Journal of Probability and Statistics
Volume 2015 (2015), Article ID 972035, 7 pages
http://dx.doi.org/10.1155/2015/972035

Research Article

Recurrence Relation and Accurate Value on Inverse Moment of Discrete Distributions

ChunYuan Wang and Wuyungaowa

Department of Mathematics, College of Sciences and Technology, Inner Mongolia University, Hohhot 010021, China

Received 6 March 2015; Accepted 7 June 2015

Academic Editor: Zacharias Psaradakis

Copyright © 2015 ChunYuan Wang and Wuyungaowa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Properties of the generalized hypergeometric series functions are employed to get the recurrence relation for inverse moments and inverse factorial moments of some discrete distributions. Meanwhile, with the existence of the recurrence relations, the accurate value for inverse moment of discrete distributions can thus be obtained.

1. Introduction and Preliminaries

Kumar and Consul [1] develop a recursive relation upon the negative moments of power series distribution. The recurrence relation for the negative moments of the Poisson distribution was first derived by Chao and Strawderman [2], after which it is shown by Kumar and Consul [1] as a special case of their result. Using the recurrence relation for the negative moments of the Lagrangian binomial distribution, Kumar and Consul [1] have established the binomial and negative binomial distributions.

Ahmad and Saboor [3] proved many properties of the hypergeometric series. Besides, the following series have been provided by Ahmad and Saboor [3] and

is generalized hypergeometric series, where , , , . If , then

is hypergeometric series.

In this paper, the recurrence relation for negative moments along with negative factorial moments of some discrete distributions can be obtained. These relations have been derived with properties of the hypergeometric series.

In the next part, some necessary definitions have been introduced.

Let be a generalized negative binomially distributed random variable with parameters , and the probability mass function is

where , , , and and a constant.

Let be a generalized Poisson distribution with parameters , and the probability mass function is

where , , and .

Let the random variable be equipped with a generalized poisson-negative-binomial distribution with parameters , , and ; the probability mass function is

where , , , , , , , and and a constant.

Let be a generalized logarithmic distributed random variable with parameters and ; the probability mass function is

where , , and .

2. The Recurrence Relation for Inverse Moments of Some Discrete Distributions

In this section, some recurrence relations for inverse moments of some discrete distributions can be obtained with the properties of the generalized hypergeometric series functions.

Theorem 1. Let be a generalized negative binomial random variable with parameters , , for , , , and probability mass function is defined in (3), and then the inverse moment of first order is given by

where , , and and a constant.

Proof. Since is a generalized negative binomial random variable with parameters , , then

Theorem 2. Let be a generalized negative binomially distributed random variable with parameters and , and probability mass function is defined in (3). Then the following relation holds:

where , , , , and .

Proof. From Theorem 1, we have

using the identity (see [4, page 85])

For , , , , , and we have

and we know that

Then (12) becomes

Using another identity (see [4, page 71])

and for , , , and we have

as

From (14) and (16) we have

Rearranging we get

By collating, we get the result (9).

Theorem 3. Let the random variable be equipped with a generalized Poisson distribution with parameters and , and probability mass function is defined in (4). Then

and , , and .

Proof. Using the identity (see [4, page 84])

By the same as Theorem 2 calculating, we get the result (20).

Theorem 4. Let the random variable be equipped with a generalized Poisson-negative-binomial distribution with parameters , , and , and probability mass function is defined in (5); then

where , , , , and .

Proof. Using identities (11) and (15), by the same as Theorem 2 calculating, we get the result (22).

Theorem 5. Let be a generalized logarithmic distributed random variable with parameters and , and probability mass function is defined in (6). Then the following recurrence relation holds:

where , , , , and .

Proof. Using identities (11) and (15), by the same as Theorem 2 calculating, we get the result (23).

3. Recurrence Relation for Inverse Factorial Moments of Discrete Distributions

In this section, some recurrence relations for inverse factorial moments of some discrete distributions can be obtained with the properties of the generalized hypergeometric series functions.

Theorem 6. Let be a generalized negative binomial random variable with parameters , , for , , , and probability mass function is defined in (3), and then the factorial inverse moment of first order is given by

where and , .

Proof. Since is a generalized negative binomial random variable with parameters , , then

Theorem 7. Let equipped with generalized negative binomial probability distribution be defined in (3) with parameters and ; suppose is the th negative factorial moment of . Then the relation

holds for , and , , .

Proof. From Theorem 6, we have

Using the identity (see [4, page 71])

and for , , , and we have

Using the identity (see [4, page 71])

for , , , and  ; rearranging we have

Substituting in (29), we get

Consider and using (28), for , , , and we have

Substituting in (32), we get

Rearranging we have

and by collating, we get the result (26).

Theorem 8. Let the random variable equipped with a generalized Poisson distribution be defined in (4) with parameters and , and is the th negative factorial moment of . Then the relation

holds for , , , .

Proof. Using the identity (see [4, pages 82 and 84])

by the same as Theorem 7 calculating, we get the result (36).

Theorem 9. Let equipped with generalized logarithmic series distribution be defined in (6) with parameters and , and is the th inverse factorial moment of . Then the relation

holds for , , , .

Proof. Using the identities (28) and (30), by the same as Theorem 7 calculating, we get the result (38).

4. The Accurate Value for Inverse Moments of Some Discrete Distributions

In this section, some accurate values for inverse moments of some discrete distributions can be obtained with their recurrence relations.

Theorem 10. Let be a generalized negative binomially distributed random variable with parameters and , and probability mass function is defined in (3). Then the accurate value for inverse moment of first order is given by

where , , , .

Proof. From (9), we have

and let ; we get

and hence

and so on; repeat the above steps; we can get

and, from (10), we have

so we obtain the result (39).

Theorem 11. Let the random variable be equipped with a generalized Poisson distribution with parameters and , and probability mass function is defined in (4), and then the accurate value for inverse moment of first order is given by

where , , and .

The proof is the same as Theorem 10, omitted here.

Theorem 12. Let the random variable be equipped with a generalized poisson-negative-binomial distribution with parameters , , and , and probability mass function is defined in (5), and then the accurate value for inverse moment of first order is given by

where , , , and .

The proof is the same as Theorem 10, omitted here.

Theorem 13. Let be a generalized logarithmic distributed random variable with parameters and , and probability mass function is defined in (6). Then the accurate value for inverse moment of first order is given by

where , , , and .

The proof is the same as Theorem 10, omitted here.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by the National Natural Science Foundation of China under Grant 11461050 and Natural Science Foundation of Inner Mongolia 2012MS0118.

References

A. Kumar and P. C. Consul, “Negative moments of a modified power series distribution and bias of the maximum likelihood estimator,” Communications in Statistics, vol. 8, no. 2, pp. 151–166, 1979.

M. T. Chao and W. E. Strawderman, “Negative moments of positive random variables,” Journal of the American Statistical Association, vol. 67, no. 338, pp. 429–431, 1972.

M. Ahmad and A. Saboor, “Properties of a newly defined hypergeometric power series function,” Pakistan Journal of Statistics, vol. 25, no. 2, pp. 227–233, 2009.

E. D. Rainville, Special Functions, Chelsea, New York, NY, USA, 1960.