Life Testing Situations Research Articles

Sample Size for Selecting the Better of Two Weibull Populations

Suppose an experimenter wishes to select the better of two Weibull populations. In a life testing situation he may have some time t* for which the device must perform properly and he would consider the better population to be the one with the larger reliability at time t*. It is reasonable for the experimenter to test n items from each of the two populations until there are r(r < n) failures from each and then select as better the population for which the maximum likelihood estimate of the reliability at time t* is the larger. This paper presents the sample sizes needed to achieve a desired level for the probability of correct selection with such a decision rule.

Best Linear Unbiased Prediction of Order Statistics in Location and Scale Families

Abstract In many life testing, reliability and replacement policy situations, it is desirable to predict the time of future failures from times of the early failures in the same sample. Interval prediction in this setting has been treated by several authors. In this article, the authors give best linear unbiased (blu) point predictors of future failures when the underlying distribution is known up to location and scale. Analogous large sample procedures based on sample quantile theory are provided. The results are illustrated for the exponential and Pareto distributions.

A Preliminary Test Procedure in Life Testing

Abstract This article provides a new estimator of the exponential mean life which makes use of a preliminary test of hypothesis on the scale parameter for certain life test situations where a guess of the parameter value is available from past studies. Expressions for the bias and mean square error of the proposed estimator are derived. Generalization of this result to a class of weighted estimators is also considered.

A Preliminary Test Procedure in Life Testing

Abstract This article provides a new estimator of the exponential mean life which makes use of a preliminary test of hypothesis on the scale parameter for certain life test situations where a guess of the parameter value is available from past studies. Expressions for the bias and mean square error of the proposed estimator are derived. Generalization of this result to a class of weighted estimators is also considered.

The Bayesian Predictive Distribution in Life Testing Models

Prediction intervals for future observations in life testing situations have been derived by Hewitt (1968), Nelson (1968), Lawless (1971, 1972). Expected-cover tolerance regions have been obtained. Here we give a Bayesian approach to such situations and use the concept of the Bayesian predictive distribution. Both the exponential and the two-parameter exponential distributions are considered.

Life Tests Under Dependent Competing Causes of Failure

In a life testing situation the failure of an individual, either a living organism or an inanimate object, may be classified into one of k(< 1) mutually exclusive classes, usually causes of failure. One often has dependent causes of failure in actual physical situations, i.e., the theoretical lifetime of an individual failing from one cause may be correlated with the theoretical lifetime of the same individual failing from a different cause. This paper i) discusses some properties of a bivariate Weibull distribution and ii) is concerned with estimating, by the method of maximum likelihood, the unknown parameters of life distributions belonging to two particular parametric families, viz., bivariate normal and bivariate Weibull, when the causes of failure are dependent. An example involving the failure of small electrical appliances is analyzed and compared with an analysis which assumes the causes of failure to be independent.

Distributions of some censored rank statistics under Lehmann alternatives for the two-sample case

Several recent papers have considered the distributions of commonly used rank statistics for the two-sample problem under the nonparametric Lehmann alternative that one underlying distribution is a power of the other. Shorack (1 966) presented recursive expressions for generating the distribution of the Mann-Whitney statistic. Steck (1969), dealing with Smirnov two-sample tests, gave a determinant form for the distribution of the one-sided Smirnov statistic under the Lehmann alternative. Davies (1971) compared some small and large sample properties of a Lehmann test with a test proposed by Savage (1956) for the two-sample problem. Here we present results for the distributions of some rank statistics under Lehmann alternatives when sampling is censored on the occurrence of a given order statistic in one of the samples. Shorack (1968) gave recursive formulae for the distributions of other nonparametric test statistics when censoring is based on the occurrence of a given order statistic in the combined samples. Censoring of these types occur, for example, in life testing situations where testing time may be costly. We first consider the distribution of a linear function of the ranks in a given sample and establish some difference relations. These are then used to generate the power of the median and precedence tests and to establish recursive relations for the distribution of a modified Wilcoxon statistic. Finally, the distribution of the censored one-sided Smirnov statistic is derived and used to obtain some new results for the case when the two populations have identical distributions.

Bayesian Concepts for the Estimation of Reliability in the Weibull Life-Testing Model

Since in many life-testing situations it is not unlikely to note the unpredictable fluctuation of the scale parameter in a failure model, it is justifiable to consider such a parameter as a random variable and, thus, appeal to a Bayesian analysis. In specific, let 0 denote the random variable associated with the scale parameter and 0 its realization. Obviously, a Bayesian analysis depends on the utilization of prior information which, in this case, we assume to exist either in the form of a prior distribution of 0 or a sequence of sufficient statistics from past experiments. For the ordinary Bayes approach, we appeal to a well-known transformation to generalize the results of Bhattacharya (1967) for the one-parameter exponential model so as to include the flexibility provided by the shape parameter ? in the Weibull distribution. In fact, the Weibull failure model has an increasing ( > 1) or decreasing ( < 1) failure rate and, thus, is likely to describe the life-span of items with variable failure rates. The empirical Bayes estimation technique was largely motivated by Robbins (1955), who assumed the existence of a prior distribution for an unknown parameter but not the knowledge of its form. Instead, he substitutes past information which he assumes to exist as a result of the repetitive nature in the problem of estimation. Thus, in the absence of knowledge concerning the form of the prior distribution, we appeal to an empirical Bayes approach to estimate the scale parameter. By using this estimate, an estimate of the reliability function is made possible.

Analysis of Accelerated Life Test Data-Part III: Product Comparisons and Checks on the Validity of the Model and Data

This is Part III of a three-part series presenting statistical methods for planning and analyzing temperature-accelerated life tests with the Arrhenius model when all test units are run to failure. These methods are presented so they can be profitably used by individuals with a limited statistical background. In Part I, the Arrhenius model is described and graphical methods for analysis of such data are given. In Part II, numerical methods for analysis of such data are given and optimum and standard test plans are presented and compared. In Part III, graphical and numerical methods for comparing different products are given and methods for assessing the validity of the data and the assumptions of the Arrhenius model also are given. These methods are illustrated throughout with accelerated life test data on insulation. While the methods are presented for the Arrhenius model, they can be used for planning and analyzing many other accelerated life test situations.

Analysis of Accelerated Life Test Data - Part II: Numerical Methods and Test Planning

This is Part II of a three-part series presenting statistical methods for planning and analyzing temperature-accelerated life tests with the Arrhenius model, when all test units are run to failure. These methods are presented so they can be profitably used by individuals with a limited statistical background. In Part I, the Arrhenius model is described and graphical methods for analysis of such data are given. In Part II, numerical methods for analysis of such data are given and optimum and standard test plans are presented and compared. In Part III, graphical and numerical methods for comparing different products as well as methods for assessing the validity of the data and the assumptions of the Arrhenius model are also given. These methods are illustrated throughout with accelerated life-test data on insulation. While the methods are presented for the Arrhenius model, they can be used for planning and analyzing many other accelerated life-test situations.

Graphical Analysis of Accelerated Life Test Data with the Inverse Power Law Model

In this paper, graphical methods are presented for analyzing accelerated life test data with the inverse power law model, when all test units are run to failure. The inverse power law model is described, and graphical methods for estimating its parameters from such complete data are given. These methods are illustrated with accelerated test data on time to breakdown of an insulating fluid. While the methods are presented with the inverse power law model, they can be used for analyzing many other accelerated life test situations. These methods are presented so that they can be used by individuals with a limited statistical background.

Analysis of Accelerated Life Test Data - Part I: The Arrhenius Model and Graphical Methods

This is Part I of a three-part series presenting statistical methods for planning and analyzing temperature-accelerated life tests when all test units are run to failure. These methods are presented so they can be profitably used by individuals with a limited statistical background. In Part I, the Arrhenius model is described, and graphical methods for analysis of such complete data are given. In Part II, numerical methods for analysis of such data are given, and optimum and standard test plans are presented and compared. In Part III, graphical and numerical methods for comparing different products are given, and methods for assessing the validity of the data and the assumptions of the Arrhenius model are also given. These methods are illustrated throughout with accelerated life test data on insulation. While the methods are presented here with the Arrhenius model, they can be used for planning and analyzing many other accelerated life test situations.

Life Tests under Competing Causes of Failure and the Theory of Competing Risks

Suppose that in a life-testing situation the failure of an individual can be classified into one of k( >1) mutually exclusive classes, usually causes of failure. It is assumed that associated with each cause of failure there is a characteristic life distribution belonging to a specific class of distributions. A general likelihood function is obtained which allows for dependence of the causes, for both censored and uncensored data. The method of maximum likelihood is used to estimate the parameters when the underlying life distributions are independent Weibulls with equal shape constants (the exponential being a special case) and independent Weibulls with unequal shape constants. A study involving 121 patients treated for cancer of the breast by surgery and/or X-ray therapy is analyzed, and from the estimates of the parameters of the underlying life distributions (assumed to be Weibull with unequal shape constants) the relevant probabilities in competing risk theory are obtained.

An Estimation Problem in Life-Testing

This paper considers the maximum likelihood estimator (m.l.e.) of the scale parameter of the exponential distribution in a life-testing situation, where the stopping rule is a prestated time. The life-test is examined periodically and the number of items that have failed since the previous inspection are counted. Approximations to the m.l.e. are considered along with their properties.

An Estimation Problem in Life-Testing

This paper considers the maximum likelihood estimator (m.l.e.) of the scale parameter of the exponential distribution in a life-testing situation, where the stopping rule is a prestated time. The life-test is examined periodically and the number of items that have failed since the previous inspection are counted. Approximations to the m.l.e. are considered along with their properties.

Consideration of power for some two sample tests with censoring based on a given order statistic

SUMMARY This paper deals with some properties of three nonparametric tests which can be used for detecting a difference in location of two populations wlhen samples are censored on the occurrence of a given order statistic. The null distributions of the test statistics are presented together with some tables of critical values. Exact expressions are obtained for the powers of the tests under exponential and rectangular alternatives. Finally, the expected durations of the tests are compared. A number of tests which permit early termination of an experiment have been proposed for detecting a difference in the location of two populations. These tests have important applications in life-testing situations where testing time may be costly. The criterion for termination may be a preselected period of time from the start of the experiment or the occurrence of a given order statistic in one of the samples or the two samples combined. In this paper we consider three nonparametric tests which may be used to provide a one-sided test for difference in location of two populations. The common feature of the tests is that the experiment is terminated on or before the occurrence of a preselected order statistic in one of the samples. We first consider the distributions of the test statistics under the null hypothesis that the two samples have been drawn randomly from populations with identical cumulative distribution functions. Some tables of critical values are presented. Expressions are given for the test powers under the alternative of a translation in an exponential distribution and for changes in the location and scale parameters of a rectangular distribution. Finally some results are given for the expected durations of the tests.

Distribution-Free Tolerance Intervals for Continuous Symmetrical Populations

For continuous symmetrical distributions and specified tolerance interval characteristics, distribution-free tolerance intervals are presented that require substantially smaller sample sizes than those required for arbitrary continuous populations. In the case of one-sided tolerance intervals, about twice as many sample values are required for the general situation as for the symmetrical situation. For two-sided intervals, the additional sample values required for the general situation vary from about 35 percent to about 85 percent in the cases considered, depending on the specified tolerance interval characteristics. These intervals furnish at least rough protection against all possible violations of the symmetry assumption. In addition, some specialized one-sided and two-sided tolerance intervals are developed for life-test situations involving continuous symmetrical populations. Results for continuous symmetrical populations with known central median values are also presented; the life-testing versions of these intervals are especially useful.

Estimating Future from Past in Life Testing

Let $\theta_p$ represent the unique 100$p$ per cent point of a continuous statistical population, while $x_r$ is the $r$th largest value of a sample of size $n$ from this population $(r = 1, \cdots, n)$. This paper considers estimation of $\theta_p$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $(n + 1)p$. Also considered is estimation of $x_R$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $R$. The results are of a nonparametric nature and based on expected value considerations. These estimation procedures may be useful for life-testing situations where time to failure is the variable and some of the items tested have not yet failed when observation is discontinued. Then $\theta_p$ and $x_R$ can be estimated for $p$ and $R$ values which extend a moderate way into the region where sample data is not available. Estimation of the $x_R$ value which would be obtained by continuing to observe the experiment represents a prediction of the future from the past. The results of this paper may be of value in the actuarial, population statistics, operations research, and other fields.

Asymptotic Efficiencies of a Nonparametric Life Test for Smaller Percentiles of a Gamma Distribution

Abstract This paper does not present new statistical methods. Rather, it shows that a well known nonparametric procedure which is valid under extremely general conditions also has high efficiency and other favorable properties for certain types of life testing situations. This nonparametric method is the sign test and its confidence interval and point estimate analogs. Sign test procedures and methods of applying them have already received extensive coverage in the literature (e.g., [2]). Consequently, in this paper emphasis is placed on fields of application instead of on methods of application. However, the basic methods for obtaining tests, confidence intervals, and point estimates of the sign test type are restated for convenience. The properties investigated are concerned with the estimation and testing of smaller population percentage points for the case of a sample from a gamma probability distribution. The investigations show that appropriate use of the sign test type of nonparametric tests and es...

Estimating population mean, variance, and percentage points from truncated data

Abstract Consider a set of observations which are obtained by truncating a sample of known size. The truncation procedure consists of deleting a known number of the largest sample values and a known number of the smallest sample values. One problem considered is the use of this data to estimate certain of the population percentage points for which the corresponding sample data was deleted. Another problem is to estimate the population mean and standard deviation. This paper presents solutions to these problems which are valid for a rather general class of continuous statistical populations. The results obtained should be applicable to most practical cases of a continuous type. A sample analog of the percentage point estimation procedure has interesting uses for life testing situations. Namely, the first time at which a specified number of additional items of a sample will have failed can be predicted from the values of the items which have already failed.