H. Leon Harter

CRC handbook of tables for the use of order statistics in estimation

Introduction Quasi-Ranges of Samples from a Normal Population Point Estimation of the Population Standard Deviation s Interval Estimation of the Population Standard Deviation s The Range of Samples from a Rectangular Population Point Estimation of the Population Standard Deviation s Interval Estimation of the Population Standard Deviation s Expected Values, Variances and Covariances of Order Statistics of Samples from Various Populations Normal Population Exponential, Weibull and Gamma Populations Other Populations One or Two Order Statistics from an Exponential Population Point Estimation of One or Two Parameters Interval Estimation of the Parameter s Singly Censored Samples from Populations Related to the Exponential Weibull Populations with Known Location and Shape Parameters Type I Extreme-Value Population with Known Scale Parameters Type II Extreme-Value Population with Known Shape Parameters Pareto and Limited Populations with Known Location Parameters Single Order Statistics from Populations Related to the Exponential Weibull Populations with Known Location and Shape Parameters Type I Extreme-Value Population with Known Scale Parameters Type II Extreme-Value Population with Known Shape Parameters Limited and Pareto Populations with Known Location Parameters Doubly Censored Samples from Various Populations Maximum-Likelihood Estimation of Two Parameters of Normal Population Location-Maximum-Likelihood Estimation of Three Parameters of Lognormal Population Maximum-Likelihood Estimation of Three Parameters of Weibull and Gamma Populations Maximum-Likelihood Estimation of Two Parameters of Logistic Populations Maximum-Likelihood Estimation of Two Parameters of Type I Extreme-Value Populations Maximum-Likelihood Estimation of Two Parameters of Log-Gamma Populations Maximum-Likelihood Estimation of Three Parameters of Generalized Extreme-Value Populations Remarks on Interval Estimation of Parameters of Above Populations Appendices Subject Index Author Index

Local-Maximum-Likelihood Estimation of the Parameters of Three-Parameter Lognormal Populations from Complete and Censored Samples

Abstract The natural logarithm of the likelihood function is written down for the m − r order statistics remaining after censoring the n − m largest and the r smallest observations of a sample of size n(0 ≤ r < m ≤ n) from a three-parameter lognormal population. Its first partial derivatives with respect to the parameters, when equated to zero, yield the likelihood equations, and the negatives of its second partial derivatives with respect to the parameters are the elements of the information matrix. Algebraic solution of the likelihood equations is impossible, so it is necessary to resort to iteration on an electronic computer. The iterative procedure proposed is applicable to special cases in which one or two of the parameters are known as well as to the most general case in which all three parameters are unknown. A modification of the procedure allows circumvention of a certain anomaly which sometimes occurs in maximum-likelihood estimation of the parameters of a three-parameter lognormal population fr...

A New Table of Percentage Points of the Pearson Type III Distribution

Recently the U. S. Water Resources Council has proposed standardization of the analysis of peak flood discharges by fitting a Pearson Type III distribution to the logarithms of the data. This action has served to draw attention to the inadequacy of available tables of percentage points of the Pearson Type III distribution and the need for better tables. Many tables of percentage points of the related chi-square distribution are available in the literature, perhaps the most comprehensive being those published by the author in 1964. These could be used to obtain percentage points of the Pearson Type III distribution, but it would be much more convenient to have a table from which percentage points of the latter distribution could be read directly for uniformly spaced values of the skewness coefficient. The author has therefore, by a modification of the programs used to compute his 1964 tables of percentage points of the chi-square distribution, obtained percentage points, corresponding to cumulative probabi...

Maximum-Likelihood Estimation of the Parameters of a Four-Parameter Generalized Gamma Population from Complete and Censored Samples

Consider the four-parameter generalized Gamma population with location parameter c, scale parameter a, shape/power parameter b, and power parameter p (shape parameter d = bp) and probability density function f(x; c, a, b, p) = p(x — c) bp–1 exp {–[(x – c)/a] p }/a bp Γ(b), where a, b, p > 0 and x ≥ c ≥ 0. The likelihood equations for parameter estimation are obtained by equating to zero the first partial derivatives, with respect to each of the four parameters, of the natural logarithm of the likelihood function for a complete or censored sample. The asymptotic variances and covariances of the maximum-likelihood estimators are found by inverting the information matrix, whose components are the limits, as the sample size n → ∞, of the negatives of the expected values of the second partial derivatives of the likelihood function with respect to the parameters. The likelihood equations cannot be solved explicitly, but an iterative procedure for solving them on an electronic computer is described. The results ...

Circular Error Probabilities

Abstract A problem which often arises in connection with the determination of probabilities of various miss distances of bombs and missiles is the following: Let x and y be two normally and independently distributed orthogonal components of the miss distance, each with mean zero and with standard deviations σ x and σ y , respectively, where for convenience one labels the components so that σ x ≥σ y . Now for various values of c = σ y /σ x , it is required to determine (1) the probability P that the point of impact lies inside a circle with center at the target and radius Kσ x , and (2) the value of K such that the probability is P that the point of impact lies inside such a circle. Solutions of (1), for c = 0.0(0.1) 1.0 and K = 0.1 (0.1) 5.8, and (2), for the same values of c and P = 0.5, 0.75, 0.9, 0.95, 0.975, 0.99, 0.995, 0.9975, and 0.999, are given along with some hypothetical examples of the application of the tables.

Exact Confidence Bounds, Based on One Order Statistic, for the Parameter of an Exponential Population

For a one-parameter negative exponential population, reasonably good interval estimates of the parameter σ may be obtained from one suitably chosen order statistic. The coefficients of the mth order statistic xm in exact confidence bounds for σ are found by taking the negative reciprocals of the natural logarithms of percentage points of the Beta distribution. The interval between exact lower and upper confidence bounds, each associated with confidence 1 – P, is, of course, an exact central confidence interval (confidence 1 – 2P). Results have been computed for several values of m, clustered about the value which yields the most efficient point estimator, for sample size n = 1(1)20(2)40 and P = .0001, .0005, .001, .005, .01, .025, .05, .1(.1).5. The definition of efficiency commonly used for point estimators is extended to confidence bounds and confidence intervals. The following tables are included, together with a description of the method of computation and a brief discussion of possible uses: (1) a ta...

Conditional Maximum-Likelihood Estimators, from Singly Censored Samples, of the Scale Parameters of Type II Extreme-Value Distributions

Use of the functional relationships between the exponential and the Type II asymptotic distributions of largest and smallest values enables one to obtain conditional maximum-likelihood estimators, from singly censored samples, of the scale parameters (characteristic largest and characteristic smallest values) of the Type II asymptotic distributions of largest and smallest values, F 1(y; vn , K) = exp [–(y/vn )−K ] and F 2(x; v 1, K) = 1 – exp [– (x/v 1)−K ], by simple transformations of the corresponding estimator, , of the scale parameter of the exponential distribution, based on the first m order statistics of a sample of size n. Use is made of the fact that v n , | K = and v1, | K , where 2m /θ has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on the scale parameters, vn , and v 1, of the Type II asymptotic distributions of largest and smallest values. The probability densities of v n , | K and v1 K, each of which for given m is the same for any n > m, are obtained by...

Criteria for Best Substitute Interval Estimators, with an Application to the Normal Distribution

Abstract The following criteria for determining best substitute interval estimators of a given class for a population parameter are considered in this paper: (1) minimizing the expected length of the confidence interval (maximizing the effectiveness); (2) minimizing the sum of the mean (absolute) deviations of the upper and lower confidence bounds from the true value (maximizing the effectivity); and (3) minimizing the sum of the mean squared deviations of the upper and lower confidence bounds from the true value (maximizing the efficiency). Modified forms of these criteria can also be applied to one-sided confidence intervals. Both theoretical and practical reasons are given for preferring criterion (3) to the others. A second main point of the paper is that exact confidence intervals, when available, are superior (and often far superior) to approximate ones. Both points are illustrated by a study of confidence bounds, based on one quasi-range, for the standard deviation of a normal population.

MTBF Confidence Bounds Based on MIL-STD-781C Fixed-Length Test Results

Exact confidence bounds are available for the parameter ( of an exponential distribution in the case of Type II (failure) censored samples, but not in the case of Type I (time) censored samples. In the latter case, the conventional confidence bounds are..

Use of Tables of Percentage Points of Range and Studentized Range

A description is given of new tables, more extensive and more accurate than any previously published, of the percentage points of the range and of the studentized range, for samples from a normal population. The purpose of this paper is to call attention to these tables and to illustrate their use. The following examples of the use of the tables are given: (1) an application of the percentage points of the range to tests of hypotheses concerning the standard deviation of a normal population; (2) an application of the percentage points of the range to rejection of outliers; and (3) an application of the percentage points of the studentized range to multiple comparisons tests on means of samples from a normal population.