On the robustness of the linear hypothesis test procedure in the singular linear model with implied restrictions

The linear hypothesis test procedure is considered in the restricted linear modelsM _r = {y, Xβ |Rβ = 0, σ ²V} andM _r ^* = {y, Xβ |ARβ = 0, σ ²V}. Necessary and sufficient conditions are derived under which the statistic providing anF-test for the linear hypothesisH ₀:Kβ=0 in the modelM _r ^* (M_r) continues to be valid in the modelM _r (M _r ^* ); the results obtained cover the case whereM _r ^* is replaced by the general Gauss-Markov modelM = {y, Xβ, σ ²V}.     

On misspecification of the dispersion matrix in mixed linear models

The general mixed linear model can be written y = Xβ + Zu + e, where β is a vector of fixed effects, u is a vector of random effects and e is a vector of random errors. In this note, we mainly aim at investigating the general necessary and sufficient conditions under which the best linear unbiased estimator for \varvec r(\varvec l, \varvec m) = \varvec l\varvec ￠\varvec b+\varvec m\varvec ￠\varvec u{\varvec \varrho}({\varvec l}, {\varvec m}) = {\varvec l}{\varvec '}{\varvec \beta}+{\varvec m}{\varvec '}{\varvec u} is also optimal under the misspecified model. In addition, we offer approximate conclusions in some special situations including a random regression model.     

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.     

Necessary sample sizes for categorial data

This paper deals with the problem how to determine the necessary sample size for the estimation of the parameter π=(π1,...,πk) (πj ≥ 0, Σjπj=1) based on the vector f=(f1,...,fk) of relative frequencies with sample size n. The vector n-f has a multinomial distribution. For a given precision c, 0≤c≤1, and a given confidence number β, 0≤β≤1, there exists a smallest positive integer N⁰=N⁰(β, c, k) with P{|fj−πj|≤c; j=1, ...,k}≥β for all sample sizes n≥N⁰ and for all π. As results are given in this paper exact upper bounds for N⁰ and an improved asymptotical upper bound for N⁰ which is derived from the asymptotical multinormal approximation for the distribution of f.     

Dispersive ordering—Some applications and examples

A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F ⁻¹ and G ⁻¹ be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤ _disp X) if G ⁻¹(β)−G ⁻¹(α)≤F ⁻¹(β)−F ⁻¹(α), for all 0<α≤β<1. This means that the difference between any two quantiles of G is smaller than the difference between the corresponding quantiles of F. A consequence of Y≤ _disp X is that |Y ₁−Y ₂| is stochastically smaller than |X ₁−X ₂| and this in turn implies var(Y)≤var(X) as well as E[|Y ₁−Y ₂|]≤E[|X ₁−X ₂|], where X ₁, X ₂ (Y ₁, Y ₂) are two independent copies of X(Y). In this review paper, we give several examples and applications of dispersive ordering in statistics. Examples include those related to order statistics, spacings, convolution of non-identically distributed random variables and epoch times of non-homogeneous Poisson processes. This work was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University. Subhash Kochar is thankful to Dr. B. Khaledi for many helpful discussions.     

Regression analysis using order statistics

In this article, we study the joint distribution of X and two linear combinations of order statistics, a ^T Y ₍₂₎ and b ^T Y ₍₂₎, where a = (a ₁, a ₂) ^T and b = (b ₁, b ₂) ^T are arbitrary vectors in R ² and Y ₍₂₎ = (Y ₍₁₎, Y ₍₂₎) ^T is a vector of ordered statistics obtained from (Y ₁, Y ₂) ^T when (X, Y ₁, Y ₂) ^T follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and Loperfido (Test 17:370–380, 2008).     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

The multivariate linear model with multivariatet and intra-class covariance structure

The prediction distribution of future responses from a multivariate linear model with error having a multivariatet-distribution and intra-class covariance structure has been derived. The distribution depends on ρ, the intra-class correlation coefficient. For unknown ρ, the marginal likelihood function of ρ has been obtained and the prediction distribution has been approximated by the estimate of ρ. As an application, a β-expectation tolerance region for the model has been constructed.     

Estimation of the SURE model

The paper presents the essentials of the SURE model and the estimation of its parameters β and ω. Two alternative compact representations of the model are being used. The parameter β is estimated by least squares (LS), generalized least squares (GLS) and maximum likelihood (ML) (under normality). For ω two estimators are being considered, viz an LS-related estimator and a maximum likelihood estimator (under normality). Attention is being given to the study of asymptotic properties of all estimators examined. It turns out that the LS-related and ML estimators of ω follow the same asymptotic (normal) distribution. Efficiency comparisons for the various estimators of β conclude the paper.     

Prediction intervals for growth curves

The growth curve model Y^n×p = A^{n×p ξ mtimes;k}B^k×p+ E^nxp, where Y is an observation matrix, &sigma is a matrix of unknown parameters, A is a known matrix of rank m, B is a known matrix of rank k with 1'= (1, …, 1) as its first row, and the rows of E are independent each distributed as N^p(0,Σ,) is considered. The problem of constructing the prediction intervals for future observations using the above model is considered and approximate intervals assuming different structures on σ are derived. The results are illustrated with several data sets.     

The number of records within a random interval of the current record value

LetX ₁,X ₂, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ _n ⁻ , μ _n ⁺ be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ _n ⁻ , μ _n ⁺ are studied in the present paper. Some statistical applications connected with these variables are also provided.     

Improved estimation of the population parameters when some additional information is available

Estimation of population parameters is considered by several statisticians when additional information such as coefficient of variation, kurtosis or skewness is known. Recently Wencheko and Wijekoon (Stat Papers 46:101–115, 2005) have derived minimum mean square error estimators for the population mean in one parameter exponential families when coefficient of variation is known. In this paper the results presented by Gleser and Healy (J Am Stat Assoc 71:977–981, 1976) and Arnholt and Hebert (, 2001) were generalized by considering T (X) as a minimal sufficient estimator of the parametric function g(θ) when the ratio t²=[ g(q) ]^-2Var[ T(X ) ]{\tau^{2}=[ {g(\theta )} ]^{-2}{\rm Var}[ {T(\boldsymbol{X} )} ]} is independent of θ. Using these results the minimum mean square error estimator in a certain class for both population mean and variance can be obtained. When T (X) is complete and minimal sufficient, the ratio τ² is called “WIJLA” ratio, and a uniformly minimum mean square error estimator can be derived for the population mean and variance. Finally by applying these results, the improved estimators for the population mean and variance of some distributions are obtained.     

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.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.     

A characterization of the dilation order and its applications

LetX andY be two random variables with finite expectationsE X andE Y, respectively. ThenX is said to be smaller thanY in the dilation order ifE[ϕ(X-E X)]≤E[ϕ(Y-E Y)] for any convex functionϕ for which the expectations exist. In this paper we obtain a new characterization of the dilation order. This characterization enables us to give new interpretations to the dilation order, and using them we identify conditions which imply the dilation order. A sample of applications of the new characterization is given. Partially supported by MURST 40% Program on Non-Linear Systems and Applications. Partially supported by “Gruppo Nazionale per l'Analisi Funzionale e sue Applicazioni”—CNR.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

On the BLUEs in Two Linear Models via C. R. Rao's Pandora's Box

We consider the estimation of the parameters in two partitioned linear models, denoted by 𝒜 = {y, X ₁ β ₁ + X ₂ β ₂, V _𝒜} and ? = {y, X ₁ β ₁ + X ₂ β ₂, V _?}, which we call full models. Correspondingly, we define submodels 𝒜₁ = {y, X ₁ β ₁, V _𝒜} and ?₁ = {y, X ₁ β ₁, V _?}. Using the so-called Pandora's Box approach introduced by Rao (1971 Rao , C. R. ( 1971 ). Unified theory of linear estimation . Sankhy?, Ser. A 33 : 371 – 394 . [Corrigendum (1972), 34, p. 194, 477.]  [Google Scholar], we give new necessary and sufficient conditions for the equality between the best linear unbiased estimators (BLUEs) of X ₁ β ₁ under 𝒜₁ and ?₁ as well as under 𝒜 and ?. In our considerations we will utilise the Frisch–Waugh–Lovell theorem which provides a connection between the full model 𝒜 and the reduced model 𝒜 _r  = {M ₂ y, M ₂ X ₁ β ₁, M ₂ V _𝒜 M ₂} with M ₂ being an appropriate orthogonal projector. Moreover, we consider the equality of the BLUEs under the full models assuming that they are equal under the submodels.     

Sequential design of computer experiments for the estimation of a probability of failure

This paper deals with the problem of estimating the volume of the excursion set of a function f:ℝ ^d →ℝ above a given threshold, under a probability measure on ℝ ^d that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of f and aim at performing evaluations of f as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

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.