Applications of a new power normal family

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson-Hilferty power transformation setting.     

An Approximate Normalizing Power Transformation for the FDistribution

A power transformation of the Fdistribution is presented, yielding simple normal approximations for both probabilities and quantiles of the distribution. The transformation proposed is shown to produce the well-known Wilson-Hilferty cube root transformation (Wilson and Hilferty, 1931) for the chi-square distribution as a limiting case, as well as the Fisher logarithmic transformation (Fisher, 1924, 1925) for equal degrees of freedom. A numerical assessment of the accuracy achieved for approximating tail probabilities and a comparison with some of the existing approximate procedures are given.     

Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.     

Transforming linear functions to normality:optimal component powers

This paper considers the analysis of linear models where the response variable is a linear function of observable component variables. For example, scores on two or more psychometric measures (the component variables) might be weighted and summed to construct a single response variable in a psychological study. A linear model is then fit to the response variable. The question addressed in this paper is how to optimally transform the component variables so that the response is approximately normally distributed. The transformed component variables, themselves, need not be jointly normal. Two cases are considered; in both cases, the Box-Cox power family of transformations is employed. In Case I, the coefficients of the linear transformation are known constants. In Case II, the linear function is the first principal component based on the matrix of correlations among the transformed component variables. For each case, an algorithm is described for finding the transformation powers that minimize a generalized Anderson-Darling statistic. The proposed transformation procedure is compared to likelihood-based methods by means of simulation. The proposed method rarely performed worse than likelihood-based methods and for many data sets performed substantially better. As an illustration, the algorithm is applied to a problem from rural sociology and social psychology; namely scaling family residences along an urban-rural dimension.     

Better Articulating Normal Curve Theory for Introductory Mathematical Statistics Students: Power Transformations and Their Back-Transformations

This article addresses a gap in many, if not all, introductory mathematical statistics textbooks, namely, transforming a random variable so that it better mimics a normal distribution. Virtually all such textbooks treat the subject of variable transformations, which furnishes a nice opportunity to introduce and study this transformation-to-normality topic, a topic students frequently encounter in subsequent applied statistics courses. Accordingly, this article reviews variable power transformations of the Box–Cox type within the context of normal curve theory, as well as addresses their corresponding back-transformations. It presents four theorems and a conjecture that furnish the basics needed to derive equivalent results for all nonnegative values of the Box–Cox power transformation exponent. Results are illustrated with the exponential random variable. This article also includes selected pedagogic tools created with R code.     

PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE

This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.     

Families of Multivariate Distributions Involving “Triangular” Transformations

Attention is initially focused on certain pseudo-normal distributions. These are multivariate distributions in which one coordinate variable has a normal distribution and the distribution of the remaining variables is determined by a specific triangular transformation model involving normally distributed components. A remarkably flexible family of models is obtainable in this fashion. Some examples are described. In addition, models involving non-normal component distributions are discussed together with their relationship with those models obtainable by means of the beta-generalized-Rosenblatt construction. Inferential questions regarding these models will be the subject of a separate report.     

On approximating the distribution of indefinite quadratic forms

This paper provides a simple methodology for approximating the distribution of indefinite quadratic forms in normal random variables. It is shown that the density function of a positive definite quadratic form can be approximated in terms of the product of a gamma density function and a polynomial. An extension which makes use of a generalized gamma density function is also considered. Such representations are based on the moments of a quadratic form, which can be determined from its cumulants by means of a recursive formula. After expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, one can obtain an approximation to its density function by means of the transformation of variable technique. An explicit representation of the resulting density approximant is given in terms of a degenerate hypergeometric function. An easily implementable algorithm is provided. The proposed approximants produce very accurate percentiles over the entire range of the distribution. Several numerical examples illustrate the results. In particular, the methodology is applied to the Durbin–Watson statistic which is expressible as the ratio of two quadratic forms in normal random variables. Quadratic forms being ubiquitous in statistics, the approximating technique introduced herewith has numerous potential applications. Some relevant computational considerations are also discussed.     

Tukey's gh Distribution for Multiple Imputation

Tukey proposed a class of distributions, the g-and-h family (gh family), based on a transformation of a standard normal variable to accommodate different skewness and elongation in the distribution of variables arising in practical applications. It is easy to draw values from this distribution even though it is hard to explicitly state the probability density function. Given this flexibility, the gh family may be extremely useful in creating multiple imputations for missing data. This article demonstrates how this family, as well as its generalizations, can be used in the multiple imputation analysis of incomplete data. The focus of this article is on a scalar variable with missing values. In the absence of any additional information, data are missing completely at random, and hence the correct analysis is the complete-case analysis. Thus, the application of the gh multiple imputation to the scalar cases affords comparison with the correct analysis and with other model-based multiple imputation methods. Comparisons are made using simulated datasets and the data from a survey of adolescents ascertaining driving after drinking alcohol.     

Comparison of Statistical Methods for Pretest–Posttest Designs in Terms of Type I Error Probability and Statistical Power

The pretest–posttest design is widely used to investigate the effect of an experimental treatment in biomedical research. The treatment effect may be assessed using analysis of variance (ANOVA) or analysis of covariance (ANCOVA). The normality assumption for parametric ANOVA and ANCOVA may be violated due to outliers and skewness of data. Nonparametric methods, robust statistics, and data transformation may be used to address the nonnormality issue. However, there is no simultaneous comparison for the four statistical approaches in terms of empirical type I error probability and statistical power. We studied 13 ANOVA and ANCOVA models based on parametric approach, rank and normal score-based nonparametric approach, Huber M-estimation, and Box–Cox transformation using normal data with and without outliers and lognormal data. We found that ANCOVA models preserve the nominal significance level better and are more powerful than their ANOVA counterparts when the dependent variable and covariate are correlated. Huber M-estimation is the most liberal method. Nonparametric ANCOVA, especially ANCOVA based on normal score transformation, preserves the nominal significance level, has good statistical power, and is robust for data distribution.     

A Methodology to Implement Box-Cox Transformation When No Covariate is Available

Box-Cox transformation is one of the most commonly used methodologies when data do not follow normal distribution. However, its use is restricted since it usually requires the availability of covariates. In this article, the use of a non-informative auxiliary variable is proposed for the implementation of Box-Cox transformation. Simulation studies are conducted to illustrate that the proposed approach is successful in attaining normality under different sample sizes and most of the distributions and in estimating transformation parameter for different sample sizes and mean-variance combinations. Methodology is illustrated on two real-life datasets.     

The uniform power distribution

This paper introduces a generalization of the normal distribution: the uniform power distribution. It is a symmetric and unimodal family of distributions, defined on the real line, and is closely related to the exponential power family. The exponential power family was introduced to allow the modelling of kurtosis. The uniform power family matches the exponential power family with respect to the range of kurtosis. However, whereas the exponential is somewhat difficult to work with, the contrary is true for the uniform power family.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

An extended exponentiated-G-negative binomial family with threshold effect

In this paper, we formulate a very flexible family of models which unifies most recent lifetime distributions. The main idea is to obtain a cumulative distribution function to transform the baseline distribution with an activation mechanism characterized by a latent threshold variable. The new family has a strong biological interpretation from the competitive risks point of view and the Box–Cox transformation provides an elegant manner to interpret the effect on the baseline distribution to obtain this alternative model. Several structural properties of the new model are investigated. A Bayesian analysis using Markov Chain Monte Carlo procedure is developed to illustrate with a real data the usefulness of the proposed family.     

A higher order approximation to a percentage point of the non–central t-distribution

A higher order approximation formula for a percentage point of the noncentral t–distribution with v degrees of freedom is given up to the order o(v^-3), using the Cornish-Fisher expansion for the statistic based on a lin-ear combination of a normal random variable and a chi-random variable. The upper confidence limit and the confidence interval for the non–centrality parameter are given. Numerical results are also obtained.     

Transforming the exponential by minimizing the sum of the absolute differences

This work presents an optimal value to be used in the power transformation to transform the exponential to normality for statistical process control (SPC) applications. The optimal value is found by minimizing the sum of absolute differences between two distinct cumulative probability functions. Based on this criterion, a numerical search yields a proposed value of 3.5142, so the transformed distribution is well approximated by the normal distribution. Two examples are presented to demonstrate the effectiveness of using the transformation method and its applications in SPC. The transformed data are almost normally distributed and the performance of the individual charts is satisfactory. Compared to charts that use the original exponential data and probability control limits, the individual charts constructed using the transformed distribution are superior in appearance, ease of interpretation and implementation by practitioners.     

Letters to the Editor

Although several authors have indicated that the median test has low power in small samples, it continues to be presented in many statistical textbooks, included in a number of popular statistical software packages, and used in a variety of application areas. We present results of a power simulation study that shows that the median test has noticeably lower power, even for the double exponential distribution for which it is asymptotically most powerful, than other readily available rank tests. We suggest that the median test be “retired” from routine use and recommend alternative rank tests that have superior power over a relatively large family of symmetric distributions.     

ON THE FOURTH ROOT TRANSFORMATION OF CHI-SQUARE

We show that, within the family of power transformations of a Chisquare variable, the square and fourth roots minimize Pearson's index of kurtosis. Two new transtormations of the fourth root, a symmetrized-truncated version and its linear combination with the square root are also studied. The first transformation shows a considerable improvement over the fourth root while the second one turns out to be even more accurate than Hilferty-Wilson's cube root transformation.     

Regression estimators in extreme and median ranked set samples

The ranked set sampling (RSS) method as suggested by McIntyre (1952) may be modified to come up with new sampling methods that can be made more efficient than the usual RSS method. Two such modifications, namely extreme and median ranked set sampling methods, are considered in this study. These two methods are generally easier to use in the field and less prone to problems resulting from errors in ranking. Two regression-type estimators based on extreme ranked set sampling (ERSS) and median ranked set sampling (MRSS) for estimating the population mean of the variable of interest are considered in this study and compared with the regression-type estimators based on RSS suggested by Yu & Lam (1997). It turned out that when the variable of interest and the concomitant variable jointly followed a bivariate normal distribution, the regression-type estimator of the population mean based on ERSS dominates all other estimators considered.     

Limit theorems for intermediate and central order statistics under nonlinear normalization

In this paper the work of Pancheva (1984) for extreme order statistics under nonlinear normalization is extended to order statistics with variable ranks. Two new results are proved. The first is that under nonlinear normalization, the nondegenerate type (family of types) of the distribution functions with two finite growth points is a possible weak limit of any central order statistic with regular rank sequence. The second result is that the possible nondegenerate weak limits of any central order statistic with regular rank under the traditionally linear normalization and under the power normalization are the same. Finally, the class of all possible weak limits for lower and upper intermediate order statistics is derived under power normalization from the corresponding weak limits of extremes under power normalization.