Statistical inference for the shape parameter change-point estimator in negative associated gamma distribution

In this paper, the change-point estimator for the shape parameter is proposed in a negative associated gamma random variable sequence. Suppose that X1,…,Xn are negative associated random variables satisfying that X1,…,X[nτ0] are identically distributed with Γ(x;ν1,λ), and that X[nτ0]+1,…,Xn are identically distributed with Γ(x;ν2,λ); the change point τ0 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 OP convergence rate of the change-point estimator is presented under the local alternative hypothesis condition. MSC:62F12, 62G10.


Introduction
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 gammadistributed random variable X with a shape parameter ν and a scale parameter λ is given by where I(·) is the indicator function, (·) is a function with (p) = ∞  e -x x p- dx. 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 ν = , the gamma distribution is an exponential distribution with parameter λ; when λ =   , the gamma distribution is a chi-squared distribution, with ν 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  ). 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 X  , X  , . . . , X n be a negative associated sequence that satisfies the conditions that X  , . . . , X [nτ  ] have the common distribution (x; ν  , λ), and that X [nτ  ]+ , . . . , X n have the common distribution (x; ν  , λ), where τ  is an unknown parameter called the change point; ν  , ν  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 [nτ  ] in the sequence. Noticing that λX ∼ (x; ν, ) and its distribution is not related to the scale parameter, logarithm transformations may be made for {X i , i = , . . . , n} as follows. Let It can be shown that the mean of Y  is μ  = EY  = (ν  ) and the mean of Y [nτ  ]+ is μ  = EY [nτ  ]+ = (ν  ), where (ν) is the derivation of ln (ν); that is, (ν) can be expressed, as in [, p.], by ln X i , http://www.journalofinequalitiesandapplications.com/content/2013/1/161 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 τ  as For convenience, throughout this paper, c, c  , . . . represent a constant which is independent of n and may take different values in different expressions.
The paper is arranged as follows. In Section , the change-point estimator τ is proposed based on the CUSUM method by an appropriate logarithm transformation for {X i , i = , . . . , n}, and its constancy and convergence rate are investigated. The proofs of theorems are given in Section .

Main results
Theorem  Assume that X  , X  , . . . , X n is a negative associated random variable sequence satisfying the conditions that X  , . . . , where l(n) is a slowly varying function with lim n→∞ l(n) = +∞.
Theorem  Assume that the conditions of Theorem  hold, then τ is a strong consistent estimator of τ  , and Next, we will study the O P convergence rate of τ under the local alternative hypothesis; that is, ρ  is not a constant independent of n, but it depends on n and is denoted by ρ n . Noticing that if ρ n is large, the change-point estimation is usually quite precise. In practice it may be more important to construct confidence intervals for τ  when ρ n is small. We hence assume that ρ n -→  as n -→ ∞. 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 μ i = (ν i ), i = , . Then, by the mean theorem, μ n (under the local alternative hypothesis, denoting μ  as μ n ) can be expressed as whereν lies between ν  and ν  . Hence, with some added conditions, μ n is equal to ρ n in practice. http://www.journalofinequalitiesandapplications.com/content/2013/1/161 Theorem  Assume that X  , X  , . . . , X n is a negative associated random variable sequence, and X  , . . . , X [nτ  ] are identically distributed by (x; ν  , λ), and X [nτ  ]+ , . . . , X n are identically distributed by (x; ν  , λ). If μ n satisfies (  ) Remark  Theorems  and  give the weak and strong consistency and convergence rates for the change-point estimator τ of the shape parameter in a gamma distribution. In Theorem , the O P convergence rate of the change-point estimator τ 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 τ . Having this O P value, we can study the limiting distribution of τ . This will be the subject of a future paper.

Proof of the theorem
To prove the above theorems, we first consider the following lemmas. Lemma  Let {Z j , j ∈ N} be a negative associated sequence with zero mean satisfying β p = sup j∈N E|Z j | p < ∞ for some p ≥ . Denoting S a,k = k- j= Z a+j , then there exist constants C p , K p ≥  related to p, for all a, n ∈ N , such that  Proof of Theorem  Noticing that ln(λx) is an increasing positive function and X  , . . . , X n are negative associated sequences, we have from Lemma  that {Y i , i = , . . . , n} are negative associated sequences. Without loss of generality, assuming that ν  > ν  , by the increasing character of (ν) in (, ∞), we know that μ  = μ μ  > . By simple computation, it can be shown that

Lemma  Let
and Hence, From the triangle inequality, it can easily be shown that namely, Noticing that |U k  | ≤ |U k |, hence we have

then by () and (), it follows that
(   ) http://www.journalofinequalitiesandapplications.com/content/2013/1/161 Hence, Since Y  , Y  , . . . , Y n are the negative associated variables, by the Markov inequality and Lemma , ∀r > , we have Hence, if we choose g  (n) = n   l - (n), where l(n) is a slowly varying function satisfying lim n→∞ l(n) = +∞, then combining ()-() we have, as n → ∞, that is, τ is the weak consistent estimator of τ  , and Proof of Theorem  From () to () we have, for ∀ε > , By the Borel-Cantelli lemma, we obtain that τ is the strongly consistent estimator of τ , and a.s. for some  < δ <   .
Proof of Theorem  To this end, we choose a value  < θ <   such that τ ∈ (θ , θ ). By () and ()-() (g  (n) = ), it is easily found that τ is a consistent estimator of τ  . Therefore, for every ε > , P( τ / ∈ (θ , θ )) < ε. Thus, we now have only to examine the behavior of U k when M -→ ∞. For every M > , define Then we have Furthermore, we obtain (   ) http://www.journalofinequalitiesandapplications.com/content/2013/1/161 It can be seen from the definition of D n,M that Because Y *  , . . . , Y * n are negative associated variables, by the Markov inequality, Lemma  and (), ∀p ≥ , we obtain