A theorem on the density of the density ordinate and an alternative interpretation of the bqx-muller method

This paper obtains a theorem about the density of the density ordinate when the latter is regarded as a random variable in its own right. The theorem is used to provide an alternative interpretation of the BOX-MULLER simulation method which extends also to an alternative method for simulating from the LAPLACE density     

On estimating the box-cox transformation to normality

This paper studies four methods for estimating the Box-Cox parameter used to transform data to normality. Three of these are based on optimizing test statistics for standard normality tests (the Shapiro-Wilk. skewness, and kurtosis tests); the fourth uses the maximum likelihood estimator of the Box-Cox parameter. The four methods are compared and evaluated with a simulation study, where their performances under different skewness and kurtosis conditions are analyzed. The estimator based on optimizing the Shapiro-Wilk statistic generally gives rise to the best transformations, while the maximum likelihood estimator performs almost as well. Estimators based on optimizing skewness and kurtosis do not perform well in general.     

Estimation of p(y<x) in the gamma case

We consider estimation of P(Y<X) when X?Γr(M,λ) and Y?Γ(N,μ) are independent with M and N known. A concise representation of the UMVUE and several representations for the MLE are derived. Closed-form exact expressions of both MSE's and the bias of the MLE are obtained. Large-sample results are given and numerical comparison of the two point estimators is made. Confidence intervals are given.     

A Derivation of a Useful Consequence of the Gauss-Markov Theorem

The Gauss-Markov (GM) theorem and its many consequences are well known. It has been our experience that graduate students in linear model courses benefit substantially by deriving these consequences, all of which form natural corollaries to the GM theorem. In deriving any one lemma, the student is compelled to reuse many other GM-based lemmas. In this article we present another example of such a lemma that arises in the forward-elimination variable selection procedure. The derivation given is considered to be pedagogically useful.     

Asymptotically valid single-stage multiple-comparison procedures

We establish general conditions for the asymptotic validity of single-stage multiple-comparison procedures (MCPs) under the following general framework. There is a finite number of independent alternatives to compare, where each alternative can represent, e.g., a population, treatment, system or stochastic process. Associated with each alternative is an unknown parameter to be estimated, and the goal is to compare the alternatives in terms of the parameters. We establish the MCPs’ asymptotic validity, which occurs as the sample size of each alternative grows large, under two assumptions. First, for each alternative, the estimator of its parameter satisfies a central limit theorem (CLT). Second, we have a consistent estimator of the variance parameter appearing in the CLT. Our framework encompasses comparing means (or other moments) of independent (not necessarily normal) populations, functions of means, quantiles, steady-state means of stochastic processes, and optimal solutions of stochastic approximation by the Kiefer–Wolfowitz algorithm. The MCPs we consider are multiple comparisons with the best, all pairwise comparisons, all contrasts, and all linear combinations, and they allow for unknown and unequal variance parameters and unequal sample sizes across alternatives.     

Cost Efficiency Benchmarking for Operational Units with Multiple Cost Drivers

We consider the activity‐based costing situation, in which for each of several comparable operational units, multiple cost drivers generate a single cost pool. Our study focuses on published data from a set of property tax collection offices, called rates departments, for the London metropolitan area. We define what may be called benchmark or most efficient costs per unit of driver. A principle of maximum performance efficiency is proposed, and an approach to estimating the benchmark unit costs is derived from this principle. A validation approach for this estimation method is developed in terms of what we call normal‐like‐or‐better performance effectiveness. Application to longitudinal data on a single unit is briefly discussed. We also consider some implications for the more routine case when costs are disaggregated to subpools associated with individual cost drivers.     

THE EMPIRICAL DETERMINANTS OF INVENTORY LEVELS IN HIGH-VOLUME MANUFACTURING

This study uses survey data on several hundred automotive suppliers in North America to evaluate the determinants of inventory levels in high-volume discrete parts manufacturing. We assess the magnitude of raw materials, work-in-process, and finished goods inventories held at automotive supply plants. Inventories are shown to be jointly determined by technological and managerial factors in a manner roughly consistent with classical inventory theory. Several categories of managerial practices are found to be important. Low inventories are linked to employee problem solving and frequent communication with customers. More unexpectedly, we find the absence of inventory differences between U.S.-owned and Japanese-owned plants in North America. This contrasts with substantial differences in inventory holding between US plants and those in Japan.     

On the Generality of the PERT Average Time Formula

It is well known that the PERT average time formula approximation is related to the Beta distribution assumption. In this note, it is shown that if the formula is modified to use the median instead of the mode, then it becomes a good approximation regardless of distribution assumptions. Applications of the formula beyond project management are highlighted.     

Concepts,Theory, and Techniques ON THE USE OF THE GEOMETRIC MEAN IN LONG-TERM INVESTMENT

Application of the geometric mean to holding-period returns is discussed from a statistical theory standpoint. The population geometric mean is considered a parameter of the probability distribution of returns its relationship to moments of the distribution is discussed. The sample geometric mean and its relation to sample moments is assessed through its sampling distribution it is viewed as an estimator of the population geometric mean. For application to long-term investment where a geometric mean is maximized, the distributional properties of the geometric mean should be used. The terms statistic, approximation, and parameter are differentiated.