Statistical inference for the shape parameter change-point estimator in negative associated gamma distribution
© Tan et al.; licensee Springer. 2013
Received: 25 October 2012
Accepted: 18 March 2013
Published: 8 April 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.
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.
where γ is the Euler-Mascheroni constant, that is, . Since , hence is an increasing function in .
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
where is a slowly varying function with .
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.
where lies between and . Hence, with some added conditions, is equal to in practice.
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.
are the increasing (or decreasing) positive functions for every element, then are the negative associated variables.
Proof See Joag-Dev and Proschan . □
Proof See Su, Zhao and Wang . □
Proof See Hu and Zhang . □
From (24), (33) and (37), we know that (23) holds; that is, Theorem 3 is proved. □
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).
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