Open Access

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

Journal of Inequalities and Applications20132013:161

https://doi.org/10.1186/1029-242X-2013-161

Received: 25 October 2012

Accepted: 18 March 2013

Published: 8 April 2013

Abstract

In this paper, the change-point estimator for the shape parameter is proposed in a negative associated gamma random variable sequence. Suppose that X 1 , , X n are negative associated random variables satisfying that X 1 , , X [ n τ 0 ] are identically distributed with Γ ( x ; ν 1 , λ ) , and that X [ n τ 0 ] + 1 , , X n are identically distributed with Γ ( x ; ν 2 , λ ) ; the change point τ 0 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 O P convergence rate of the change-point estimator is presented under the local alternative hypothesis condition.

MSC:62F12, 62G10.

Keywords

change point Γ-distribution negative associated shape parameter convergence rate

1 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 gamma-distributed random variable X with a shape parameter ν and a scale parameter λ is given by
f ( x ; ν , λ ) = λ ν Γ ( ν ) x ν 1 e λ x I ( x > 0 ) ,
(1)

where I ( ) is the indicator function, Γ ( ) is a Γ function with Γ ( p ) = 0 e x x p 1 d x .

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 = 1 , the gamma distribution is an exponential distribution with parameter λ; when λ = 1 2 , 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 [1] proposed a statistic for testing a change in the one-parameter exponential family; Hsu [2] considered a change point for the scale parameter of gamma random variables, assuming that the shape parameter was constant; Diaz [3] posed the Bayesian test regarding the scale parameter change point for the independent gamma variables; Gupta and Ramanayake [4], Ramanayake and Gupta [5] discussed a linear trend change for the exponential distribution; Ramanayake [6] 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. [7]), 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 [8]; Bai and Perron [9]; Braun et al. [10]; Chen et al. [11]). 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 1 , X 2 , , X n be a negative associated sequence that satisfies the conditions that X 1 , , X [ n τ 0 ] have the common distribution Γ ( x ; ν 1 , λ ) , and that X [ n τ 0 ] + 1 , , X n have the common distribution Γ ( x ; ν 2 , λ ) , where τ 0 is an unknown parameter called the change point; ν 1 , ν 2 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 [ n τ 0 ] in the sequence. Noticing that λ X Γ ( x ; ν , 1 ) and its distribution is not related to the scale parameter, logarithm transformations may be made for { X i , i = 1 , , n } as follows. Let
Y i = ln λ X i , i = 1 , 2 , , n .
(2)
It can be shown that the mean of Y 1 is μ 1 = E Y 1 = Ψ ( ν 1 ) and the mean of Y [ n τ 0 ] + 1 is μ 2 = E Y [ n τ 0 ] + 1 = Ψ ( ν 2 ) , where Ψ ( ν ) is the derivation of ln Γ ( ν ) ; that is,
Ψ ( ν ) = d [ ln ( Γ ( ν ) ) ] d ν = Γ ( ν ) Γ ( ν ) .
Ψ ( ν )
can be expressed, as in [[12], p.16], by
Ψ ( ν ) = γ + 0 + e t e ν t 1 e t d t ,

where γ is the Euler-Mascheroni constant, that is, γ = 0 + e x ln x d x . Since Ψ ( ν ) = 0 + t 1 e t e ν t d t > 0 , hence Ψ ( ν ) is an increasing function in ( 0 , + ) .

Define
U k = i = 1 k Y i k n i = 1 n Y i , ρ 0 = ν 1 ν 2 , μ 0 = μ 1 μ 2 .
(3)
Since
U k = i = 1 k ln λ X i k n i = 1 n ln λ X i = i = 1 k ln X i k n i = 1 n ln X i ,
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 0 as
τ ˆ = 1 n min { k : | U k | = max 1 j n | U j | } .
(4)

For convenience, throughout this paper, c , c 1 , 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 τ ˆ is proposed based on the CUSUM method by an appropriate logarithm transformation for { X i , i = 1 , , n } , and its constancy and convergence rate are investigated. The proofs of theorems are given in Section 3.

2 Main results

Theorem 1 Assume that X 1 , X 2 , , X n is a negative associated random variable sequence satisfying the conditions that X 1 , , X [ n τ 0 ] are identically distributed with Γ ( x ; ν 1 , λ ) , and X [ n τ 0 ] + 1 , , X n are identically distributed with Γ ( x ; ν 2 , λ ) . Let
k 0 = [ n τ 0 ] , k ˆ = [ n τ ˆ ] , k = [ n τ ] for some 0 < τ < 1 ,
(5)
where [ A ] denotes the integer part of a number A. If the ρ 0 = ν 1 ν 2 is a non-zero constant, then τ ˆ is a consistent estimator of τ 0 and
| τ ˆ τ 0 | = o P ( n 1 2 l ( n ) ) ,
(6)

where l ( n ) is a slowly varying function with lim n l ( n ) = + .

Theorem 2 Assume that the conditions of Theorem  1 hold, then τ ˆ is a strong consistent estimator of τ 0 , and
| τ ˆ τ 0 | = o ( n 1 2 + δ ) , a.s. for some 0 < δ < 1 2 .
(7)

Next, we will study the O P convergence rate of τ ˆ under the local alternative hypothesis; that is, ρ 0 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 0 when ρ n is small. We hence assume that ρ n 0 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 = Ψ ( ν i ) , i = 1 , 2 . Then, by the mean theorem, μ n (under the local alternative hypothesis, denoting μ 0 as μ n ) can be expressed as
μ n = μ 1 μ 2 = Ψ ( ν ˜ ) ( ν 1 ν 2 ) = Ψ ( ν ˜ ) ρ n ,

where ν ˜ lies between ν 1 and ν 2 . Hence, with some added conditions, μ n is equal to ρ n in practice.

Theorem 3 Assume that X 1 , X 2 , , X n is a negative associated random variable sequence, and X 1 , , X [ n τ 0 ] are identically distributed by Γ ( x ; ν 1 , λ ) , and X [ n τ 0 ] + 1 , , X n are identically distributed by Γ ( x ; ν 2 , λ ) . If μ n satisfies
μ n 0 , n μ n ,
(8)
then
| τ ˆ τ 0 | = O P ( 1 n μ n 2 ) .
(9)

Remark 1 Theorems 1 and 2 give the weak and strong consistency and convergence rates for the change-point estimator τ ˆ of the shape parameter in a gamma distribution. In Theorem 3, the O P convergence rate of the change-point estimator τ ˆ 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 ˆ . Having this O P value, we can study the limiting distribution of τ ˆ . 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 A 1 , A 2 , , A m be disjoint subsets of { 1 , 2 , , n } , and let a i = ( A i ) be the number of elements in A i , i = 1 , 2 , , m . Assume that Z 1 , Z 2 , , Z n are negative associated variables, then if
f i : R a i R , i = 1 , 2 , , m
(10)

are the increasing (or decreasing) positive functions for every element, then f 1 ( Z j , j A 1 ) , , f m ( Z j , j A m ) are the negative associated variables.

Proof See Joag-Dev and Proschan [13]. □

Lemma 2 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 2 . Denoting S a , k = j = 0 k 1 Z a + j , then there exist constants C p , K p 1 related to p, for all a , n N , such that
E | S 1 , n | p C p n p 2 1 j = 1 n E | Z j | p ; E ( max 1 k n | S a , k | ) p K p β p n p 2 .
(11)

Proof See Su, Zhao and Wang [14]. □

Lemma 3 Let { Z n , n 1 } be a negative associated sequence, if { b k , k 1 } is an increasing number serial, then ε > 0 and m n , we have
(12)
(13)

Proof See Hu and Zhang [15]. □

Proof of Theorem 1 Noticing that ln ( λ x ) is an increasing positive function and X 1 , , X n are negative associated sequences, we have from Lemma 1 that { Y i , i = 1 , , n } are negative associated sequences. Without loss of generality, assuming that ν 1 > ν 2 , by the increasing character of Ψ ( ν ) in ( 0 , ) , we know that μ 0 = μ 1 μ 2 > 0 . By simple computation, it can be shown that
E U k 0 = k 0 ( n k 0 ) n ( μ 1 μ 2 ) = n τ 0 ( 1 τ 0 ) μ 0 ,
(14)
and
E U k = { k ( n k 0 ) n μ 0 , k k 0 , ( n k ) k 0 n μ 0 , k > k 0 , = { n τ ( 1 τ 0 ) μ 0 , k k 0 , n ( 1 τ ) τ 0 μ 0 , k > k 0 .
(15)
Hence,
| E U k 0 | | E U k | = { n ( 1 τ 0 ) ( τ 0 τ ) μ 0 , k k 0 , n τ 0 ( τ τ 0 ) μ 0 , k > k 0 , n τ μ 0 | τ τ 0 | ,
(16)

where τ = min { τ 0 , 1 τ 0 } .

From the triangle inequality, it can easily be shown that
| U k | | U k 0 | 2 max 1 k n | U k E U k | + | E U k | | E U k 0 | ,
namely,
| E U k 0 | | E U k | 2 max 1 k n | U k E μ k | + | U k 0 | | U k | .
Noticing that | U k 0 | | U k ˆ | , hence we have
| E U k 0 | | E U k ˆ | 2 max 1 k n | U k E U k | .
(17)
Let Y i = Y i E Y i , then by (16) and (17), it follows that
n τ μ 0 | τ τ 0 | 2 max 1 k n | U k E U k | = 2 max 1 k n | i = 1 k Y i k n i = 1 n Y i | = 2 max 1 k n | n k n i = 1 k Y i k n i = k + 1 n Y i | 2 max 1 k n | i = 1 k Y i | + 2 max 1 k n | i = k + 1 n Y i | .
(18)
Hence,
P ( g 1 ( n ) | τ ˆ τ 0 | > ε ) = P ( | τ ˆ τ 0 | > ε g 1 ( n ) ) P ( 2 n τ μ 0 { max 1 k n | i = 1 k Y i | + max 1 k n | i = k + 1 n Y i | } > ε g 1 ( n ) ) P ( max 1 k n | i = 1 k Y i | > n τ μ 0 4 g 1 ( n ) ε ) + P ( max 1 k n | i = k + 1 n Y i | > n τ μ 0 4 g 1 ( n ) ε ) = ˆ A 1 + A 2 .
(19)
Since Y 1 , Y 2 , , Y n are the negative associated variables, by the Markov inequality and Lemma 2, r > 2 , we have
A 1 E ( max 1 k n | i = 1 k Y i | ) r / ( n τ μ 0 ε 4 g 1 ( n ) ) r = 4 r ( ε τ μ 0 ) r g 1 r ( n ) n r E ( max 1 k n | i = 1 k Y i | ) r 4 r ( ε τ μ 0 ) r g 1 r ( n ) n r K r β r n r 2 4 r K r β r ( ε τ μ 0 ) r ( n 1 2 g 1 ( n ) ) r ,
(20)
where β r = max { E | Y 1 | r , E | Y [ n τ 0 ] + 1 | r } is a constant independent of n. Similar arguments give that
A 2 4 r K r β r ( ε τ μ 0 ) r ( n 1 2 g 1 ( n ) ) r .
(21)
Hence, if we choose g 1 ( n ) = n 1 2 l 1 ( n ) , where l ( n ) is a slowly varying function satisfying lim n l ( n ) = + , then combining (19)-(21) we have, as n ,
P ( g 1 ( n ) | τ ˆ τ 0 | > ε ) 0 ,
that is, τ ˆ is the weak consistent estimator of τ 0 , and
| τ ˆ τ 0 | = o P ( n 1 2 l ( n ) ) .

 □

Proof of Theorem 2 From (19) to (21) we have, for ε > 0 ,
n = 1 P ( g 2 ( n ) | τ ˆ τ 0 | > ε ) n = 1 4 r K r β r ( ε τ μ 0 ) r ( n 1 2 g 2 ( n ) ) r
if we choose g 2 ( n ) = n 1 2 δ for some 0 < δ < 1 2 . Let r > 1 δ , then we have, as n ,
n = 1 P ( g 2 ( n ) | τ ˆ τ 0 | > ε ) < .
(22)
By the Borel-Cantelli lemma, we obtain that τ ˆ is the strongly consistent estimator of τ, and
| τ ˆ τ 0 | = o ( n 1 2 + δ ) a.s. for some  0 < δ < 1 2 .

 □

Proof of Theorem 3 To this end, we choose a value 0 < θ < 1 2 such that τ ( θ , 1 θ ) . By (8) and (19)-(21) ( g 1 ( n ) = 1 ), it is easily found that τ ˆ is a consistent estimator of τ 0 . Therefore, for every ε > 0 , P ( τ ˆ ( θ , 1 θ ) ) < ε . Thus, we now have only to examine the behavior of U k over those k for which n θ k n ( 1 θ ) . To prove | τ ˆ τ 0 | = O P ( 1 n μ n 2 ) , we shall prove that
P ( | τ ˆ τ 0 | > M n μ 0 2 ) 0 ,
(23)
when M . For every M > 0 , define
D n , M = { k : n θ k n ( 1 θ ) , | k k 0 | > M μ n 2 } .
Then we have
P ( | τ ˆ τ 0 | > M n μ 0 2 ) P ( τ ˆ ( θ , 1 θ ) ) + P ( | τ ˆ τ 0 | > M n μ n 2 , τ ˆ ( θ , 1 θ ) ) ε + P ( sup k D n , M | U k | | U k 0 | ) .
(24)
Since
(25)
Noticing that U k + U k 0 0 implies U k E U k + U k 0 E U k 0 E U k E U k 0 E U k 0 , which in turn implies that
U k E U k 1 2 E U k 0 or U k 0 E U k 0 1 2 E U k 0 .
(26)
Since E U k 0 > 0 , we have
| U k E U k | 1 2 E U k 0 or | U k 0 E U k 0 | 1 2 E U k 0 .
Furthermore, we obtain
B 2 P ( sup k D n , M | U k E U k | 1 2 E U k 0 ) + P ( | U k 0 E U k 0 | 1 2 E U k 0 ) = ˆ D 1 + D 2 .
(27)
It can be seen from the definition of D n , M that
D 1 P ( sup n θ k n ( 1 θ ) | U k E U k | 1 2 E U k 0 ) = P ( sup n θ k n ( 1 θ ) | i = 1 k Y k n i = 1 n Y | 1 2 n τ 0 ( 1 τ 0 ) μ n ) P ( max 1 k n | i = 1 k Y i k n i = 1 n Y i | 1 2 n τ 0 ( 1 τ 0 ) μ n ) P ( max 1 k n | i = 1 k Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) + P ( max 1 k n | k n i = 1 n Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) P ( max 1 k n | i = 1 k Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) + P ( | i = 1 n Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) = ˆ E 1 + E 2 .
(28)
Because Y 1 , , Y n are negative associated variables, by the Markov inequality, Lemma 2 and (8), p 2 , we obtain
E 1 = P ( max 1 k n | i = 1 k Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) E ( max k | i = 1 k Y i | p ) / ( 1 4 n τ 0 ( 1 τ 0 ) μ n ) p 4 p [ n τ 0 ( 1 τ 0 ) μ n ] p K p β p n p 2 c 1 1 ( n 1 2 μ n ) p 0 ( as  n μ n 2 ) .
(29)
Similar arguments give
E 2 = P ( | i = 1 n Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) 4 p [ n τ 0 ( 1 τ 0 ) μ n ] p C p n p 2 1 i = 1 n E | Y | p c 2 1 ( n 1 2 μ n ) p 0 ( as  n μ n 2 ) .
(30)
Combining (28)-(30), we see that
D 1 0 as  n μ n 2 .
(31)
Similar arguments as those for D 1 give
D 2 = P ( | i = 1 k 0 Y i k 0 n i = 1 n Y i | 1 2 n τ 0 ( 1 τ 0 ) μ n ) P ( | i = 1 k 0 Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) + P ( | i = 1 n Y i | 1 4 n τ 0 ( 1 τ 0 ) μ n ) c 3 1 ( n 1 2 μ n ) p 0 as  n μ n 2 .
(32)
Combining (27), (31) and (32), we obtain
B 2 0 .
(33)
Now consider B 1 . Because of symmetry, we only consider the case of k k 0 . The event U k U k 0 0 implies that
Therefore,
B 1 P ( max k D n , M | 1 k 0 k i = k + 1 k 0 Y i + 1 n i = 1 n Y i | τ μ n ) P ( max k D n , M | 1 k 0 k i = k + 1 k 0 Y i | 1 2 τ μ n ) + P ( max k D n , M | i = 1 n Y i | 1 2 n τ μ n ) = ˆ F 1 + F 2 .
(34)
From the Markov inequality, Lemma 2 and (8), we obtain
F 2 2 p C P ( n τ μ n ) p n p 2 c 4 ( 1 n μ n 2 ) p 2 0 .
(35)
Denoting σ 1 2 = Var ( Y 1 ) , from Lemma 3 and (8), we obtain
F 1 = P ( max n θ k k 0 M μ n 2 | 1 k 0 k i = k + 1 k 0 Y i | 1 2 τ μ n ) = P ( max M μ n 2 k 0 k k 0 n θ | 1 k 0 k j = 1 k 0 k Y k 0 + 1 j | 1 2 τ μ n ) = P ( max M μ n 2 t k 0 n θ | 1 t j = 1 t Y k 0 + 1 j | 1 2 τ μ n ) 4 4 ( τ μ n ) 2 { j = 1 M μ n 2 Var ( Y k 0 + 1 j ) ( M μ n 2 ) 2 + 8 j = M μ n 2 + 1 k 0 n θ Var ( Y k 0 + 1 j ) j 2 } 16 σ 1 2 ( τ ) 2 μ n 2 { μ n 2 M + 8 j = M μ n 2 + 1 k 0 n θ 1 j ( j 1 ) } = 16 σ 1 2 ( τ ) 2 μ n 2 { μ n 2 M + 8 ( μ n 2 M 1 k 0 n θ 1 ) } = 16 σ 1 2 ( τ ) 2 ( 9 M 8 ( k 0 n θ 1 ) μ n 2 ) 0 ( as  n μ n 2 , M ) .
(36)
Combining (34)-(36), we have
B 1 0 .
(37)

From (24), (33) and (37), we know that (23) holds; that is, Theorem 3 is proved. □

Declarations

Acknowledgements

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).

Authors’ Affiliations

(1)
School of Mathematics, Hefei University of Technology
(2)
Department of Statistics and Finance, University of Science and Technology of China
(3)
Department of Mathematics and Computer Science, Tongling University

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© Tan et al.; licensee Springer. 2013

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