Improved estimation for a model arising in reliability and competing risks

Let (Z1,M1),..., (Zn,Mn) be independent and identically distributed 1 - (p + 1) random vectors from the exponential-multinomial distribution which has density function f(z,m[theta]) = [lambda] exp(-[lambda]z) [Pi]j=1p([theta]j/[lambda])mj for z > 0 and m = (m1,...,mp) with mj [set membership, variant] {0,1} and m1p = 1, and where 1k denotes a k - 1 vector of 1's. The parameter [theta] = ([theta]1,...,[theta]p) has [theta]j > 0 and [lambda] = [theta]1p. This density function arises by observing a series system or a competing risks model with p sources of failure with the lifetime of the ith component or source of failure being exponential with mean 1/[theta]i, and where the random variable Z denotes system lifetime, while the ith component of M is a binary random variable denoting whether the ith component failed. It can also arise from the Marshall-Olkin multivariate exponential distribution. The problem of estimating [theta] with respect to the quadratic loss function L(a, [theta]) = ||a - [theta]||2/||[theta]||2, where ||v||2 = vv' for any 1 - k vector v, is considered. Equivariant estimators are characterized and it is shown that any estimator of form cN/T, where T = [Sigma]i=1nZi and N = [Sigma]i=1nMi, is inadmissible whenever c (n - 2)/n. Since the maximum likelihood and uniformly minimum variance unbiased estimators correspond to cN/T with c = 1 and c = (n - 1)/n, respectively, then they are inadmissible. An adaptive estimator, which possesses a self-consistent property, is developed and a second-order approximation to its risk function derived. It is shown that this adaptive estimator is preferable to the estimators cN/T with c = (n - 2)/(n + p - 1) and c = (n - 2)/n. The applicability of the results to the Marshall-Olkin distribution is also indicated.