John Gurland

A Class of Distributions Applicable to Accidents

Abstract This paper presents an extension of the mathematical model used to justify accident proneness. It assumes that the distribution of accidents incurred by an individual in non-overlapping intervals is a correlated bivariate Poisson (C.B.P.). On compounding this correlated bivariate Poisson through a Gamma distribution an extended bivariate negative binomial or, more precisely, a compound correlated bivariate Poisson (C.C.B.P.) distribution is obtained. Recurrence relations and expressions for the required probabilities are illustrated for two sets of data. The C.C.B.P. proved to fit as well as the bivariate negative binomial when the estimate of B12 was close to zero, and much better than the latter distribution when the estimate of B12 was not close to zero.

Size and Power of Tests for Equality of Means of Two Normal Populations with Unequal Variances

Abstract A method is developed for obtaining the size and power of a wide class of tests which includes solutions to the Behrens-Fisher problem proposed by various authors. A new test in this class is also developed which controls the size effectively and has high power. A comparison is made with other known tests as regards control of size. The power of the proposed test as well as that of Welch-Aspin is also obtained.

How Pooling Failure Data May Reverse Increasing Failure Rates

Abstract Although mixtures of decreasing failure rate (DFR) distributions are always DFR, some mixtures of increasing failure rate (IFR) distributions can also be ultimately DFR. In this article various types of discrete and continuous mixtures of IFR distributions are considered, and conditions are developed for such mixtures to be ultimately DFR. These conditions lead to an interesting result—that certain mixtures of IFR distributions, even those with very rapidly increasing failure rates (e.g., Weibull, truncated extreme), ultimately become DFR distributions. It is common practice to pool data from several different IFR distributions to enlarge sample size, for instance. The results of this article sound a warning that such pooling may actually reverse the IFR property of the individual samples to a DFR property.

Goodness of Fit Tests for the Gamma and Exponential Distributions

Goodness of fit tests based on generalized minimum x 2 techniques are developed for the gamma and exponential distributions. The power of these tests has been found for several alternative families of distributions by utilizing the asymptotic non-null distribution of the test statistic. The tests behave very well for the types of alternatives considered here. Applications to some failure data of Proschan (1963) are included for illustrative purposes.

Estimation of parameters in the beta binomial model

This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970,Appl. Statist.,19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of “ones” has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.

Reversal of Increasing Failure Rates When Pooling Failure Data

Because mixtures of exponential distributions (with constant failure rate) have the decreasing failure rate (DFR) property, as shown by Proschan in the sixties, it is not unexpected that mixtures of distributions that have a mildly increasing failure rate (IFR) also have this property. What is, perhaps, surprising is that mixtures of IFR distributions with rapidly increasing failure rate may also behave in this manner. Two striking examples of such mixtures are presented here and illustrated graphically. This phenomenon could be the cause of concern, in practice, when it is suspected that a sample is based on pooled data.

How Many Classes in the Pearson Chi-Square Test?

Abstract The asymptotic non-null distribution is obtained for the modified form of the Pearson chi-square statistic studied by Dahiya and Gurland [3]. By utilizing this result the power is obtained for specific alternative distributions in testing for normality. This enables recommendations to be made as to the number of class intervals to be employed in applying the aforementioned modification of the Pearson chi-square test of normality.

Combinations of Unbiased Estimators of the Mean Which Consider Inequality of Unknown Variances

Abstract The problem considered in this paper is how to combine estimators of the common mean from two samples corresponding to normal populations with different unknown variances. Attention is confined to the case where it is known that the variance of one specific population exceeds that of the other. Three classes of unbiased estimators are presented, one of which is based on a preliminary test of significance regarding the ratio of the population variances. The gain achieved by utilizing the knowledge that the ratio of variances exceeds one is investigated by comparing the efficiencies of these estimators with an estimator presented by Graybill and Deal [1] in which no restriction on the ratio of variances is present.

A Test of Fit for the Negative Binomial and Other Contagious Distributions

Abstract A test of fit for the negative binomial and other contagious distributions is presented here. The test, is free of certain disadvantages of the Pearson chi-square test ordinarily used. The proposed test statistic, here called XF 2, is constructed from estimators given by Hinz and Gurland [5] which are obtainable through weighted least squares and which were shown to be highly efficient in wide regions of the parameter space. The power of the proposed test procedure is given for some alternatives and a comparison is made with the corresponding power of the Pearson chi-square test.

Estimation of Parameters on Some Extensions of the Katz Family of Discrete Distributions Involving Hypergeometric Functions

A two-parameter family of discrete distributions developed by Katz (1963) is extended to three- and four-parameter families whose probability generating functions involve hypergeometric functions. This extension contains other distributions appearing in the literature as particular cases. Various methods of estimating the parameters are investigated and their asymptotic efficiency relative to maximum likelihood estimators compared.