Exact likelihood inference for exponential distributions under generalized progressive hybrid censoring schemes
Introduction
In the last decade, various models of hybrid censoring have been proposed (for a recent review, see [5]). Based on truncated life testing as proposed first in [18], Type-I hybrid and Type-II hybrid censored order statistics have been extensively discussed. In particular, likelihood inference for the scale and location parameter of two-parameter exponential distributions with distribution and density function as in have received great attention and the exact distribution of the MLEs has been addressed in many articles. Starting with Chen and Bhattacharyya [9], the method of conditional moment generating function has been successively applied to derive the density function of the MLE of in the scale model (for a simplified version, see also [12]). Childs et al. [12] also proposed Type-II hybrid censoring and established respective results for the distribution of the MLE. Since both hybrid censoring procedures face some drawbacks, Chandrasekar et al. [8] proposed extensions called generalized Type-I and Type-II hybrid censoring schemes which alleviate some negative effects (see also [5, Section 4]). The model has been further discussed w.r.t. competing risks [24], [21] and step-stress models [27]. The Fisher information in generalized hybrid censoring schemes has been addressed in [25]. Further extensions called unified hybrid censoring scheme and flexible hybrid censoring scheme have been proposed in [6] and in [26], respectively. Initiated by Childs et al. [11] and Kundu and Joarder [23], some of the above models have also been discussed in the model of progressive censoring (see also [2, Chapters 5 and 14]). Similar results for the exact (conditional) distribution of the MLEs have been obtained in these papers as well as in [10], [19]. Although these models exhibit more complicated decision rules for generating the data, the method of generating function worked and yielded explicit expressions for the density and distribution functions of the MLEs in the exponential case. However, the resulting expressions were complicated due to alternating sums and truncated gamma functions, and, thus, difficult to implement on a computer.
In this paper, we extend the generalized Type-I and Type-II hybrid censoring schemes to progressively censored data. It has to be noticed that generalized Type-I progressive hybrid censoring has already been discussed recently in [13] whereas the Type-II version is new and has not been addressed so far. In contrast to the moment generating function method applied in [13], we apply the spacings’ based approach due to Cramer and Balakrishnan [14] for Type-I progressive hybrid censored data to derive rather compact expressions for the density functions of the MLE of the scale parameter. It should be noted that the same method has recently be successfully applied to Type-II progressively hybrid censored order statistics by Cramer et al. [15]. In comparison to the representations obtained by the moment generating function approach, the resulting expressions in terms of B-splines can be easily and efficiently implemented on a computer. Moreover, they give some insight into the structure of the distributions. Furthermore, it is worth noting that the structure of the formulas remains the same when the procedure is based on Type-II censored data only. Hence, there is no significant simplification in this (simpler) setting.
The present paper is structured as follows. In Section 2, we introduce generalized Type-I and Type-II progressive hybrid censoring and present the MLEs for the location and scale parameters for two-parameter exponential distributions. Using these results, we establish in Sections 3 Exact distribution of the MLE for Type-I GPHCS, 4 Exact distribution of the MLE for Type-II GPHCS expressions for the density functions of the MLEs. We discuss both the case of a known and unknown location-parameter . In the latter case, we derive the joint density function of the bivariate MLE . The methods proceed by volume computations of intersections of simplices with half-spaces as already used in [14], [15], respectively.
Section snippets
Models and likelihood inference
In order to present the new censoring schemes extending the models presented in [8] to progressively Type-II censored data, we first present some basic results. For an exponentially distributed i.i.d. sample , a fixed value , with , and for a prefixed censoring scheme , we consider a progressively Type-II censored lifetest leading to the data In such a lifetest, successively failure times are observed but at the th failure time units are
Exact distribution of the MLE for Type-I GPHCS
In this section, we consider the Type-I GPHCS-model presented in Section 2.1 and establish formulas for the density of the MLE for known and unknown, respectively. In the latter case, we determine a bivariate density function for . First, we derive distributional results for the generalized progressively Type-I hybrid censored samples conditioned on . In order to obtain exact distributional results for the MLEs, we consider both known and unknown location parameter .
From the
Exact distribution of the MLE for Type-II GPHCS
In this section, we address the model of generalized Type-II progressive hybrid censoring as described in Section 2.2. Similarly to the case of Type-I GPHCS, we make use of already known results. However, the situation is more involved and we have to perform additional calculations which take into account the two random counters and simultaneously. Considering this more complex case, we find that the desired characterization in terms of B-spline functions is still possible but more
References (27)
- et al.
On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints
Statist. Probab. Lett.
(2014) - et al.
Bounds for means and variances of progressive type II censored order statistics
Statist. Probab. Lett.
(2001) - et al.
Hybrid censoring: Models, inferential results and applications (with discussions)
Comput. Statist. Data Anal.
(2013) - et al.
Exact likelihood inference for an exponential parameter under generalized progressive hybrid censoring scheme
Stat. Methodol.
(2015) - et al.
On some exact distributional results based on Type-I progressively hybrid censored data from exponential distributions
Stat. Methodol.
(2013) - et al.
Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring
J. Statist. Plann. Inference
(2012) - et al.
Analysis of Type-II progressively hybrid censored data
Comput. Statist. Data Anal.
(2006) - et al.
On simple calculation of the Fisher information in hybrid censoring schemes
Statist. Probab. Lett.
(2009) - et al.
Progressive Censoring: Theory, Methods, and Applications
(2000) - et al.
The Art of Progressive Censoring. Applications to Reliability and Quality
(2014)
Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution
J. Stat. Comput. Simul.
Statistical Inference
Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring
Naval Res. Logist.
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