Statistical Inference on the Basis of Sequential Order Statistics under a Linear Trend for Conditional Proportional Hazard Rates

This paper deals with systems consisting of independent and heterogeneous exponential components. Since failures of components may change lifetimes of surviving components because of load sharing, a linear trend for conditionally proportional hazard rates is considered. Estimates of parameters, both point and interval estimates, are derived on the basis of observed component failures for s(≥ 2) systems. Fisher information matrix of the available data is also obtained which can be used for studying asymptotic behaviour of estimates. The generalized likelihood ratio test is implemented for testing homogeneity of s systems. Illustrative examples are also given.


Introduction
Let X 1 , · · · , X n be independent and identically distributed (i.i.d.) random variables with a common distribution function (DF), say F , and abbreviated by X 1 , · · · , X n i.i.d.
∼ F . Denote in magnitude order of X 1 , · · · , X n by X 1:n ≤ · · · ≤ X n:n , which are known as order statistics (OSs). The theory of OSs has been widely studied in literature specially in system reliability analyses. For example, lifetimes of known r-out-of-n systems coincide to X r:n where X 1 , · · · , X n stand for component lifetimes; For more information, see Barlow and Proschan [3], David and Nagaraja [11] and references therein. In order to introduce more flexible models for analysing practical systems, various generalizations of OSs such as fractional order statistics and generalized order statistics have been proposed. The former is useful for providing more flexible tools and the later is a setting to unify similar results (David and Nagaraja [11], p. 21). In this paper, we deal with another unified concept, called sequential order statistics (SOS). There is also another motivation in reliability analyses for implementing SOS. Specifically, when component lifetimes are i.i.d., the OSs are suitable for describing the r-out-of-n system lifetime. Thus failing a component does not effect the DFs of lifetimes of surviving components. As motivated by Cramer and Kamps [7], in practice the failure of a component may result in a higher load on the surviving components and hence causes lifetime distributions change. This property may be due to load sharing and/or common working environments and hence dependent component lifetimes. More precisely, suppose that F j and f j , for j = 1, · · · n, denote the common DF and probability density function (PDF) of the component lifetimes when n − j + 1 components are jointly working. Then, the components begin to work independently at time t = 0 with the common DF F 1 . When at 463 time x 1 , the first component failure occurs, the remaining n − 1 components are working with the (left truncated) common DF F 2 at x 1 . This process continues up to n − r + 1 components with the common DF F r work until the r-th failure occurs at time x r and hence the whole system fails. This system is called sequential r-out-ofn system (or dynamic system) and the system lifetime is then r-th observed component failure time, denoted by X (r) . In the literature, (X (1) , · · · , X (r) ) is called SOSs. Let x = (x 1 , . . . , x r ) be the observed values from SOSs (X (1) , · · · , X (r) ) with DFs (F 1 , . . . , F r ) and PDFs (f 1 , . . . , f r ) of component lifetimes. The joint PDF of  (1972) focuses directly on the hazard rate function. The simple member of the family is the proportional hazard rate model. Different kinds of proportional hazard models may be obtained by making different assumptions about the baseline survival function, or equivalently, the baseline hazard rate function. Let F 0 (.) be a absolutely continuous DF with a corresponding PDF f 0 (.). The hazard rate function is defined by h 0 (t) = f 0 (t)/F 0 (t) for t > 0, whereF 0 (t) = 1 − F 0 (t) is the survival function of the DF F 0 (.). If X is a member of proportional hazard family with the baseline DF F 0 (.), then the survival function of X becomes F (t; θ) =F θ 0 (t), t ∈ S, where θ is the proportional parameter and F 0 (.) is the baseline DF and S is the support of the baseline DF. In this case, the hazard rate function of X is given by h(t; θ) = f (t; θ)/F (t; θ) = θh 0 (t) for t > 0.
In this paper, we consider the problem of estimating the parameters on the basis of s (≥ 2) independent SOSs samples under a proposed linear trend conditional proportional hazard rates (LTCPHR) model, defined bȳ F j (t) =F aj 0 (t) for j = 1, · · · , r, where aj = a × j, a > 0 and F 0 (t) is the underlying DF. Remember that the hazard rate function of the DF F defined by h(t) = f (t)/F (t) for t > 0, where f (t) = ∂F (t)/∂t is the probability density function (PDF) of the DF F (t). Therefore, h j (t) = ajh 0 (t), for j = 1, · · · , r, is the proportional hazard rate function of the DF F j , where h 0 (t) is hazard rate function of the baseline DF F 0 for all t.
The LTCPHR model is a new defined statistical concept for modelling engineering systems in which components share the system load. In fact, impact of failing a component on the surviving components are modeled via hazard rate components. Notice that F i is the common component distribution function when n − i + 1 components are jointly working. The connection between F i is done by assuming a proportional hazard rate among them. Hence, it is called conditionally proportional hazard rate models. In this paper, we consider the problem of the estimation parameters of the LTCPHR model with independent multiple SOS samples coming from heterogeneous exponential populations. Thus, this paper is organized as follows: In Section 2, the maximum likelihood estimates (MLEs) of parameters are derived and the generalized likelihood ratio test (GLRT) is used for testing homogeneity of the parent exponential populations. In Section 3, we analyse a simulated data set. Finally, some concluding remarks are given in Section 4.

Statistical inference for the LTCPHR model parameters
In this section, we obtain MLEs of LTCPHR model parameters. Also, GLRT is derived for testing homogeneity of populations. To do these, two scenarios, namely, (i) the parameter a is known; and (ii) the parameter a is unknown, are considered.

Maximum likelihood estimation
Suppose that we observed s (≥ 2) independent heterogeneous SOS samples. The available data may be represented as where the i-th row of the matrix x in (2) denotes the SOS sample coming from the i-th system. The likelihood function (LF) of the available data given by (2) is then derived from (4) as j denote the survival function, DF and PDF of jth component lifetime of the i−th dynamic system, respectively. For more details, see Cramer and Kamps [8,9] and Hashempour and Doostparast [13]. Suppose that the baseline DF of the i-th dynamic system (i = 1, · · · , s) follows the exponential distribution with the mean σ i , i.e.
It should be mentioned that the baseline DFs for the considered LTCPHR model for component lifetimes are heterogeneous with different scale parameters. Therefore under the earlier mentioned LTCPHR model in Section 1, we haveF j , · · · , s and j = 1, · · · , r. Then, Equation (3) yields the LF of the available data as where a > 0, and m j = (n − j + 1)ja − (n − j)(j + 1)a with convention (n − r)(r + 1)a ≡ 0. For sake of brevity, we assumed that the proportional parameter a are the same among the s sequential r-out-of-n systems. Following Cramer and Kamps [8] and Hashempour and Doostparast [13], two cases are considered in sequel: (i) a is known, and (ii) a is unknown.

Case I: The parameter a is known
Suppose that the parameter a in Equation (5) is known. If σ 1 = · · · = σ s , the ML estimate of the common mean of the s baseline exponential populations, say σ 0 , is derived by maximizing (5) with respect to σ 0 aŝ If baseline exponential populations are heterogeneous, the (unique) ML estimate of σ i (i = 1, · · · , s) is derived from Equation (5) Stat., Optim. Inf. Comput. Vol. Under the LTCPHR with the one-parameter exponential baseline DF, we have where Γ(m, n) calls for the gamma distribution with the shape and the scale parameters m and n, respectively. From Equation (8) and for i = 1, · · · , s,σ i ∼ Γ(r, σ i /r), and then E(σ i ) = σ i and V ar(σ i ) = σ 2 i /r. Notice that the ML estimateσ 0 in Equation (6) is the arithmetic mean of the ML estimatesσ i for the mean populations given by Equation (7)

Case II: The parameter a is unknown
Suppose that the parameter a in Equation (5) is unknown. In this case, calculations are complicated. The logarithm of LF in Equation (5) can be written as The ML estimates of the parameters which are shown byσ i andâ for i = 1, . . . , s, (if exist) are obtained (numerically) by solving the following likelihood equations: and From Equations (10) and (11), we havê andσ Equations (12) and (13) (14) is not necessary negative definite on the parameter space. Therefore, one needs to use numerically methods for maximizing the LF (9) with respect to a and σ 1 , · · · , σ s by using Equations (12) and (13).

Generalized likelihood ratio test
In this section, we consider the problem of homogeneity testing on the basis of independent SOS samples from different exponential populations, i.e.,

Lemma 1 (Balakrishnan and Nevzorov [2])
Let Z 1 , · · · , Z k be independent random variables and Z i ∼ Γ(a i , 1), for i = 1, · · · , k. Then (U 1 , · · · , U k ) ∼ Here, D(a 1 , · · · , a k ) stands for the Dirichlet distribution with PDF . The Jacobian transformation is then The joint PDF of (V 1 , · · · , V s ) under the homogeneity hypothesis H 0 in (15) is derived from Equations (8) and (19) and Lemma 1 as for In practice, one may use numerical methods such as Monte Carlo simulation to derive the threshold c in the rejection region (18). For more details, see Hashempour [14].

Remark 1
It is easy to verify that the distribution family (5) is invariant with respect to the group of the scale transformations Also, the problem of hypotheses testing (15) remains invariant under G in (22) sinceḠ(Ω) = Ω and From Equation (8), one can see that 2r (σ i /σ i ) ∼ χ 2 2r , where χ 2 ν stands for the chi-square distribution with ν degrees of freedom. So, an equi-tailed confidence interval at level 100γ% for σ i (i = 1, · · · , s) is where χ 2 ν,p calls for the p-th quantile of the χ 2 ν -distribution.

STATISTICAL INFERENCE ON THE BASIS OF SEQUENTIAL ORDER STATISTICS
Note that the observed Fisher Information (FI), denoted by I jk (σ 1 , · · · ,σ s ), on the basis of the available SOSs data (2) is equal to minus of the Hessian matrix (HM) at the point of MLEs, i.e.

Case II: The parameter a is unknown
It is easy to verify that the unique ML estimates of the parameters under the null hypothesis H 0 arê wherem 0,j = (n − j + 1)jâ 0 − (n − j)(j + 1)â 0 , with conventionâ 0 (r + 1) ≡ 0. Therefore, the GLRT statistic for the hypotheses testing problem (15) is wherem j = (n − j + 1)α j − (n − j)α j+1 . The logarithm of the GLRT statistic Λ 2 in Equation (27) reads The null hypothesis where −2 log Λ 2 follows asymptotically chi-square distribution. Exact distribution of the statistic log Λ 2 in Equation (28) under the null hypothesis H 0 is complicated and we could not obtained an explicit expression. This remains as an open problem. In practice, one may use numerical methods such as the Monte Carlo simulation to derive the threshold c in the rejection region (29). Bedbur [5] obtained the uniformly most powerful unbiased tests under the conditionally proportional hazard rates model based on multiple homogeneous SOS samples from a common exponential distribution.

An illustrative example
In order to assess the performance of the derived estimates in the preceding sections a simulation study was conducted. In the case of unknown parameter a, N = 10 4 multiple SOS samples were generated for some selected values of parameters. For this purpose, we assume that a = 0.9, 1, 1.1, n = 10, 20, s = 4, r = 4, 6, 15   Table 1 are summarized as follow; • The MSEs are decreasing in n and r.
• The Biases are negligible and negative.
• The MSE of the obtained numerical estimator of parameter a is increasing in a.

Conclusions and further remarks
In this paper, based on independent SOSs coming from heterogeneous exponential populations under a linear trend model, the MLEs parameters were obtained on the basis of multiple SOS samples. The GLR tests were derived for testing homogeneity of the exponential populations. Some open problems were also mentioned. The results of this paper may be extended in some directions. For example, derivation of the uniformly most powerful scale-invariant test (if exist) is worth for further consideration.