| Abstrakt: |
We formulate a general multi-stage sampling strategy for minimum risk point estimationof a function g(β)of an unknown scale parameter β(>0)in a Gamma(α,β) population where α(>0)is assumed known. We work under a squared error lossfunction plus the cost of sampling and begin a key theoretical structure assuming conditions (i) A1 and A2 on g(.) plus (ii) A3–A5 on choices of multi-stage estimation strategies. The main result (Theorem 3.1) shows that such a MRPE strategy satisfies the asymptotic risk efficiency property (3.9). Next, we exhibit purely sequential, accelerated sequential and three-stage estimation strategies enjoying both the first-order(f.o.) efficiency and asymptotic risk-efficiency properties under generality followed by a number of examples of practical importance including (i) estimation of a survival probability (Illustration 1) in an Expo(β) distribution, (ii) estimation of a density (Illustration 2) in an Expo(β) distribution at x=a,and (iii) estimation (Illustration 3) and comparisons (Illustration 4) of survival probabilities in a Gamma(α,β) distribution. The breadth of the general theory is emphasized by addressing related problems of practical importance including estimation of (iv) the availability parameter (Illustration 5), and (v) a survival probability (Illustration 6) in a mixture of Gamma distributions. In each situation, we validate the assumptions A1–A5. We supplement with summaries from moderate sample size performances of the proposed estimation strategies using simulations (Tables 1, 2, 3, 4, 5and 6) followed by data analyses of (a) remission times from bladder cancer patients (Illustrations 1 and 4: Table 7) plus (b) survival times from breast cancer patients (Illustration 6: Table 8). Overall, the moderate sample performances look highly encouraging. |
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