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|θ) = λ exp(-λz) Πj=1p( θj λ)m for z > 0 and m = (m1,...,mp) with mj ∈ {0,1} and m1p = 1, and where 1k denotes a k × 1 vector of 1's. The parameter θ = (θ1,...,θp) has θj > 0 and λ = θ1p. 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 θi, 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 θ with respect to the quadratic loss function L(a, θ) = {norm of matrix}a - θ{norm of matrix}2/{norm of matrix}θ{norm of matrix}2, where {norm of matrix}v{norm of matrix}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 = Σi=1nZi and N = Σi=1nMi, is inadmissible whenever c < (n-2) (n + p -1) or 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. © 1991.