Improved estimation of the mean in one-parameter exponential families with known coefficient of variation. (English) Zbl 1057.62020
Summary: The value for which the mean square error of a biased estimator \(aT\) for the mean \(\mu\) is less than the variance of an unbiased estimator \(T\) is derived by minimizing \(MSE(aT)\). The resulting optimal value is \(1/[1+c(n)v^2]\), where \(v=\sigma/\mu\) is the coefficient of variation. When \(T\) is the UMVUE \(\overline X\), then \(c(n) =1/n\), and the optimal value becomes \(1/(n+v^2)\) [D. T. Searls, J. Am. Stat. Assoc. 59, 1225–1226 (1964; Zbl 0124.09904)].
Whenever prior information about the size of \(v\) is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determine \(v\).
Whenever prior information about the size of \(v\) is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determine \(v\).
Citations:
Zbl 0124.09904References:
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