John E. Angus

The probability integral transform and related results

A simple proof of the probability integral transform theorem in probability and statistics is given that depends only on probabilistic concepts and elementary properties of continuous functions. This proof yields the theorem in its fullest generality. A similar theorem that forms the basis for the inverse method of random number generation is also discussed and contrasted to the probability integral transform theorem. Typical applications are discussed. Despite their generality and far reaching consequences, these theorems are remarkable in their simplicity and ease of proof.

On computing MTBF for a k-out-of-n:G repairable system

It is often necessary to calculate the MTBF (mean time between failures) quickly in order to make timely design decisions. An important system for which such calculations must be made is a k-out-of-n:G parallel system with unlimited repair and exponential interfailure and repair times at the unit level. Although a general formula is known, it is not easily remembered or derived. A method for deriving a formula for MTBF in this situation that is easily reproduced quickly by remembering a few simple concepts is presented. >

Goodness-of-fit tests for exponentiality based on a loss-of-memory type functional equation

Abstract Three goodness-of-fit tests for exponentiality based on the functional equation characterization 1−F(2x)=[1−F(x)]2 for every x⩾0 are proposed and shown to compare well to several popular tests against common alternative cdfs. Small-sample critical values for α=0.10,0.05 are developed for the two superior test statistics and the asymptotic null-distributions are characterized.

Bootstrap one-sided confidence intervals for the log-normal mean

SUMMARY Boostrap procedures for computing lower and upper confidence limits for the mean of a log-normal distribution based on complete samples are presented. The procedures are based on an approximate pivotal statistic and are shown to yield confidence bounds that are often nearly equal to the optimal (uniformly most accurate unbiased) bounds that require complex numerical algorithms.

Improved Confidence Statements for the Binomial Parameter

Abstract In the computation of two-sided confidence intervals for the binomial parameter p (using the binomial mass function), it is known that such intervals achieve a confidence coefficient that in general is not equal to the confidence level 1 – α, say. In this article we present some general results on the confidence coefficient and tabulate them for selected pairs (α, n = number of trials). We treat only the nominal equal tail probability case because it is the most commonly taught and used.

A note on pricing Asian derivatives with continuous geometric averaging

A general expression is derived for the price of a European‐style Asian contingent claim in which the terminal value depends on both the underlying asset price and the continuous geometric average of the price of the underlying asset over the life of the claim. Specific formulas are derived for Asian call, put, and binary options, as well as for the average strike binary options.

Optimal mutation probability for genetic algorithms

We derive the value of the mutation probability which maximizes the probability that the genetic algorithm finds the optimum value of the objective function under simple assumptions. This value is compared with the optimum mutation probability derived in other studies. An empirical study shows that this value, when used with a larger scaling factor in linear scaling, improves the performance of the genetic algorithm. This feature is then added to a model developed by Hinton and Nowlan which allows certain bits to be guessed in an effort to increase the probability of finding the optimum solution.

Asymptotic theory for bootstrapping the extremes

This paper considers asymptotic analysis of bootstrap distributions for the extremes from an iid sample. In contrast to the case of almost sure convergence to a fixed (normal) distribution in the case of the sample mean (finite variance case), the bootstrap distribution of an extreme tends in distribution to a random probability measure. These results are similar to the result for the bootstrap distribution of the sample mean in the infinite variance case where the underlying random variables are in the domain of attraction of a stable law with index αe(0,2).

Modeling Genetic Regulatory Networks using Gene Expression Profiling and State-Space Models

We describe a Bayesian network approach to infer genetic regulatory interactions from microarray gene expression data. This problem was introduced in Chapter 7 and an alternative Bayesian network approach was presented in Chapter 8. Our approach is based on a linear dynamical system, which renders the inference problem tractable: the E-step of the EM algorithm draws on the well-established Kalman smoothing algorithm. While the intrinsic linearity constraint makes our approach less suitable for modeling non-linear genetic interactions than the approach of Chapter 8, it has two important advantages over the method of Chapter 8. First, our approach works with continuous expression levels, which avoids the information loss inherent in a discretization of these signals. Second, we include hidden states to allow for the effects that cannot be measured in a microarray experiment, for example: the effects of genes that have not been included on the microarray, levels of regulatory proteins, and the effects of mRNA and protein degradation.

Optimal Preventive Maintenance Rate for Best Availability With Hypo-Exponential Failure Distribution

The optimal rate of periodic preventive maintenance to achieve the best availability is studied for Markov systems with multiple degraded operational stages, where the time-to-failure has a hypo-exponential distribution. An analytical expression is developed for the availability of such systems having n operational stages, and a necessary and sufficient condition is derived for a non-trivial optimal rate of periodic maintenance to exist. Numerical procedures for finding the optimal rate of periodic maintenance are given, and examples are presented.