Donald R. Barr

Mean and Variance of Truncated Normal Distributions

Abstract Maximum likelihood estimators for the mean and variance of a truncated normal distribution, based on the entire sample from the original distribution, are developed. The estimators are compared with the sample mean and variance of the censored sample, considering only data remaining after truncation. The full- and censored-sample estimators are compared using simulation. It is seen that, surprisingly, the censored-sample estimators generally have smaller mean square error than have the full-sample estimators.

A Kolmogorov-Smirnov Test for Censored Samples

Modifications of the two-sided one-sample Kolmogorov-Smirnov “goodness-of-fit” test, for use with censored and truncated samples, are suggested. Tables of the distributions of the modified statistics are given. Applications to life testing and reliability estimation problems are discussed.

A comparison of multivariate normal generators

Three methods for generating outcomes on multivariate normal random vectors with a specified variance-covariance matrix are presented. A comparison is made to determine which method requires the least computer execution time and memory space when utilizing the IBM 360/67. All methods use as a basis a standard Gaussian random number generator. Results of the comparison indicate that the method based on triangular factorization of the covariance matrix generally requires less memory space and computer time than the other two methods.

Using Confidence Intervals to Test Hypotheses

When separate 100(1-alpha) percent confidence intervals on two population means overlap, the means may or may not be significantly different at the 100-alpha percent level. This phenomenon is explored analytically and numerical examples are given...

An Approximate Test of Markov Chain Lumpability

Abstract Under certain conditions the state space of a discrete parameter Markov chain may be partitioned to form a smaller lumped chain that retains the Markov property. The problem of formulating lump-ability hypotheses when the transition probability matrix P is not known and is possibly of large dimension is discussed. An approximate test of these hypotheses is described, based on well-known nonparametric methods. The procedure is illustrated with an application to a Markov manpower model.

A Class of General Reliability Growth Prediction Models

In structuring reliability growth prediction models as Markov chains it is seen that the computation of the reliability after n trials and possible associated repairs R, may be accomplished with any of several different methods. This paper considers a class of models that accommodates variations in several important factors, such as the interdependences of assignable cause failure modes, inclusion of an inherent failure mode, the repair policy, and the distribution of initial states of the system.

On the Distribution of a Difference of Two Scaled Chi-Square Random Variables

by tending to make the animals remain in the protected area, the number of surviving animals is increased from 263 to 784. It can be noted that since the death rate, is between P21 and P31, i.e. 0.1 < death rate, <0.5 and the birth rate is bounded by M22 and M33, i.e. birth rate equals 0.1, extinction is inevitable. Any efforts to reduce the death rate or to increase the birth rate would change the transition matrix. Further reduction in death rates may be feasible by increased enforcement or patrols, increased hunting license fees, etc. By way of further example, another transition matrix P could be defined as P = .5 .2 .3

A Model for Analyzing Artillery Registration Procedures

This paper develops a model that extends and generalizes the methods of the bio-assay analysis by incorporating differing stopping rules, samples per level, and level magnitudes, and applies it to the problem of artillery registration in a decision-theoretic setting. It finds that significant improvements in the procedure currently used by the US Army and Marine Corps field artillery units (in terms of accuracy, timeliness, and ammunition expenditures) may well be possible. The method of analysis is applicable in a wide range of contexts involving “calibration” problems where the variance of the population being tested is known.

Strong Optimality of the Shoot-Adjust-Shoot Strategy

This paper shows that seemingly different “adjustment” procedures are equivalent, if viewed in appropriate coordinate systems. It extends previous results concerning sequential adjustments that are constrained to be linear functions of observed impact points to the class of translation-invariant procedures. It also reviews properties of the optimal sequential adjustment procedure, including some related to stochastic approximation.

Identification of Failing Mechanical Systems through Spectrometric Oil Analysis

The results obtained from spectrometric analyses may be viewed as being stochastic in nature, so that decisions based upon them are necessarily accompanied by uncertainties. A class of rules is developed for making decisions concerning whether a mechanical system may be failing, based upon spectroscopic analyses of the systems oil over a period of time. Some considerations that went into the development of these rules, including studies of past analysis records and experiments, are presented. Identification procedures of the type suggested should perform well in connection with a computerized analysis system, at least insofar as routinely monitoring the well behaved systems, while calling the attention of appropriate personnel to possibly discrepant systems.