The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

It is well known that the ordinary least squares estimator of Xβ in the general linear model E _y = Xβ, cov y = σ² V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.     

A Note on Fitting a Regression without an Intercept Term

By entering the data (y _i ,x _i ) followed by (–y _i ,–x _i ), one can obtain an intercept-free regression Y = Xβ + ε from a program package that normally uses an intercept term. There is no bias in the resultant regression coefficients, but a minor postanalysis adjustment is needed to the residual variance and standard errors.     

Bayesian Hierarchical Kernelized Probabilistic Matrix Factorization

We present a Bayesian analysis framework for matrix-variate normal data with dependency structures induced by rows and columns. This framework of matrix normal models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of prediction uncertainty, model structure search, and extensions to multidimensional arrays. Compared with Bayesian probabilistic matrix factorization, which integrates a Gaussian prior for single row of the data matrix, our proposed model, namely Bayesian hierarchical kernelized probabilistic matrix factorization, imposes Gaussian Process priors over multiple rows of the matrix. Hence, the learned model explicitly captures the underlying correlation among the rows and the columns. In addition, our method requires no specific assumptions like independence of latent factors for rows and columns, which obtains more flexibility for modeling real data compared to existing works. Finally, the proposed framework can be adapted to a wide range of applications, including multivariate analysis, times series, and spatial modeling. Experiments highlight the superiority of the proposed model in handling model uncertainty and model optimization.     

A Numerical Likelihood-Based Approach to Combining Correlation Matrices

Numerical approaches to developing accurate and efficient approximations to combined likelihoods of population correlation matrices in meta-analysis under normality assumptions for the data are studied. The likelihood is expressed as a multiple integral over the unit cube in (p ? 1)-dimensional space, where p is the row and column dimensionality of the correlation matrix. Three types of computation are proposed as ways to calculate the likelihood for any population correlation matrix P. As an application, inference is explored concerning intercorrelations among math, spatial and verbal scores in a SAT exam. Comparisons are made with conventional methods.     

A geometric process warranty model using a combination policy

This paper proposes a geometric process warranty model. Assume that a combination policy (W, T) is applied after selling a product, so that a free warranty is offered in [0, W), followed by a pro-rata warranty in [W, T). Assume further the successive operating times (repair times) of the product form a decreasing (increasing) geometric process. The average cost rate of the product to the manufacturer and a consumer can be derived respectively. For exponential distribution case, the explicit formulas of the average cost rate are obtained, and an finite algorithm for determination of an optimal combination policy is suggested.     

High breakdown point robust estimators with missing data

In this paper, we propose a new procedure to estimate the distribution of a variable y when there are missing data. To compensate the presence of missing responses, it is assumed that a covariate vector x is observed and that y and x are related by means of a semi-parametric regression model. Observed residuals are combined with predicted values to estimate the missing response distribution. Once the responses distribution is consistently estimated, we can estimate any parameter defined through a continuous functional T using a plug in procedure. We prove that the proposed estimators have high breakdown point.     

The sampling distribution of the W estimator of the number of valid signatures on a petition

Abstract

A new non linear estimator, W, for the number of valid, unique signatures on a petition has been shown better, for the cases enumerated and with certain restrictions, than a popular Goodman-type statistic, G. This article extends those results with relaxed conditions by developing the exact probability mass function and mean of W and a close approximation of the variance (Var(W)). If the proportion of valid signatures among unique and duplicated signatures is the same, then Var(W) is approximately a function of the means and variances of the two sample statistics. Using the delta method, we estimate Var(W), with the resulting approximation shown to be good, even when the condition of equal proportions does not hold. We compare W to G and establish which estimator is preferred for different intervals of the design parameters. Data from a Washington State petition illustrate the findings.     

Partly linear models on Riemannian manifolds

In partly linear models, the dependence of the response y on (x ^T, t) is modeled through the relationship y=x ^T β+g(t)+?, where ? is independent of (x ^T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO₂ pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.     

Best Quadratic Unbiased Prediction in a General Linear Model with Stochastic Regression Coefficients

In this article, we discuss on how to predict a combined quadratic parametric function of the form β′ H β + hσ² in a general linear model with stochastic regression coefficients denoted by y  =  X β +  e . Firstly, the quadratic predictability of β′ H β + hσ² is investigated to obtain a quadratic unbiased predictor (QUP) via a general method of structuring an unbiased estimator. This QUP is also optimal in some situations and therefore we hope it will be a fine predictor. To show this idea, we apply the Lagrange multipliers method to this problem and finally reach the expected conclusion through permutation matrix techniques.     

Extending Product Method of Estimation to Positive Correlation Case in Surveys

The product method of estimation (Murthy, 1964) complements the ratio method when the study variate, y, and an auxiliary variate, x, have negative correlation. However, such cases are not frequent in survey practice. This paper suggests a simple transformation of x in the more common situation of positive correlation between y and x, to permit a product method of estimation rather than a ratio method. This leads to the advantage that the bias and mean square error have exact expressions. The technique developed by Quenouille (1956) and applied by Shukla (1976) is used for making the estimator unbiased. The minimum variance situation is investigated. Two numerical examples are included. The case of negative correlation is also examined.     

Zonal Polynomials and Hypergeometric Functions of Quaternion Matrix Argument

We define zonal polynomials of quaternion matrix argument and deduce some impor-tant formulae of zonal polynomials and hypergeometric functions of quaternion matrix argument. As an application, we give the distributions of the largest and smallest eigenvalues of a quaternion central Wishart matrix W ～ ?W(n, Σ), respectively.     

Monotonicity Conditions and Inequality Imputation for Sample-Selection and Non-Response Problems

ABSTRACT

Under a sample selection or non-response problem, where a response variable y is observed only when a condition δ = 1 is met, the identified mean E(y|δ = 1) is not equal to the desired mean E(y). But the monotonicity condition E(y|δ = 1) ≤ E(y|δ = 0) yields an informative bound E(y|δ = 1) ≤ E(y), which is enough for certain inferences. For example, in a majority voting with δ being the vote-turnout, it is enough to know if E(y) > 0.5 or not, for which E(y|δ = 1) > 0.5 is sufficient under the monotonicity. The main question is then whether the monotonicity condition is testable, and if not, when it is plausible. Answering to these queries, when there is a ‘proxy’ variable z related to y but fully observed, we provide a test for the monotonicity; when z is not available, we provide primitive conditions and plausible models for the monotonicity. Going further, when both y and z are binary, bivariate monotonicities of the type P(y, z|δ = 1) ≤ P(y, z|δ = 0) are considered, which can lead to sharper bounds for P(y). As an empirical example, a data set on the 1996 U.S. presidential election is analyzed to see if the Republican candidate could have won had everybody voted, i.e., to see if P(y) > 0.5, where y = 1 is voting for the Republican candidate.     

Testing sphericity using small samples

It is shown here that with small sample sizes, the null distribution of the ellipticity, or sphericity, likelihood criterion W(n, p) can be obtained very accurately, either by computation using the Meijer function, or by Monte Carlo simulation. Testing in repeated measures design can now be carried out with much more accuracy.     

A recipe for bivariate copulas

ABSTRACT

We give conditions on a ? ?1, b ∈ ( ? ∞, ∞), and f and g so that C_{a, b}(x, y) = xy(1 + af(x)g(y))^b is a bivariate copula. Many well-known copulas are of this form, including the Ali–Mikhail–Haq Family, Huang–Kotz Family, Bairamov–Kotz Family, and Bekrizadeh–Parham–Zadkarmi Family. One result is that we produce an algorithm for producing such copulas. Another is a one-parameter family of copulas whose measures of concordance range from 0 to 1.     

Inverse prediction for heteroscedastic response using mixed models software

Distributions of a response y (height, for example) differ with values of a factor t (such as age). Given a response y_* for a subject of unknown t_*, the objective of inverse prediction is to infer the value of t_* and to provide a defensible confidence set for it. Training data provide values of y observed on subjects at known values of t. Models relating the mean and variance of y to t can be formulated as mixed (fixed and random) models in terms of sets of functions of t, such as polynomial spline functions. A confidence set on t_* can then be had as those hypothetical values of t for which y_* is not detected as an outlier when compared to the model fit to the training data. With nonconstant variance, the p-values for these tests are approximate. This article describes how versatile models for this problem can be formulated in such a way that the computations can be accomplished with widely available software for mixed models, such as SAS PROC MIXED. Coverage probabilities of confidence sets on t_* are illustrated in an example.     

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.     

Equivariant estimators of the covariance matrix

Given a Wishart matrix S [S ∽ W_p(n, Σ)] and an independent multinomial vector X [X ∽ N_p (μ, Σ)], equivariant estimators of Σ are proposed. These estimators dominate the best multiple of S and the Stein-type truncated estimators.     

Adjustment with three continuous variables

Correlation is not causation. Spurious association between X and Y may be due to a confounding variable W. Statisticians may adjust for W using a variety of techniques. This article presents the results of simulations conducted to assess the performance of these techniques under various, elementary, data-generating processes. The results indicate that no technique is best overall and that specific techniques should be selected based on the particulars of the data-generating process. Here, we show how causal graphs can guide the selection or design of techniques for statistical adjustment. R programs are provided for researchers interested in generalization.     

Testing linear hypotheses in high-dimensional regressions

For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative hypothesis requires complex analytic approximations, and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say p≤20. On the other hand, assuming that the data dimension p as well as the number q of regression variables are fixed while the sample size n grows, several asymptotic approximations are proposed in the literature for Wilk's Λ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension p and a large sample size n. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null hypothesis and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large p and large n context, but also for moderately large data dimensions such as p=30 or p=50. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in multivariate analysis of variance which is valid for high-dimensional data.