A GENERALIZED EMERSON RECURRENCE RELATION

The Emerson (1968, Biometrics 24 , 695–701) recurrence relation has many important applications in statistics. However, the original derivation applied only to discrete distributions. In the following, a simple derivation is given that generalizes the Emerson recurrence relation to any distribution for which the necessary expectations exist. A modern application is outlined.     

Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

Fully efficient estimation of coefficients of correlation in the presence of imputed survey data

Marginal imputation, that consists of imputing items separately, generally leads to biased estimators of bivariate parameters such as finite population coefficients of correlation. To overcome this problem, two main approaches have been considered in the literature: the first consists of using customary imputation methods such as random hot‐deck imputation and adjusting for the bias at the estimation stage. This approach was studied in Skinner & Rao 2002 . In this paper, we extend the results of Skinner & Rao 2002 to the case of arbitrary sampling designs and three variants of random hot‐deck imputation. The second approach consists of using an imputation method, which preserves the relationship between variables. Shao & Wang 2002 proposed a joint random regression imputation procedure that succeeds in preserving the relationships between two study variables. One drawback of the Shao–Wang procedure is that it suffers from an additional variability (called the imputation variance) due to the random selection of residuals, resulting in potentially inefficient estimators. Following Chauvet, Deville, & Haziza 2011 , we propose a fully efficient version of the Shao–Wang procedure that preserves the relationship between two study variables, while virtually eliminating the imputation variance. Results of a simulation study support our findings. An application using data from the Workplace and Employees Survey is also presented. The Canadian Journal of Statistics 40: 124–149; 2012 © 2011 Statistical Society of Canada     

The Construction and Use of Approximations for Missing Quarterly Observations: A Model-Based Approach

We use a model-based approach to derive quarterly figures on several variables for the aggregate labor market in the Netherlands that are only observed annually. These approximations are conditional expectations derived from univariate and bivariate quarterly time series models for the series under consideration. They are subsequently used as proxies to estimate and analyze the structural labor market equations. Attention is given to the properties of estimation procedures based on proxy variables.     

Estimating Archimedean Copulas in High Dimensions

Abstract. This article presents a novel estimation procedure for high‐dimensional Archimedean copulas. In contrast to maximum likelihood estimation, the method presented here does not require derivatives of the Archimedean generator. This is computationally advantageous for high‐dimensional Archimedean copulas in which higher‐order derivatives are needed but are often difficult to obtain. Our procedure is based on a parameter‐dependent transformation of the underlying random variables to a one‐dimensional distribution where a minimum‐distance method is applied. We show strong consistency of the resulting minimum‐distance estimators to the case of known margins as well as to the case of unknown margins when pseudo‐observations are used. Moreover, we conduct a simulation comparing the performance of the proposed estimation procedure with the well‐known maximum likelihood approach according to bias and standard deviation.     

Instrumental variable estimation in ordinal probit models with mismeasured predictors

Researchers in the medical, health, and social sciences routinely encounter ordinal variables such as self‐reports of health or happiness. When modelling ordinal outcome variables, it is common to have covariates, for example, attitudes, family income, retrospective variables, measured with error. As is well known, ignoring even random error in covariates can bias coefficients and hence prejudice the estimates of effects. We propose an instrumental variable approach to the estimation of a probit model with an ordinal response and mismeasured predictor variables. We obtain likelihood‐based and method of moments estimators that are consistent and asymptotically normally distributed under general conditions. These estimators are easy to compute, perform well and are robust against the normality assumption for the measurement errors in our simulation studies. The proposed method is applied to both simulated and real data. The Canadian Journal of Statistics 47: 653–667; 2019 © 2019 Statistical Society of Canada     

On generalized moment identity and its applications: a unified approach

In this paper, we obtain a generalized moment identity for the case when the distributions of the random variables are not necessarily purely discrete or absolutely continuous. The proposed identity is useful to find the generator which has been used for the approximation of distributions by Stein's method. Apparently, a new approach is discussed for the approximation of distributions by Stein's method. We bring the characterization based on the relationship between conditional expectations and hazard measure in our unified framework. As an application, a new lower bound to the mean-squared error is obtained and it is compared with Bayesian Cramer–Rao bound.     

Bounds for moments through a general orthogonal expansion in a pre-hilbert space-I

Bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube.     

Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach

The exact density distribution of the non‐linear least squares estimator in the one‐parameter regression model is derived in closed form and expressed through the cumulative distribution function of the standard normal variable. Several proposals to generalize this result are discussed. The exact density is extended to the estimating equation (EE) approach and the non‐linear regression with an arbitrary number of linear parameters and one intrinsically non‐linear parameter. For a very special non‐linear regression model, the derived density coincides with the distribution of the ratio of two normally distributed random variables previously obtained by Fieler almost a century ago, unlike other approximations previously suggested by other authors. Approximations to the density of the EE estimators are discussed in the multivariate case. Numerical complications associated with the non‐linear least squares are illustrated, such as non‐existence and/or multiple solutions, as major factors contributing to poor density approximation. The non‐linear Markov–Gauss theorem is formulated on the basis of the near exact EE density approximation.     

Adaptive Nonparametric Comparison of Regression Curves

In this article, we develop a new and novel kernel density estimator for a sum of weighted averages from a single population based on utilizing the well defined kernel density estimator in conjunction with classic inversion theory. This idea is further developed for a kernel density estimator for the difference of weighed averages from two independent populations. The resulting estimator is “bootstrap-like” in terms of its properties with respect to the derivation of approximate confidence intervals via a “plug-in” approach. This new approach is distinct from the bootstrap methodology in that it is analytically and computationally feasible to provide an exact estimate of the distribution function through direct calculation. Thus, our approach eliminates the error due to Monte Carlo resampling that arises within the context of simulation based approaches that are oftentimes necessary in order to derive bootstrap-based confidence intervals for statistics involving weighted averages of i.i.d. random variables. We provide several examples and carry forth a simulation study to show that our kernel density estimator performs better than the standard central limit theorem based approximation in term of coverage probability.     

Gibbs sampling for marginal posterior expectations

In earlier work (Gelfand and Smith, 1990 and Gelfand et al, 1990) a sampling based approach using the Gibbs sampler was offered as a means for developing marginal posterior densities for a wide range of Bayesian problems several of which were previously inaccessible. Our purpose here is two-fold. First we flesh out the implementation of this approach for calculation of arbitrary expectations of interest. Secondly we offer comparison with perhaps the most prominent approach for calculating posterior expectations, analytic approximation involving application of the LaPlace method. Several illustrative examples are discussed as well. Clear advantages for the sampling based approach emerge.     

Bounds for moments through a general orthogonal expansion in a pre-hilbert space-II

In an earlier article [Canad. J. Statist., Vol, 3, No. 1, 1975, 13–34] bounds are obtained for the product moments of an arbitrary finite number of ordered random variables. These bounds are obtained with the help of a representation of an arbitrary function in terms of a complete orthonormal system in a pre-Hilbert space of square integrable functions defined in a k-dimensional unit cube. These results are extended to symmetric cases in this article.     

Measures of dependence for the multivariate t distribution with applications to the stock market

Two measures of dependence for multivariate t and Cauchy random variables are developed based on Kullback–Leibler number. The mutual information number T(X) is obtained in a closed expression form, as well as its asymptotic distribution. A dependence coefficient ρ₁, is defined (based on the Kullback–Leibler number) with the properties of ρ₁=0 indicating independence and ρ₁=1indicating degeneracy. Two real life examples from the stock market are used to analyze the level of dependence and correlation among stocks.     

Existence of every possible distribution for any sample order statistic

Given are any stated value for sample size and an arbitrary but specified univariate distribution. Let an arbitrary but specified order statistic for a univariate random sample of this size be considered. It is shown that a statistical population always exists, for yielding the sample, such that the distribution of the order statistic is exactly the specified distribution. Some asymptotic implications of these results are outlined.     

The log-beta Weibull regression model with application to predict recurrence of prostate cancer

We study the properties of the called log-beta Weibull distribution defined by the logarithm of the beta Weibull random variable (Famoye et al. in J Stat Theory Appl 4:121–136, 2005; Lee et al. in J Mod Appl Stat Methods 6:173–186, 2007). An advantage of the new distribution is that it includes as special sub-models classical distributions reported in the lifetime literature. We obtain formal expressions for the moments, moment generating function, quantile function and mean deviations. We construct a regression model based on the new distribution to predict recurrence of prostate cancer for patients with clinically localized prostate cancer treated by open radical prostatectomy. It can be applied to censored data since it represents a parametric family of models that includes as special sub-models several widely-known regression models. The regression model was fitted to a data set of 1,324 eligible prostate cancer patients. We can predict recurrence free probability after the radical prostatectomy in terms of highly significant clinical and pathological explanatory variables associated with the recurrence of the disease. The predicted probabilities of remaining free of cancer progression are calculated under two nested models.     

The exact proability that the roots of quadratic,cubic, and quartic equations are all real if the equation coefficients are random

Suppose that the coefficients of a polynomial equation are Independent random variables defined on subsets of real numbers, The purpose of this paper is to find the exact probability that all roots of a random polynomial equation are real. Since a polynomial equation of degree higher than four with arbitrary coefficients cannot be solved algrebraically, this paper will consider quadratic, cubic and quartic equations only. The general results are obtained in each case, Also, a number of special cases are furnished.     

LTD and RTI dependence orderings

The authors show how the approach of Capéra à & Genest (The Canadian Journal of Statistics, 1990) can be used to order bivariate distributions with arbitrary marginals by their degree of dependence in the LTD (left‐tail decreasing) or RTI (right‐tail increasing) sense. Some properties of these new orderings are given, along with applications to Archimedean copulas, order statistics and compound random variables.     

A cluster‐sample approach for Monte Carlo integration using multiple samplers

A computational problem in many fields is to estimate simultaneously multiple integrals and expectations, assuming that the data are generated by some Monte Carlo algorithm. Consider two scenarios in which draws are simulated from multiple distributions but the normalizing constants of those distributions may be known or unknown. For each scenario, existing estimators can be classified as using individual samples separately or using all the samples jointly. The latter pooled‐sample estimators are statistically more efficient but computationally more costly to evaluate than the separate‐sample estimators. We develop a cluster‐sample approach to obtain computationally effective estimators, after draws are generated for each scenario. We divide all the samples into mutually exclusive clusters and combine samples from each cluster separately. Furthermore, we exploit a relationship between estimators based on samples from different clusters to achieve variance reduction. The resulting estimators, compared with the pooled‐sample estimators, typically yield similar statistical efficiency but have reduced computational cost. We illustrate the value of the new approach by two examples for an Ising model and a censored Gaussian random field. The Canadian Journal of Statistics 41: 151–173; 2013 © 2012 Statistical Society of Canada     

How to Select Representative Samples

We give a formal definition of a representative sample, but roughly speaking, it is a scaled‐down version of the population, capturing its characteristics. New methods for selecting representative probability samples in the presence of auxiliary variables are introduced. Representative samples are needed for multipurpose surveys, when several target variables are of interest. Such samples also enable estimation of parameters in subspaces and improved estimation of target variable distributions. We describe how two recently proposed sampling designs can be used to produce representative samples. Both designs use distance between population units when producing a sample. We propose a distance function that can calculate distances between units in general auxiliary spaces. We also propose a variance estimator for the commonly used Horvitz–Thompson estimator. Real data as well as illustrative examples show that representative samples are obtained and that the variance of the Horvitz–Thompson estimator is reduced compared with simple random sampling.     

On generalized variance of product of powered components and multiple stable Tweedie models

A flexible family of multivariate models, named multiple stable Tweedie (MST) models, is introduced and produces generalized variance functions which are products of powered components of the mean. These MST models are built from a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are also real independent Tweedie variables, with the same dispersion parameter equal to the fixed component. In this huge family of MST models, generalized variance estimators are explicitly pointed out by maximum likelihood method and, moreover, computably presented for the uniform minimum variance and unbiased approach. The second estimator is brought from modified Lévy measures of MST which lead to some solutions of particular Monge–Ampère equations.