Results on double sample estimation for the binomial distribution. (English) Zbl 0545.62050

Cohen, Arthur; Sackrowitz, Harold B.

Ann. Stat. 12, 1109-1116 (1984).

The authors consider a sequence of i.i.d. Bernoulli variables with parameter p, 0\(\leq p\leq 1\). A complete class result is obtained for double sample estimation and sequential estimation. The complete class result can be applied to prove that if the loss is a linear combination of squared error and cost of sampling then any estimation procedure (with its sampling rule) where the sample proportion is the terminal decision is inadmissible except for the case where the total sample is a single observation.
If the terminal decision loss function is squared error divided by p(1-p) and the overall loss is a linear combination, then any double sample procedure in which the estimate is the sample proportion is inadmissible.
These last two results demonstrate the inadmissibility of the Miller- Freund procedure for various linear combination loss functions. Finally, it is shown that the sample proportion as a true double sample estimate is not unbiased.

Reviewer: V.Mammitzsch

MSC:

62L12	Sequential estimation
62C07	Complete class results in statistical decision theory
62C05	General considerations in statistical decision theory
62C15	Admissibility in statistical decision theory

Keywords:

vector loss function; unbiasedness; binomial distribution; sequence of i.i.d. Bernoulli variables; double sample estimation; inadmissibility of the Miller-Freund procedure; linear combination loss functions