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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.

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 producer's and consumer's 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.

AIIE Transactions, 6, pp. 135-142. When a MIL-STD-IOSD 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. Eq.uations are given to calculate the long term cost of sampling inspection using these expected values and appropriate cost parameters.

Management Science, 24, 2, pp. 210-216. Applications in operations research often employ models which contain linear functions. These linear functions may have some components (coefficients and variables) which are random. (For instance, linear functions in mathematical programming often represent models of processes which exhibit randomness in resource availability, consumption rates, and activity levels.) Even when the linearity assumptions of these models is unquestioned, the effects of the randomness in the functions is of concern. Methods to accomodate, or at least estimate for a linear function the implications of randomness in its components typically make several simplifying assumptions. Unfortunately, when components are known to be random in a general, multivariate dependent fashion, concise specification of the randomness exhibited by the linear function is, at best, extremely complicated, usually requiring severe, unrealistic restrictions on the density functions of the random components. Frequent stipula- tions include assertion of normality, or of independence-yet, observed data, accepted collat- eral theory and common sense may dictate that a symmetric distribution with infinite domain limits is inappropriate, or that a dependent structure is definitely present. (For example, random resource levels may be highly correlated due to economic conditions, and non- negative for physical reasons.) Often, an investigation is performed by discretizing the random components at point quantile levels, or by replacing the random components by their means-methods which give a deterministic "equivalent" model with constant terms, but possibly very misleading results. Outright simulation can be used, but requires considerable time investment for setup and debugging (especially for generation of dependent sequences of pseudorandom variates) and gives results with high parametric specificity and computation cost. This paper shows how to use elementary methods to estimate the mean and variance of a linear function with arbitrary multivariate randomness in its components. Expressions are given for the mean and variance and are used to make Tchebycheff-type probability state- ments which can accomodate and exploit stochastic dependence. Simple estimation examples are given which lead to illustrative applications with (dependent-) stochastic programming models.

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.

This article introduces a replacement life-test procedure for the exponential failure rate: Failure-free operation of a unit for at least tk consecutive time units is designated a “success”; the acceptance test is passed if and only if the first success is encountered before k unit failures have been recorded. Test plans are presented and the test is compared with the usual time-truncated test and to the truncated sequential probability ratio test. It is shown that this new test has smaller expected test time than the time truncated test when the true failure rate is small relative to the null hypothesized failure rate. Consistency and unbiasedness are proved and methods for making inferences on failure rate are described.

In 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 ϑ i 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 ϑ i . Also standard tests of significance are now applicable.

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

Military Standard 105D has been almost universally adopted by government and private consumers for the lot-by-lot sampling inspection of product which may be inspected on a dichotomoun basis The plan specifies, for each lot size, a random sample size and set of acceptance numbers (maximum allowable number of defectives in each sample). The acceptance numbers are based upon the binomial distribution and depend upon the quality required by the purchaser. Where several consecutive lots are submitted, a shift to less severe (“reduced”) inspection or more severe (“tightened”) inspection is specified when the ongoing quality is very high or low. Further experience permits a return to normal sampling from either of these states This paper examines the long range costs of such a sampling scheme. The three inspection types are considered as three distinct Markov chains, with periodic transitions from chain to chain. The expected sample size and the expected proportion of rejected product are determined as a function of the two parameters under control of the manufacturer, lot size and product quality. Some numerical examples are given which illustrate how to compute the overall cost of sampling inspection. Suggestions are made concerning the choice of parameters to minimize this cost.

The occurrence of early failures in a fixed-sample acceptance test, where the sample observations are obtained sequentially, presents an interesting decision problem. It may be desirable to abandon the test at an early stage if the conditional probability of passing is small and the testing cost is high. This paper presents a stopping rule based on the maximum-likelihood estimate of total costs involved in the decision to continue beyond an early failure. A Bernoulli model, an exponential model, and a Weibull model are examined.

The maximum likelihood estimator and minimum variance unbiased estimator of the probability of zero failures in m binomial trials were compared on the basis of expected squared estimation error. The choice of estimator proved to be a function of the region of parameter space. However, the m. l. e. was preferred throughout the regions likely to be encountered in practice (where the probability of zero failures exceeded 0.5).

We Have found the well-known game of bingo to be an excellent device for illustrating to students various laws of probability and a source for hypergeo-metric probability problems to supplement the familiar playing card examples. In particular, solution of the question “What is the expected length of a game of bingo as a function of the number of players?” provides students an opportunity to develop the expectation of a random variable theoretically and then to verify the result by computer simulation of the game.

Minimum variance unbiased estimators of reliability are developed for a series, parallel, and standby redundant system where the components are from a single exponentially distributed population. It is assumed that sampling information has been censored with respect to number of failures or test time.

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.

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).

A method is suggested for obtaining a multiple linear regression equation which permits the variance, as well as the mean, of normally distributed random variables, Y to be a function of known constants X1 … Xp. The method is applicable to large samples, and yields an approximation of maximum likelihood estimates of regression coefficients, which may be used to construct confidence intervals for the parameters of the normal distribution and tolerance intervals for individual Y. A likelihood ratio test may be used to test this model against the usual homoscedastic least squares model. Two exmples of this type of analysis on data from the literature are presented.