Saul Blumenthal

Estimating Population Size with Exponential Failure

Abstract We are given J observations obtained by truncated sampling of a population of N items which fail independently according to the exponential distribution, where both N and the scale parameter of the exponential are unknown. Estimates of N are developed, and compared. These are conditional and unconditional maximum likelihood estimates, and a class of Bayes modal estimates. On the basis of second-order asymptotic properties, one of the Bayes estimates is singled out as most desirable. This estimator is also good for estimating mean life, but for estimating failure rate, the maximum likelihood estimates are preferable.

ESTIMATION OF THE LARGER OF TWO NORMAL MEANS

Let Xi1, Xi2, …, Xin, i = 1, 2, be a pair of random samples from populations which are normally distributed with means θi, and common known variance τ2. The problem is to estimate the function φ(θ1, θ2) = maximum (θ1, θ2). In this paper we consider five different estimators (or sets of estimators) for φ(θ1, θ2) and evaluate their biases and mean square errors. The estimators are (i) φ(X1, X2), where Xi is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of φ(θ1, θ2) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X1 + X2)/2 when |X1 – X2| is small, and estimate by φ(X1, X2) when |X1 − X2| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, ...

Multinomial Sampling With Partially Categorized Data

When sampling categorically, some observations are known to belong in a given set of categories, but it is not known in which member or members of this set they belong. The problem is estimation of the probabilities of the individual categories. Maximum likelihood estimates are developed, and the bias and variance of these estimates are studied. An example of the application of the results is given.

Estimating the Complete Sample Size from an Incomplete Poisson Sample

Abstract Maximum likelihood estimators and a modified maximum likelihood estimator are developed for estimating the zero class from a truncated Poisson sample when the available sample size itself is a random variable. All the estimators considered here are asymptotically equivalent in the usual sense; hence their asymptotic properties are investigated in some detail theoretically as well as by making use of Monte Carlo experiments. One modified estimator appears to be best with respect to the chosen criteria. An example is given to illustrate the results obtained.

A Sequential Screening Procedure

Given n items with common known distribution of time to failure, where n is unknown Failures are observed as they occur and a stopping rule is given so that if there were J failures up to the time of stopping, there is confidence (1 – c) (given) that the remaining number of items (n – J) is no greater than k (given), regardless of n. Properties of the rule are studied and tables for its use are given.

Estimating population size with truncated sampling

Let X1, X2,…,Xn be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ∊ (X−R) and not otherwise. The number J of observable X’s is binomial, N, Q, Q = 1−P(X ∊ R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the [Ncirc]’s of the form [Ncirc] = N + α√N + β + op(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ2 (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β’s differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estima...

Robustness of Stein's Two-Stage Procedure for Mixtures of Normal Populations

Abstract Explicit expressions for the level of significance and power of Steins two-stage hypothesis testing procedure are derived when the normal population is mixed with another normal population differing in the mean. Approximate expansions for these expressions have been obtained, and numerical studies of the size have been carried out using the expansions. Some numerical indication of the accuracy of the approximations has also been obtained. Steins two-stage procedure is quite robust against mixtures of normal populations differing in location parameters. Similar conclusions can be drawn regarding the coverage probabilities of confidence intervals generated by this procedure.

Estimating the Number of Items on Life Test

If n items are on life test where n is unknown, and failures are observed either until time T has elapsed or until r failures have occurred, then an estimate of n can be obtained. Both maximum likelihood and Bayes estimates are obtained and both known and unknown failure distributions are considered.

Sequential estimation of the largest normal mean when the variance is unknown

Given a observations from each of k populations whose distributions differ by a location parametar,the value of the largest parameter is to be estimated using the largest value of the k sample means. It is desired to design, a sampting rule which guarantees that the Mean Squarad Error (M.S.E.) of the estimate does not exceed a given bound when the distributions have a common but unknown scale parameter. A sequential sampling scheme is devised based on an estimate of the scale parameter and a “least favorable” configuration of the location parameters. The sample size characteristics of the sampling plan are studied under mild restrictions on the distributions involved. The M.S.E. of the resulting estimator is studied under the additional assumption of normality. A brief discussion is given of an alternate sequential plan which uses informationin the sample regarding the configuracion of the Locaeion parameters.