By M. Ghosh, G. Meeden (auth.)
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Additional resources for Bayesian Methods for Finite Population Sampling
Since the inequality holds for all y in the parameter space and is strict for at least one, the proof is complete. 0 Note that this result is quite general; there are no restrictions on the function to be estimated, the loss function and the design. L(y) denote the population mean. Here we consider the problem of estimating the population mean with squared error loss. Let 8sm denote the estimator which is just the sampie mean, Le. 5) The admissibility of the sampie mean will be demonstrated for two different parameter spaces.
K let cy(i) = number of yj's which equal bi , and for a sampie s cy(i, s) = number of yj's in y(s) which equal bi , and finally for a data point Z = (s, zs) cz(i, s) = number of zj's in Zs which equal bio In the first stage we take as the prior 7fl the distribution which puts mass l/k on the k points (b1, ... ,bdT, ... ,(bk, ... ,bkf and zero mass elsewhere. Under this prior the only points in our sampie space, Z(Y(b),p), which receive positive marginal probability are those where all the observed values are identical.
1. Z)} . 6) tEs Now if 7f is such that for a given Z and for each j f/. z) = Z = 8sm (z) then 8tr (z) = Z = 8sm (z). So if we can produce a finite sequence of priors such that at each step, and for every Z with positive probability at the given step, the conditional expectation of each unobserved unit given the observed data is just the sampie mean the admissibility of 8sm will be proved. The way the argument will proceed is that first we will take care of all data points where all the observed units take on the same value.
Bayesian Methods for Finite Population Sampling by M. Ghosh, G. Meeden (auth.)