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Statistical inference for the shape parameter change-point estimator in negative associated gamma distribution
Journal of Inequalities and Applications volume 2013, Article number: 161 (2013)
In this paper, the change-point estimator for the shape parameter is proposed in a negative associated gamma random variable sequence. Suppose that are negative associated random variables satisfying that are identically distributed with , and that are identically distributed with ; the change point is unknown. The weak and strong consistency, and the weak and strong convergence rate of the change-point estimator, are given by the CUSUM method. Furthermore, the convergence rate of the change-point estimator is presented under the local alternative hypothesis condition.
The gamma distribution occurs frequently in a variety of applications, especially in reliability, in survival analysis and in modeling income distributions. The density of a gamma-distributed random variable X with a shape parameter ν and a scale parameter λ is given by
where is the indicator function, is a Γ function with .
The family of gamma distributions includes the chi-squared distribution, exponential distribution and Erlang distribution. For example, the gamma distribution is an Erlang distribution with a positive integer ν. When the shape parameter , the gamma distribution is an exponential distribution with parameter λ; when , the gamma distribution is a chi-squared distribution, with 2ν degrees of freedom. The shape parameter is especially of interest in reliability theory because the gamma distribution is either a decreasing failure rate (DFR), a constant or an increasing failure rate (IFR) according to whether the shape parameter is negative, zero or positive. The shape parameter also plays an important role in renewal theory when modeling arrival times of events.
As for the gamma distribution parameter change-point problems, Kander and Zacks  proposed a statistic for testing a change in the one-parameter exponential family; Hsu  considered a change point for the scale parameter of gamma random variables, assuming that the shape parameter was constant; Diaz  posed the Bayesian test regarding the scale parameter change point for the independent gamma variables; Gupta and Ramanayake , Ramanayake and Gupta  discussed a linear trend change for the exponential distribution; Ramanayake  proposed some tests for detecting a change in the shape parameter of gamma distributions assuming that λ is constant. The strong consistency and convergence rate of the change-point estimator have been investigated by applying the moving averages method (Tan et al. ), assuming that there is at most one change point.
Change-point analysis is widely used in fields such as quality control, economics and finance, biostatistics and so on (see Page ; Bai and Perron ; Braun et al. ; Chen et al. ). Change-point problems have also received considerable attention due to the wide variety of applications and recent developments in computational methods. There is a considerable body of literature on change-point analysis that assume that the random variables being considered are independent.
Let be a negative associated sequence that satisfies the conditions that have the common distribution , and that have the common distribution , where is an unknown parameter called the change point; , are the shape parameters before and after change, respectively. In this paper, we assume that the scale parameter does not change, but the shape parameter is susceptible to change at an unknown time in the sequence. Noticing that and its distribution is not related to the scale parameter, logarithm transformations may be made for as follows. Let
It can be shown that the mean of is and the mean of is , where is the derivation of ; that is,
can be expressed, as in [, p.16], by
where γ is the Euler-Mascheroni constant, that is, . Since , hence is an increasing function in .
are not related to the scale parameter λ, then if we know in advance or by test that there is a change in the shape parameter, we may define the estimator of the change point as
For convenience, throughout this paper, represent a constant which is independent of n and may take different values in different expressions.
The paper is arranged as follows. In Section 2, the change-point estimator is proposed based on the CUSUM method by an appropriate logarithm transformation for , and its constancy and convergence rate are investigated. The proofs of theorems are given in Section 3.
2 Main results
Theorem 1 Assume that is a negative associated random variable sequence satisfying the conditions that are identically distributed with , and are identically distributed with . Let
where denotes the integer part of a number A. If the is a non-zero constant, then is a consistent estimator of and
where is a slowly varying function with .
Theorem 2 Assume that the conditions of Theorem 1 hold, then is a strong consistent estimator of , and
Next, we will study the convergence rate of under the local alternative hypothesis; that is, is not a constant independent of n, but it depends on n and is denoted by . Noticing that if is large, the change-point estimation is usually quite precise. In practice it may be more important to construct confidence intervals for when is small. We hence assume that as . It can be seen that the results obtained in the above theorems cannot be applied here, and we need to establish stronger results than those obtained in the above theorems.
Notice that , . Then, by the mean theorem, (under the local alternative hypothesis, denoting as ) can be expressed as
where lies between and . Hence, with some added conditions, is equal to in practice.
Theorem 3 Assume that is a negative associated random variable sequence, and are identically distributed by , and are identically distributed by . If satisfies
Remark 1 Theorems 1 and 2 give the weak and strong consistency and convergence rates for the change-point estimator of the shape parameter in a gamma distribution. In Theorem 3, the convergence rate of the change-point estimator of the shape parameter is proposed under the local alternative condition, and it is one of the necessary conditions for studying the limiting distribution of . Having this value, we can study the limiting distribution of . This will be the subject of a future paper.
3 Proof of the theorem
To prove the above theorems, we first consider the following lemmas.
Lemma 1 Let be disjoint subsets of , and let be the number of elements in , . Assume that are negative associated variables, then if
are the increasing (or decreasing) positive functions for every element, then are the negative associated variables.
Proof See Joag-Dev and Proschan . □
Lemma 2 Let be a negative associated sequence with zero mean satisfying for some . Denoting , then there exist constants related to p, for all , such that
Proof See Su, Zhao and Wang . □
Lemma 3 Let be a negative associated sequence, if is an increasing number serial, then and , we have
Proof See Hu and Zhang . □
Proof of Theorem 1 Noticing that is an increasing positive function and are negative associated sequences, we have from Lemma 1 that are negative associated sequences. Without loss of generality, assuming that , by the increasing character of in , we know that . By simple computation, it can be shown that
From the triangle inequality, it can easily be shown that
Noticing that , hence we have
Let , then by (16) and (17), it follows that
Since are the negative associated variables, by the Markov inequality and Lemma 2, , we have
where is a constant independent of n. Similar arguments give that
Hence, if we choose , where is a slowly varying function satisfying , then combining (19)-(21) we have, as ,
that is, is the weak consistent estimator of , and
Proof of Theorem 2 From (19) to (21) we have, for ,
if we choose for some . Let , then we have, as ,
By the Borel-Cantelli lemma, we obtain that is the strongly consistent estimator of τ, and
Proof of Theorem 3 To this end, we choose a value such that . By (8) and (19)-(21) (), it is easily found that is a consistent estimator of . Therefore, for every , . Thus, we now have only to examine the behavior of over those k for which . To prove , we shall prove that
when . For every , define
Then we have
Noticing that implies , which in turn implies that
Since , we have
Furthermore, we obtain
It can be seen from the definition of that
Because are negative associated variables, by the Markov inequality, Lemma 2 and (8), , we obtain
Similar arguments give
Combining (28)-(30), we see that
Similar arguments as those for give
Combining (27), (31) and (32), we obtain
Now consider . Because of symmetry, we only consider the case of . The event implies that
From the Markov inequality, Lemma 2 and (8), we obtain
Denoting , from Lemma 3 and (8), we obtain
Combining (34)-(36), we have
From (24), (33) and (37), we know that (23) holds; that is, Theorem 3 is proved. □
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We are grateful to the three referees for the valuable comments and advices that led to substantial improvement of an original draft of this paper.
The work of Dr. Tan was partially supported by a grant from the National Natural Science Foundation of China (No. 11201108), the Humanities and Social Sciences Project from Ministry of Education of China (No. 12YJC910007, 11YJC790311), the Anhui Provincial Natural Science Foundation (No. 1208085QA12) and the twelfth Five-Year plan National Science and Technology major Project (2012ZX10004609).
The authors declare that they have no competing interests.
C-cT conceived of the study questions which can be done, participated in the proofs and drafted the manuscript. B-qM participated in the proofs and provided the related reference. X-cZ participated in the proof of Theorems and helped to draft the manuscript. All authors read and approved the final manuscript.
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Tan, Cc., Miao, Bq. & Zhou, Xc. Statistical inference for the shape parameter change-point estimator in negative associated gamma distribution. J Inequal Appl 2013, 161 (2013). https://doi.org/10.1186/1029-242X-2013-161
- change point
- negative associated
- shape parameter
- convergence rate