Let ( Z 1, M 1),…, ( Z n , M n ) 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(−λ z) Π j=1 p( θ j λ ) m j for z > 0 and m = ( m 1,…, m p) with m j ∈ {0,1} and m1 p = 1, and where 1 k denotes a k × 1 vector of 1's. The parameter θ = ( θ 1,…, θ p ) has θ j > 0 and λ = θ 1 p . 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 θ 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 θ with respect to the quadratic loss function L( a, θ) = ‖ a − θ‖ 2/‖θ‖ 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 = Σ i=1 n Z i and N = Σ i=1 n M i , 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 c N/ 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 c N/ 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.