Parameter estimation for the Moore-Bilikam distribution under progressive type-II censoring, with application to failure times

ABSTRACT

The Moore-Bilikam distribution is convenient for survival analysis. The estimation of its parameters and its reliability function is performed by maximum likelihood, expectation-maximization, stochastic expectation-maximization, and the Bayesian method. The data are progressively censored of type II (samples are removed randomly from the experiment). Simulation shows that the expectation-maximization estimator of the parameter and the Bayesian-shrinkage estimator of the reliability function are the most efficient (with the minimum mean square error) when they are based on the Weibull and the Pareto distributions, which are specific cases of the Moore-Bilikam distribution. Bayesian and maximum likelihood estimations using the Moore-Bilikam distribution under type-II progressive censoring allow for fitting empirical failure times of an insulating fluid between two electrodes and the resistance of single carbon fibers. The associated reliability functions are estimated by each method.

KEYWORDS:

• Bayesian estimator
• expectation-maximization algorithm
• stochastic expectation-maximization algorithm

JEL CLASSIFICATION:

• C02
• C11
• C15

Disclosure statement

The authors reported no potential conflict of interest.

Notes

¹ Upper record: For a sequence $\{X_i,i\ge 1\}$ of random variables, $X_j$ is an upper record if its value is greater than all previous observations, namely if $X_j>max\{X_1,\dots ,X_{j-1}\},j\ge 2$.

$k$-upper record: The $m^{\mathrm{t}\mathrm{h}}$ upper $k$-record time is $T_m^k$, then $T_1^k=k$ and for $m\ge 2$

(12) $T_m^k=min\{j:j>T_{m-1}^k,X_j>X_{T_{m-1}^k-k+1:T_{m-1}^k}\},$(12)

where $X_{i:m}$ is the $i^{\mathrm{t}\mathrm{h}}$ order statistic from a sample of size $m$. The sequence $\{Y_{m-1}^k\}$, where $Y_{m-1}^k=X_{T_{m-1}^k:T_{m-1}^k+k-1}$, is called “sequence of the $k^{\mathrm{t}\mathrm{h}}$ upper record values.”

² $n$ independent groups with $k$ items within each group are put in a test, $r_1$ groups, and the group in which the first failure is observed are randomly removed from the test as soon as the first failure (say $X_{1:m:n:k}^r$) has occurred, $r_2$ groups and the group in which the second failure is observed are removed randomly from the test as soon as the second failure (say $X_{2:m:n:k}^r$ has occurred. Finally $r_m(m\le n)$ groups and the group in which the $m^{\mathrm{t}\mathrm{h}}$ failure is observed are removed randomly from the test as soon as the $m^{\mathrm{t}\mathrm{h}}$ failure (say $X_{m:m:n:k}^r$) has occurred. Then $X_{1:m:n:k}^r<X_{2:m:n:k}^r<\dots <X_{m:m:n:k}^r$ are said to be progressively first-failure censored (Soliman et al., 2013).