Danny Dyer

Best Linear Unbiased Estimator of the Parameter of the Rayleigh Distribution - Part I: Small Sample Theory for Censored Order Statistics

The best linear unbiased estimator of the parameter of the Rayleigh distribution using order statistics in a Type II censored sample from a potential sample of size N is considered. The coefficients for this estimator are tabled to five decimal places for N = 2(1)15 and censoring values of r1, (the number of observations censored from the left) and r2 (the number of observations censored from the right) such that r1 + r2 ? N - 2 for N = 2(1)10, r1 + r2 ? N - 3 for N = 11(1)15.

Structural probability bounds for the strong Pareto law

By using the structural density function (Fraser 1979, Ch. 7) of the parameters of a Pareto distribution, the structural distribution function of the strong Pareto law is derived. Its fractiles have been evaluated numerically for special cases, and the results are displayed through graphs from which structural one-sided probability bounds may be found. It is shown that these graphs may also be used to find structural tolerance bounds for the Pareto distribution.

On the Determination of Critical Values for Bartlett's Test

Abstract The exact critical values for Bartletts test for homogeneity of variances based on equal sample sizes from several normal populations are tabulated. It is also shown how these values may be used to obtain highly accurate approximations to the critical values for unequal sample sizes. An application is given that deals with the variability of log bids on a group of federal offshore oil and gas leases.

Unification of reliability/availability/repairability models for Markov systems

An examination is made of the structure of the general transition rate matrix from which the model transition rate matrices are obtained. An exact solution to the system-state equations is derived which depends on the eigenvalues of the model transition rate matrix. In order to obtain the exact numerical solution, an algorithm is given which requires a minimal amount of computer storage requirements. An approximate solution is derived which does not require determination of eigenvalues but, instead, is based on the representation of a Markov process by a Markov chain randomized by a Poisson process. This approximation is highly accurate with a controllable error, and its use is particularly effective for large systems. >

An information-theoretic approach to incorporating prior information in binomial sampling

The incorporation of prior information about θ, where θ is the success probability in a binomial sampling model, is an essential feature of Bayesian statistics. Methodology based on information-theoretic concepts is introduced which (a) quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and (b) allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information about θ which is embedded in any prior distribution. In effect, the most conservative prior distribution from a specified class (each member o f which carries the available prior information about θ) is that prior distribution within the class over which the likelihood function has the greatest average domination. The most conservative prior distributions from five different families of prior distributions over the interval (0,1) including the beta distribution are determined and compared for three situations:...

Best Linear Unbiased Estimator of the Parameter of the Rayleigh Distribution - Part II: Optimum Theory for Selected Order Statistics

The member of the class of best linear unbiased estimators (BLUEs) of a parameter based on k order statistics which has minimum variance is called the k-optimum BLUE. The ranks, coefficients, variance, and efficiency are given for the k-optimum BLUE of the parameter of the Rayleigh distribution for k = 2(1)4 and a sample size of N = 2(1)22. In addition, an approximate k-optimum BLUE is given for k = 2(1)4 and N ? 23.

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distri...

Comparison of point estimators of normal percentiles

There are available several point estimators of the percentiles of a normal distribution with both mean and variance unknown. Consequently, it would seem appropriate to make a comparison among the estimators through some “closeness to the true value” criteria. Along these lines, the concept of Pitman-closeness efficiency is introduced. Essentially, when comparing two estimators, the Pit-man-closeness efficiency gives the “odds” in favor of one of the estimators being closer to the true value than is the other in a given situation. Through the use of Pitman-closeness efficiency, this paper compares (a) the maximum likelihood estimator, (b) the minimum variance unbiased estimator, (c) the best invariant estimator, and (d) the median unbiased estimator within a class of estimators which includes (a), (b), and (c). Mean squared efficiency is also discussed.

The Convolution of Generalized F Distributions

Abstract The generalized F variate is the ratio of two independent gamma variates, and its distribution includes as special cases such distributions as the inverted beta, Lomax, and Snedecors F, Based on convolution, the distribution function of the sum of two independent generalized F variates is derived in terms of a Lauricella-Saran hyper-geometric function of three variables. The results are applied, with numerical examples given, to (a) a Bayesian analysis of the availability of a two-component series system and (b) a test of hypothesis on the multinormal mean vector whenever the covariance matrix has the intraclass correlation pattern.

An empirical power study of multi-sample tests for exponentiality

Results are given of an empirical power study of three statistical procedures for testing for exponentiality of several independent samples. The test procedures are the Tiku (1974) test, a multi-sample Durbin (1975) test, and a multi-sample Shapiro–Wilk (1972) test. The alternative distributions considered in the study were selected from the gamma, Weibull, Lomax, lognormal, inverse Gaussian, and Burr families of positively skewed distributions. The general behavior of the conditional mean exceedance function is used to classify each alternative distribution. It is shown that Tikus test generally exhibits overall greater power than either of the other two test procedures. For certain alternative distributions, Shapiro–Wilks test is superior when the sample sizes are small.