1

F¯i F¯i−1

,

j =0

we have pi1:n pi1−:n1

=

pi pi−1

Ei [ψ1 (J , i)],

(6)

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where n−1

 Ei [ψ1 (J , i)] =

n −1

j n −j −1

F¯i F¯i−1



j =0 n −1



=

j n −j −1 Fi−1 Fi−2

¯

¯

j =0 n−1



j =0

j n−j−1

F¯i F¯i−1

.

n−j−1 j F¯i−1 F¯i−2

j =0

Here, the nonnegative random variable J has a df belonging to the family P1 = {H1 (·|i), i ∈ Z} with corresponding pmf n−j−1 j Fi−2

h1 (j|i) = c1 (i)F¯i−1 where c1 (i) =



n−1 j=0



F¯i

ψ1 (j, i) =

¯

,

n −j −1 j Fi−2

F¯i−1

¯

j = 0, 1, . . . , n − 1,

−1

is the normalizing constant and

j .

F¯i−2

Strong unimodality of {pi } implies that pi /pi−1 is decreasing in i. Thus, since both factors on the right-hand side of (6) are nonnegative, it suffices to show that Ei [ψ1 (J , i)] is decreasing in i. First note that Theorem 2.2 implies F¯i−2 /F¯i−1 is increasing in i and thus h1 (j|i) is TP2 in (j, i) ∈ {0, 1, . . . , n − 1} × Z. This implies that h1 (j|i2 )/h1 (j|i1 ) is increasing in j which in turn implies that H1 (·|i1 ) ≤lr H1 (·|i2 ) and hence H1 (·|i1 ) ≤st H1 (·|i2 ) whenever i1 < i2 . Obviously, ψ1 (j, i) is decreasing in j and applying Theorem 2.2 once again, we see that it is also decreasing in i. Therefore, using Lemma 3.2(ii), we have the desired result. For the largest OS, we show that pni :n /pni+:n1 is increasing in i. We have pni :n = Fin:n − Fin−:n1 = Fin − Fin−1 = pi

n−1 

j n −j −1

Fi Fi−1

.

j =0

Thus, we obtain pni :n pni+:n1

=

pi pi+1

Ei [ψ2 (J , i)],

(7)

where n−1

 Ei [ψ2 (J , i)] =

j =0 n −1



n−1

j n−j−1



Fi Fi−1

= n −j −1

j

Fi+1 Fi

j =0

j n −j −1

Fi Fi−1

j =0 n−1



. n−j−1 j Fi

Fi+1

j =0

Here, the nonnegative random variable J has a df belonging to the family P2 = {H2 (·|i), i ∈ Z} with corresponding pmf n−j−1 j Fi

h2 (j|i) = c2 (i)Fi+1 where c2 (i) =

 n−1 j=0

 ψ2 (j, i) =

,

n −j −1 j Fi

Fi+1

Fi−1

j = 0, 1, . . . , n − 1,  −1

is the normalizing constant, and

n−j−1

Fi+1

.

Similarly, one can show that H2 (·|i1 ) ≤st H2 (·|i2 ) whenever i1 < i2 and ψ2 (j, i) is increasing in j and i. Therefore, the result follows by Lemma 3.2(i).  Known examples of discrete strongly unimodal distributions include the discrete uniform, binomial, Poisson and negative binomial distributions (see e.g. Dharmadhikari and Joag-Dev, 1988, p. 111, 112). In the next theorem, we directly show that if X is geometric or discrete uniform distribution, then Xr :n are strongly unimodal for all 1 ≤ r ≤ n. Theorem 3.5. All OSs from geometric and discrete uniform distributions are strongly unimodal and thus unimodal. Proof. Let Fi = 1 − (1 − θ )i+1 ,

i = 0, 1, . . .

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181

1 be the cdf of a geometric distribution with parameter 0 < θ < 1. Using the substitution u = p− i (y − Fi−1 ) in (5), we obtain

pir :n pir−:n1

1 (1 − θ ) 0 [uθ (1 − θ )i + 1 − (1 − θ )i ]r −1 [(1 − θ )i − uθ (1 − θ )i ]n−r du = 1 [vθ (1 − θ )i−1 + 1 − (1 − θ )i−1 ]r −1 [(1 − θ )i−1 − vθ (1 − θ )i−1 ]n−r dv 0  1 (1 − θ ) = du,  1 r −1 r 0 φ1,u,v (i)φ2n,−u,v (i)dv 0

where

vθ (1 − θ )i−1 + 1 − (1 − θ )i−1 θ (1 − θ )−1 [u(1 − θ ) + (1 − v)] = 1 − , uθ (1 − θ )i + 1 − (1 − θ )i (1 − θ )−i + uθ − 1 (1 − θ )i−1 − vθ (1 − θ )i−1 1 − vθ φ2,u,v (i) = = . (1 − θ )i − uθ (1 − θ )i (1 − θ )(1 − uθ ) φ1,u,v (i) =

Since φ1,u,v (i) is increasing in i and φ2,u,v (i) is constant w.r.t. i, we have proved the desired result. Now, we consider a discrete uniform distribution on SX = {1, . . . , N } with 1 ≤ N < ∞. Then, let Fi =

i N

,

i = 1, . . . , N .

By analogy with the case of a geometric distribution, we obtain pir :n pir−:n1

1 =  01 0

[u + i − 1]r −1 [N − i + 1 − u]n−r du [v + i − 2]

r −1

[N − i + 2 − v]

v

1

 =

n −r d

0

1

1 0

φ

r −1 3,u,v

r (i)φ4n,−u,v (i)dv

du,

where

v+i−2 1−v+u =1− , u+i−1 u+i−1 1−v+u N −i+2−v =1+ . φ4,u,v (i) = N −i+1−u N −i+1−u

φ3,u,v (i) =

The last expressions immediately show that the functions are increasing in i. This proves the assertion for the discrete uniform distribution.  We note that, for arbitrary strongly unimodal distributions, the method of the proof of Theorem 3.5 cannot be used for all 1 ≤ r ≤ n. Now, we give some illustrative examples. The following example provides an interesting situation in which {pm } is not unimodal, but its smallest and largest OSs are strongly unimodal which shows that the converse of Theorem 3.4 does not necessarily hold. Example 3.6. Suppose that p0 = 0.37, p1 = 0.26 and p2 = 0.37. Clearly, {pi } is not unimodal and therefore, it is not strongly unimodal. Now, take a random sample of size n = 3 from this distribution. Then, we have

(0.199394)2 = (p11:3 )2 ≥ p10:3 p12:3 = (0.749953)(0.050653), (0.199394)2 = (p31:3 )2 ≥ p30:3 p32:3 = (0.050653)(0.749953). However, by a slight modification on {pi } such as p′0 = 0.38, p′1 = 0.24 and p′2 = 0.38, the result fails as shown below,

(0.183456)2 = (p11:3 )2 ̸≥ p10:3 p12:3 = (0.761672)(0.54872), (0.183456)2 = (p31:3 )2 ̸≥ p30:3 p32:3 = (0.54872)(0.761672). The next example shows that unimodality of {pi } is not sufficient for strong unimodality of OSs. Example 3.7. Suppose that, p0 = 0.1, p1 = 0.2 and p2 = 0.7. Clearly, {pi } is unimodal but, it is not strongly unimodal. Take n = 3. Then, we have

(0.26)2 = (p31:3 )2 ̸≥ p30:3 p32:3 = (0.001)(0.973). That is, the largest OSs are not strongly unimodal either. Moreover, if we consider {1 − pi }, then a similar situation holds for the smallest OS.

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At the end of this section, we obtain some results about logconvexity of OSs. First note that if a discrete random variable X is DFR, then so is X1:n because pi1:n

¯

Fi1−:n1

 pi n =1− 1− . F¯i−1

In the next theorem, we obtain a stronger result. The proof is similar to that of Theorem 3.4 and, thus, omitted. Theorem 3.8. If {pi } is logconvex, then so is its smallest OS. The continuous version of Theorem 3.8 is easy to derive. Remark 3.9. According to Theorems 3.4 and 3.8, if X has geometric distribution, then X1:n is both logconcave and logconvex. We must note that this is not a surprising result, because, X has geometric distribution if, and only if, X1:n has geometric distribution. This fact is also true in the continuous case for the exponential distribution (see e.g. David and Nagaraja, 2003, p. 142). Finally, we give an example showing that Theorem 3.8 does not necessarily hold for other OSs even for the largest OS. Example 3.10. Let X have a discrete Pareto distribution with cdf Fi = 1 −

1 i+1

,

i = 1, 2, . . . .

Obviously, pi = [i(i + 1)]−1 is logconvex (while Fi is logconcave, cf. Theorem 2.5). However, taking n = 2, we have

(p22:2 )2 − p12:2 p32:2 = 0.008295, which is not negative. 4. Conclusions and a conjecture OSs appear in many areas of statistical theory and applications including quality control, robustness, outlier detection, and reliability analysis. Different properties of OSs have been developed extensively assuming that the random variables constructing the random sample are absolutely continuous. In the literature, it is known that the study of discrete OSs is a hard topic and, thus, has not attracted much attention yet. In particular, unimodality and strong unimodality (logconcavity) of OSs are mostly considered for the continuous case. In this paper, we have studied the strong unimodality of OSs in the discrete case. The discussion of this problem is rather simple for the continuous case than for the discrete case. Note that, in the continuous case, the density of the rth OS has the explicit form f r : n ( x) = r

n r

f (x)F r −1 (x)F¯ n−r (x)

and that logconcavity of f implies logconcavity of F and F¯ . Thus, f r :n becomes a product of logconcave functions. But, in the discrete case, the pmf of the rth OS does not have an explicit form. Thus, a mathematical proof for strong unimodality of OSs, in general, seems complicated. However, according to our numerous computations (one of which is given in the Appendix), we conjecture that Theorem 3.4 is valid for all OSs. We note that, in the continuous case, Huang and Ghosh (1982) showed that if X1,n and Xn,n are both strongly unimodal then so are Xr ,n for all 2 ≤ r ≤ n − 1. This may well be true in the discrete case. However, we leave this as an open problem. Acknowledgment The first author is grateful to the office of Graduate Studies of the University of Isfahan for their support. Appendix As mentioned before, the authors checked the behavior of OSs coming from several discrete strongly unimodal distributions. For instance, suppose that p1 = 2/11,

p2 = 5/11,

p3 = 3/11,

p4 = 1/11.

(8)

Obviously, {pi } is strongly unimodal. ( ) − ≥ programming, we calculated (pr2:n )2 − ( ) − positive. The results appear in Tables 1 and 2 for 1 ≤ r ≤ n ≤ 25.

( ) − ≥ 0 are trivial. Using computer ≤ r ≤ n ≤ 100 and observed that all values are

The cases pr1:n 2 pr0:n pr2:n 0 and r :n r :n r :n 2 p1 p3 and p3 pr2:n pr4:n for 1

pr4:n 2

p3r :n p5r :n

n

Table 1 The values of ‘‘(pr2:n )2 − pr1:n pr3:n ’’ for a random variable X with pmf (8).

M. Alimohammadi et al. / Statistics and Probability Letters 103 (2015) 176–185

r

183

n

Table 2 The values of ‘‘(pr3:n )2 − pr2:n pr4:n ’’ for a random variable X with pmf (8).

184 M. Alimohammadi et al. / Statistics and Probability Letters 103 (2015) 176–185

r

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