Exact Bounded Risk Estimation When the Terminal Sample Size and Estimator Are Dependent: The Exponential Case

摘要

Suppose that we have independent observations X_1, X_2,... having a common exponential distribution with a probability density function f(x; λ) = λ~(-1) exp(-x/λ) if x > 0, and 0 elsewhere with unknown mean λ( > 0). Sequentially estimating the mean parameter λ is an important problem, and it has attracted attentions from many authors. The area was broadly reviewed by Mukhopadhyay (1988, 1995b). Having recorded observations X_1,...,X_n, suppose that the loss function in estimating λ by the sample mean, X_n = n~(-1) ∑ from i=1 to n of X_i< is given by L_n = A(X_n - λ)~2 where A( > 0) is known. Given a preassigned number w( > 0), our goal is to make the associated risk AE_λ[(X_n - λ)~2] ≤ w for all fixed λ > 0. We refer to w as the preassigned risk-bound. The minimum required fixed sample size n would be the smallest integer ≥ n~* = Aλ~2/w, but its magnitude remains unknown. Hence, we propose a genuine two-stage sampling design, N = max{m, < BX_m~2/w > + 1}, where the pilot sample size is m( ≥ 3). Here, B ≡ B_m( > 0) is an appropriate design constant and < u > stands for the largest integer < u. We finally estimate λ by X_N, the sample mean based on the combined data from both stages. It is clear that E_λ[(X_N — λ)~2] cannot be expressed exclusively involving moments of N alone because I(N = n) and X_n are dependent random variables, for all n ≥ m. The main result (Theorem 2.1) consists of explicitly determining B so that the risk associated with (N, X_N), namely AE_λ[(X_N - λ)~2], has the preassigned risk-bound w. The performances of the proposed methodology are first investigated with the help of simulations. Illustrations are included with data from a multicenter clinical trial involving patients with acute myeloctic leukemia (AML) prepared for transplantation with a radiation-free conditioning regimen.