Two graphical displays for the detection of potentially influential subsets in regression

In the context of the general linear model Y=Xβ+ε, the matrix P_z =Z(Z^TZ)^?1 Z^T , where Z=(X: Y), plays an important role in determining least squares results. In this article we propose two graphical displays for the off-diagonal as well as the diagonal elements of P_Z . The two graphs are based on simple ideas and are useful in the detection of potentially influential subsets of observations in regression. Since P_Z is invariant with respect to permutations of the columns of Z, an added advantage of these graphs is that they can be used to detect outliers in multivariate data where the rows of Z are usually regarded as a random sample from a multivariate population. We also suggest two calibration points, one for the diagonal elements of P_Z and the other for the off-diagonal elements. The advantage of these calibration points is that they take into consideration the variability of the off-diagonal as well as the diagonal elements of P_Z . They also do not suffer from masking.     

Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

In the investigation of the restricted linear model ? _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.     

Application domains for the Delta method

The Delta method uses truncated Lagrange expansions of statistics to obtain approximations to their distributions. In this paper, we consider statistics Y=g(μ+X), where X is any random vector. We obtain domains 𝒟 such that, when μ∈𝒟, we may apply the distribution derived from the Delta method. Namely, we will consider an application on the normal case to illustrate our approach.     

Generalized Sobol sensitivity indices for dependent variables: numerical methods

The hierarchically orthogonal functional decomposition of any measurable function η of a random vector X=(X₁,?…?, X_p) consists in decomposing η(X) into a sum of increasing dimension functions depending only on a subvector of X. Even when X₁,?…?, X_p are assumed to be dependent, this decomposition is unique if the components are hierarchically orthogonal. That is, two of the components are orthogonal whenever all the variables involved in one of the summands are a subset of the variables involved in the other. Setting Y=η(X), this decomposition leads to the definition of generalized sensitivity indices able to quantify the uncertainty of Y due to each dependent input in X [Chastaing G, Gamboa F, Prieur C. Generalized Hoeffding–Sobol decomposition for dependent variables – application to sensitivity analysis. Electron J Statist. 2012;6:2420–2448]. In this paper, a numerical method is developed to identify the component functions of the decomposition using the hierarchical orthogonality property. Furthermore, the asymptotic properties of the components estimation is studied, as well as the numerical estimation of the generalized sensitivity indices of a toy model. Lastly, the method is applied to a model arising from a real-world problem.     

Questions & Answers: The Role of Mathematical Models in Clinical Research

Given any generalized inverse (X'X)^? appropriate to normal equations X'Xb ⁰ = X'y for the linear model y = Xb + e, a procedure is given for obtaining from it a generalized inverse appropriate to a restricted model having restrictions P'b = 0 for P'b nonestimable.     

Best quadratic predictions in finite populations

For a finite population and the resulting linear model Y=Xβ+e, the problem of the optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor for the population quadratic quantities, f(H)=Y′HY , is of interest and has been previously considered in the literature for the case of HX=0. However, the special case does not contain all of situations at all. So, predicting f(H) in general situations may be of particular interest. In this paper, we make an effort to investigate how to offer a good predictor for f(H), not restricted yet to the mentioned case. Permutation matrix techniques play an important role in handling the process. The expected predictors are finally derived. In addition, we mention that the resulting predictors can be viewed as acceptable in all situations.     

Spatial Regression Models Using Inter-Region Distances in a Non-Random Context

This article considers spatial data z( s ₁), z( s ₂),…, z( s _n ) collected at n locations, with the objective of predicting z( s ₀) at another location. The usual method of analysis for this problem is kriging, but here we introduce a new signal-plus-noise model whose essential feature is the identification of hot spots. The signal decays in relation to distance from hot spots. We show that hot spots can be located with high accuracy and that the decay parameter can be estimated accurately. This new model compares well to kriging in simulations.     

Managing Serials in an Institutional Repository: An Interview With Kay Teel

ABSTRACT

In this interview from March 2017, Kay Teel, metadata librarian for serials and arts resources at the Stanford University Libraries, discusses the issues involved in providing access to serials through an institutional repository.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution

Recently, exact confidence bounds and exact likelihood inference have been developed based on hybrid censored samples by Chen and Bhattacharyya [Chen, S. and Bhattacharyya, G.K. (1998). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics—Theory and Methods, 17, 1857–1870.], Childs et al. [Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319–330.], and Chandrasekar et al. [Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistics, 51, 994–1004.] for the case of the exponential distribution. In this article, we propose an unified hybrid censoring scheme (HCS) which includes many cases considered earlier as special cases. We then derive the exact distribution of the maximum likelihood estimator as well as exact confidence intervals for the mean of the exponential distribution under this general unified HCS. Finally, we present some examples to illustrate all the methods of inference developed here.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.     

Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

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.     

Improved ridge estimators in a linear regression model

In this paper, the notion of the improved ridge estimator (IRE) is put forward in the linear regression model y=X β+e. The problem arises if augmenting the equation 0=c′α+ε instead of 0=C α+? to the model. Three special IREs are considered and studied under the mean-squared error criterion and the prediction error sum of squares criterion. The simulations demonstrate that the proposed estimators are effective and recommendable, especially when multicollinearity is severe.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

A very simple proof of the multivariate Chebyshev's inequality

Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.     

A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

The n.s. conditions for the ration of two quadratic forms to have an f-distribution and its applications

Suppose that ξ and η be two random vectors and that (ξ^τ, η^τ have an elliptically contoured distribution or a multivariate normal distribution. In this article, we obtain some necessary and sufficient (N.S.) conditions such that the ratio of two quadratic forms, say ξ^τ Aξ and η^τ Bη(for some symmetric nonnegative matrices A and B), has an F-distribution. As applications, we extend the classical F-test to some dependent two group samples. Two cases are considered: elliptically contoured and normal distributions.     

The pitfall of instrumental variables in big data: What the rule of thumb can't give you

ABSTRACT

Background: Instrumental variables (IVs) have become much easier to find in the “Big data era” which has increased the number of applications of the Two-Stage Least Squares model (TSLS). With the increased availability of IVs, the possibility that these IVs are weak has increased. Prior work has suggested a ‘rule of thumb’ that IVs with a first stage F statistic at least ten will avoid a relative bias in point estimates greater than 10%. We investigated whether or not this threshold was also an efficient guarantee of low false rejection rates of the null hypothesis test in TSLS applications with many IVs.

Objective: To test how the ‘rule of thumb’ for weak instruments performs in predicting low false rejection rates in the TSLS model when the number of IVs is large.

Method: We used a Monte Carlo approach to create 28 original data sets for different models with the number of IVs varying from 3 to 30. For each model, we generated 2000 observations for each iteration and conducted 50,000 iterations to reach convergence in rejection rates. The point estimate was set to 0, and probabilities of rejecting this hypothesis were recorded for each model as a measurement of false rejection rate. The relationship between the endogenous variable and IVs was carefully adjusted to let the F statistics for the first stage model equal ten, thus simulating the ‘rule of thumb.’

Results: We found that the false rejection rates (type I errors) increased when the number of IVs in the TSLS model increased while holding the F statistics for the first stage model equal to 10. The false rejection rate exceeds 10% when TLSL has 24 IVs and exceed 15% when TLSL has 30 IVs.

Conclusion: When more instrumental variables were applied in the model, the ‘rule of thumb’ was no longer an efficient guarantee for good performance in hypothesis testing. A more restricted margin for F statistics is recommended to replace the ‘rule of thumb,’ especially when the number of instrumental variables is large.