An Empirical Distribution Function for Sampling with Incomplete Information

• Published in: The Annals of mathematical statistics 1955-12, Vol.26 (4), p.641-647
• Main Authors: Ayer, Miriam, Brunk, H. D., Ewing, G. M., Reid, W. T., Silverman, Edward
• Format: Article
• Language: English
• Subjects: Distribution functions; Estimators; Information economics; Integers; Mathematical monotonicity; Maximum likelihood estimation; Maximum likelihood estimators; Random variables; Ratios; Trials
• ISSN: 0003-4851; 2168-8990
• DOI: 10.1214/aoms/1177728423

For i = 1, 2, ⋯, n, let Niindependent trials be made of an event with probability pi, and suppose that the probabilities piare known to satisfy the inequalities p1≥ p2≥ ⋯ ≥ pn. Let aidenote the number of successes in the i-th trial, and p* ithe ratio ai/Ni(i = 1, 2, ⋯, n). Then the maximum likelihood estimates p̄1, ⋯, p̄nof the numbers p1, ⋯, pnmay be found in the following way. If p* 1≥ p* 2≥ ⋯ ≥ p* n≥ 0, then p̄i= p* i, i = 1, 2, ⋯, n. If p* k≥ p* k+1for some k(k = 1, 2, ⋯, n - 1), then$\bar{p}_k = \bar p_{k+1}$; the ratios p* k= ak/Nkand p* k+1= ak+1/Nk+1are then replaced in the sequence p* 1, p* 2, ⋯, p* nby the single ratio (ak+ ak+1) / (Nk+ Nk+1), obtaining an ordered set of only n - 1 ratios. This procedure is repeated until an ordered set of ratios is obtained which are monotone non-increasing. Then for each$i, \bar p_i$is equal to that one of the final set of ratios to which the original ratio ai/Nicontributed. It is seen that this method of calculating the$\bar p_i, \cdots, \bar p_n$depends on a grouping of observations which might very well appeal to an investigator on purely intuitive grounds. It seems of interest to note that it yields the maximum likelihood estimates of the desired probabilities. Particular examples of this situation are found in bio-assay [3] and in the proximity fuze problem discussed by M. Friedman ([1], Chapter 11). The last section is devoted to a consistency property of the maximum likelihood estimators.