Maximum likelihood estimation of polychoric correlations in r×s ×t contingency tables

This paper discusses the maximum likelihood estimation of the polychoric correlation coefficient based on observed frequencies of three polytomous ordinal variables. The underlying latent variables are assumed to have a standardized trivariate normal distribution. The thresholds and correlations are estimated simultaneously via the scoring algorithm. Some practical applications of the method are discussed. An example is reported to illustrate the theory and some technical details are presented in the Appendix.     

A distance-based model for spatial prediction using radial basis functions

In the context of local interpolators, radial basis functions (RBFs) are known to reduce the computational time by using a subset of the data for prediction purposes. In this paper, we propose a new distance-based spatial RBFs method which allows modeling spatial continuous random variables. The trend is incorporated into a RBF according to a detrending procedure with mixed variables, among which we may have categorical variables. In order to evaluate the efficiency of the proposed method, a simulation study is carried out for a variety of practical scenarios for five distinct RBFs, incorporating principal coordinates. Finally, the proposed method is illustrated with an application of prediction of calcium concentration measured at a depth of 0–20 cm in Brazil, selecting the smoothing parameter by cross-validation.     

Assessing and Improving the Performance of Nearly Efficient Unit Root Tests in Small Samples

Abstract

The development of unit root tests continues unabated, with many recent contributions using techniques such as generalized least squares (GLS) detrending and recursive detrending to improve the power of the test. In this article, the relation between the seemingly disparate tests is demonstrated by algebraically nesting all of them as ratios of quadratic forms in normal variables. By doing so, and using the exact sampling distribution of the ratio, it is straightforward to compute, examine, and compare the test' critical values and power functions. It is shown that use of GLS detrending parameters other than those recommended in the literature can lead to substantial power improvements. The open and important question regarding the nature of the first observation is addressed. Tests with high power are proposed irrespective of the distribution of the initial observation, which should be of great use in practical applications.     

The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter

Laplace transforms are used to derive an exact expression for the cdf of the sum of n i.i.d. Pareto random variables with common pdf f(x) = (α/β)(1 + x/β)^?α?1 for x > 0, where α > 0 and is not an integer, and β > 0. An attractive feature of this expression is that it involves an integral of non oscillating real-valued functions on the positive real line. Examples of values of cdfs are provided and are compared to those determined via simulations.     

On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables

The distribution of the ratio of two independent normal random variables X and Y is heavy tailed and has no moments. The shape of its density can be unimodal, bimodal, symmetric, asymmetric, and/or even similar to a normal distribution close to its mode. To our knowledge, conditions for a reasonable normal approximation to the distribution of Z = X/Y have been presented in scientific literature only through simulations and empirical results. A proof of the existence of a proposed normal approximation to the distribution of Z, in an interval I centered at β = E(X) /E(Y), is given here for the case where both X and Y are independent, have positive means, and their coefficients of variation fulfill some conditions. In addition, a graphical informative way of assessing the closeness of the distribution of a particular ratio X/Y to the proposed normal approximation is suggested by means of a receiver operating characteristic (ROC) curve.     

The Beta Product Distribution with Complex Parameters

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.     

A Transformation Characterizing the Normal Distribution

In this article we propose a new transformation of random variables (RVs) which characterizes the normal distribution. It allows us to transform n i.i.d. normal RVs whose mean and variance are unknown into new n ? 2 i.i.d. new normal variables with zero mean while maintaining the same unknown variance. This belongs to the class of transformations designed to reduce the number of unknown parameters or remove them altogether.

Some historical remarks concerning methods for removing parameters in the normal distribution are given and two possible applications of the new transformation are described.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

Confidence Bounds for Multiplicative Comparisons

Statements that are inherently multiplicative have historically been justified using ratios of random variables. Although recent work on ratios has extended the classical theory to produce confidence bounds conditioned on a positive denominator, this current article offers a novel perspective that eliminates the need for such a condition. Although seemingly trivial, this new perspective leads to improved lower confidence bounds to support multiplicative statements. This perspective is also more satisfying as it allows comparisons that are inherently multiplicative in nature to be properly analyzed as such.     

Phase I Monitoring of Polynomial Profiles

In some statistical process control applications, quality of a process or product is characterized by a relationship between two or more variables which is referred to as profile. In many practical situations, a profile can be modeled as a polynomial regression. In this article, three methods are developed for monitoring polynomial profiles in Phase I. Their performance is evaluated using power criterion. Furthermore, a method based on likelihood ratio test is developed to identify the location of shifts. Numerical simulation is used to evaluate the performance of the developed method.     

Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix

In this article we consider a set of t repeated measurements on p variables (or characteristics) on each of the n individuals. Thus, data on each individual is a p ×t matrix. The n individuals themselves may be divided and randomly assigned to g groups. Analysis of these data using a MANOVA model, assuming that the data on an individual has a covariance matrix which is a Kronecker product of two positive definite matrices, is considered. The well-known Satterthwaite type approximation to the distribution of a quadratic form in normal variables is extended to the distribution of a multivariate quadratic form in multivariate normal variables. The multivariate tests using this approximation are developed for testing the usual hypotheses. Results are illustrated on a data set. A method for analysing unbalanced data is also discussed.     

Forecasting With Nonspurious Factors in U.S. Macroeconomic Time Series

This study examines the practical implications of the fact that structural changes in factor loadings can produce spurious factors (or irrelevant factors) in forecasting exercises. These spurious factors can induce an overfitting problem in factor-augmented forecasting models. To address this concern, we propose a method to estimate nonspurious factors by identifying the set of response variables that have no structural changes in their factor loadings. Our theoretical results show that the obtained set may include a fraction of unstable response variables. However, the fraction is so small that the original factors are able to be identified and estimated consistently. Moreover, using this approach, we find that a significant portion of 132 U.S. macroeconomic time series have structural changes in their factor loadings. Although traditional principal components provide eight or more factors, there are significantly fewer nonspurious factors. The forecasts using the nonspurious factors can significantly improve out-of-sample performance.     

Asymptotic inference in discrete response models

Regression models for discrete responses have found numerous applications. We consider logit, probit and cumulative logit models for qualitative data, and the loglinear and linear Poisson model for counted data. Statistical analysis of these models relies heavily on asymptotic likelihood theory, i.e. asymptotic properties of the maximum likelihood estimator and the likelihood ratio as well as related test statistics. In practical situations, previously published conditions assuring these properties may be too strong, or it is difficult to see whether they apply. This paper contributes to a clarification of this point and characterizes to some extent situations where asymptotic theory is applicable and where it is not. In particular, sharp upper bounds on the admissible growth of regressors are given.     

Variable selection in gamma regression models via artificial bee colony algorithm

Variable selection is an important task in regression analysis. Performance of the statistical model highly depends on the determination of the subset of predictors. There are several methods to select most relevant variables to construct a good model. However in practice, the dependent variable may have positive continuous values and not normally distributed. In such situations, gamma distribution is more suitable than normal for building a regression model. This paper introduces an heuristic approach to perform variable selection using artificial bee colony optimization for gamma regression models. We evaluated the proposed method against with classical selection methods such as backward and stepwise. Both simulation studies and real data set examples proved the accuracy of our selection procedure.     

Joint confidence regions in the multivariate calibration problem

Consider a vector valued response variable related to a vector valued explanatory variable through a normal multivariate linear model. The multivariate calibration problem deals with statistical inference on unknown values of the explanatory variable. The problem addressed is the construction of joint confidence regions for several unknown values of the explanatory variable. The problem is investigated when the variance covariance matrix is a scalar multiple of the identity matrix and also when it is a completely unknown positive definite matrix. The problem is solved in only two cases: (i) the response and explanatory variables have the same dimensions, and (ii) the explanatory variable is a scalar. In the former case, exact joint confidence regions are derived based on a natural pivot statistic. In the latter case, the joint confidence regions are only conservative. Computational aspects and the practical implementation of the confidence regions are discussed and illustrated using an example.     

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.     

Bayesian Estimation of Stochastic Frontier Models with Multivariate Skew t Error Terms

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.     

A clustering approach to detect multiple outliers in linear functional relationship model for circular data

Outlier detection has been used extensively in data analysis to detect anomalous observation in data. It has important applications such as in fraud detection and robust analysis, among others. In this paper, we propose a method in detecting multiple outliers in linear functional relationship model for circular variables. Using the residual values of the Caires and Wyatt model, we applied the hierarchical clustering approach. With the use of a tree diagram, we illustrate the detection of outliers graphically. A Monte Carlo simulation study is done to verify the accuracy of the proposed method. Low probability of masking and swamping effects indicate the validity of the proposed approach. Also, the illustrations to two sets of real data are given to show its practical applicability.     

Construction of Confidence Intervals for the IVIean of a Population Containing Many Zero Values

The likelihood ratio method is used to construct a confidence interval for a population mean when sampling from a population with certain characteristics found in many applications, such as auditing. Specifically, a sample taken from this type of population usually consists of a very large number of zero values, plus a small number of nonzero values that follow some continuous distribution. In this situation, the traditional confidence interval constructed for the population mean is known to be unreliable. This article derives confidence intervals based on the likelihood-ratio-test approach by assuming (1) a normal distribution (normal algorithm) and (2) an exponential distribution (exponential algorithm). Because the error population distribution is usually unknown, it is important to study the robustness of the proposed procedures. We perform an extensive simulation study to compare the percentage of confidence intervals containing the true population mean using the two proposed algorithms with the percentage obtained from the traditional method based on the central limit theorem. It is shown that the normal algorithm is the most robust procedure against many different distributional error assumptions.     

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.