On a maximizing problem with several applications in statistical theory

Abstract

For a certain distribution, the three problems of determining

1. optimum spacings of observations in linear estimation of location and scale parameters

2. Optimum strata limits in proportionate sampling

3. optimum interval boundaries in grouping a sample are closely related to each other.

In section two of this paper, two basic lemmas are stated. Applied to problem (i) above, these lemmas enable us to make conclusions about the rate at which the efficiency of linear estimates increases with the number of observations used to form the estimate. Similarly, in problem (ii), statements can be made concerning the gain in accuracy due to increasing the number of strata.

Furthermore, the laborious procedure of calculating the optimum solutions of the above mentioned problems is replaced by a simplified approximation technique which will cause negligible loss of accuracy.