Herbert C. Rutemiller

The Efficiencies of Maximum Likelihood and Minimum Variance Unbiased Estimators of Fraction Defective in the Normal Case

This paper compares two point estimators of fraction defective of a normal distribution when both population parameters are unknown; the minimum variance unbiased estimator, (x), and the maximum likelihood estimator, (x). Using minimum mean squared error as a criterion, it is shown that the choice of estimator depends upon the true value of F(x), and the sample size. In the domain .0005 ≤ F(x) ≤ .50, the maximum likelihood estimator is generally superior even for small sample sizes, except for F(x) less than about 0.01, or greater than 0.25. Furthermore, the bias in the m.l.e. is slight over much of the domain where this estimator has smaller mean squared error. As a practical solution to the estimation problem. it is suggested that the m.v.u.e. be calculated, and if this estimate is between 0.01 and 0.25, it should be replaced with the m.l.e. This combined estimator is shown to be nearly as efficient as the better of the m.v.u.e. and m.l.e. throughout the domain of F(x).

Failure-Free Period Life Tests

A failure-free period life test (FFPLT) is defined by a pair of real numbers (t, T), 0 < t ≤ T, such that the test is passed if, and only if, a failure-free period of t consecutive time units is achieved in the T time units allowed. Failed items are replaced, and replacement time is not counted. Such tests have been introduced in Mil. Std. 781D (US. Department of Defense 1980). This test is studied in terms of a renewal process under the assumption of exponential interarrival times with constant failure rate λ. Test plans are developed for commonly used producers and consumers risks and discrimination ratios. The tests are compared to the well-known time-truncated and sequential-replacement life tests. Also presented are methods for point and interval estimation of λ based on test results. Unbiasedness and consistency are proved and other asymptotic properties are given.

Some Characteristics of a Ranking Procedure for Population Parameters Based on Chi-Square Statistics

A recurring statistical problem is the cbmplete ordering of a set of k distributions (or populations) with respect to some parameter. A related problem is the selection of the largest or smallest parameter of k distributions. In this paper we examine the characteristics of an ad hoc procedure for populations which admit a x 2 statistic, as a function of an estimator for the population parameter, and use these statistics to order the populations. Lower bounds for the probability of correct, ranking, of selecting the smallest, and of selecting the largest parameter are provided as a function of degrees of freedom and the number of populations to be ordered, k, for k = 3(1)8

Evaluation of Pr {x⩾y} When Both X and Y are from Three-Parameter Weibull Distributions

It is important in many reliability applications to determine the probability that the failure time of an element from one population will exceed that of an element from a second population. In this paper, we present a method for computer calculations of Pr {x ⩾ y} where X and Y are each from a three-parameter Weibull distribution. In addition, we provide the moments and the probability density function of the difference. Numerical examples are included.

Tables for Determining Expected Cost per Unit Under MIL-STD-105D Single Sampling Schemes

Abstract : When a MIL-STD-105D sampling scheme is used for a long period, some lots will be subjected to normal, some to reduced, and some to tightened inspection. This paper provides for several single sampling plans and various quality levels, the expected fraction of lots rejected, the expected sample size per lot, and the expected number of lots to be processed before sampling inspection must be discontinued. Equations are given to calculate the long term cost of sampling inspection using these expected values and appropriate cost parameters.

Point Estimation of the Parameter of a Truncated Exponential Distribution

This paper addresses point estimation of the failure rate parameter for a 1-parameter exponential distribution, based on random samples taken from a severely right-truncated distribution (truncation point known). The maximum likelihood estimation (MLE) leads to frequent estimates of zero failure rate, even for large samples. We suggest two alternative estimators which overcome this problem, a Bayes estimator and a composite estimator. These estimators are compared, through Monte Carlo trials, with the MLE, both in terms of s-bias and root mean square (rms) error. Based on these comparisons, the composite estimator is recommended. A numerical example illustrates the use of these estimators.

Simulation studies of the power of birnbaum's test

Z. W. Birnbaum has proposed a hypothesis test procedure which, under fairly general conditions, does not require explicit knowledge of the critical values of the test statistic. In this paper we investigate the power of the test in a variety of situations. In particular we have considered situations in which the underlying observations have normal, chi-square related, and Weibull distributions. We show that the asymptotic power of this test is identical to the classical test using the same statistic and that the Birnbaum test achieves its asymptotic power very rapidly. The Weibull and normal cases are considered both for complete and censored samples.

A Computer Program for the ASN of Curtailed Attributes Sampling Plans

A FORTRAN program is presented that computes the average sample number (ASN) for curtailed multiple sampling plans for attributes. The program can be used for both binomial (large lots) and hypergeometric (small lots) distributions. It also provides poi..

Unbiased Estimation of

Unbiased estimation of , where is the unknown parameter in a binomial distribution, is discussed. A “telescoping” procedure is presented which compares favorably, for large values of , to the minimum variance unbiased estimate (MVUE). It is also shown that a seemingly natural procedure results in an estimator dominated by the new procedure. In addition some biased estimators are compared to the unbiased estimator of the telescoping procedure.

Estimation of the probability of labor force participation of the AFDC population-at-risk

SummaryIn summary, the functional form makes quite a difference. An investigator should be quite wary of making generalizations based on any single specification or estimation technique. However, the above results have shown in striking fashion the superiority of MLE of the sigmoid specifications over the OLS estimation of the linear probability specification. Although the logistic or urban specification require iterative solution, this is no barrier on a modern digital computer, with appropriate special algorithms. A further advantage of the MLE is the asymptotic normality of the estimates of ϑi which permits large sample interval estimation, and the iteration method of scoring employed yields directly an estimate of the standard deviation of each normally distributed ϑi. Also standard tests of significance are now applicable.Perhaps most importantly, the sigmoid specifications are consistent with a probability interpretation since the estimates lie inside the unit interval, and the sigmoid shape is consistent with the assumed unimodal distribution of the participation decision.In conclusion, results reported in previous investigations of the probability of labor force participation or labor force participation rate which have relied on the least squares estimation of a linear probability specification are likely to be unreliable as to he magnitude of the response attributed to changes in explanatory variables.