A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model

Let [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions \varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models M = {\varvecy, \varvecX\varvecb, s²\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and M_r = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s²\varvecV}{{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}, respectively. A bound for the Euclidean distance between [^(\varveck)]{\widehat{\varvec{\kappa}}} and [^(\varveck)]_r{\widehat{\varvec{\kappa}}_r} is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model M{{\mathcal M}} and sub-restricted model M_r^*{{\mathcal M}_r^*} containing an essential part of the restrictions \varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating \varveck{\varvec{\kappa}}.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.     

On comparison of dispersion matrices of estimators under a constrained linear model

We introduce some new mathematical tools in the analysis of dispersion matrices of the two well-known OLSEs and BLUEs under general linear models with parameter restrictions. We first establish some formulas for calculating the ranks and inertias of the differences of OLSEs’ and BLUEs’ dispersion matrices of parametric functions under the general linear model \({\mathscr {M}}= \{\mathbf{y}, \ \mathbf{X }\pmb {\beta }, \ \pmb {\Sigma }\}\) and the constrained model \({\mathscr {M}}_r = \{\mathbf{y}, \, \mathbf{X }\pmb {\beta }\, | \, \mathbf{A }\pmb {\beta }= \mathbf{b}, \ \pmb {\Sigma }\}\), where \(\mathbf{A }\pmb {\beta }= \mathbf{b}\) is a consistent linear matrix equation for the unknown parameter vector \(\pmb {\beta }\) to satisfy. As applications, we derive necessary and sufficient conditions for many equalities and inequalities of OLSEs’ and BLUEs’ dispersion matrices to hold under \({\mathscr {M}}\) and \({\mathscr {M}}_r\).     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.     

Best quadratic unbiased estimation of the variance matrix in normal regression

In 1973 Balestra examined the linear model y=XB+u, where u is a normally distributed disturbance vector, with variance matrix Ω. Ω has spectral decomposition \(\sum\limits_{i = 1}^r {\lambda _i M_i } \) , and the matrices M_i are known. Estimation of ω is thus equivalent with estimation of the λ_i. Balestra presented the best quadratic unbiased estimator of λ_i. In the present paper a derivation will be given which is based on a procedure developed by this writer (1980).     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

L models and multiple regressions designs

Given an orthogonal model

${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$

an L model

${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$

is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L ₁, . . . , L _c ). In multiple regression designs this matrix will be of the form

${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$

with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.

     

Functional single-index quantile regression models

It is known that functional single-index regression models can achieve better prediction accuracy than functional linear models or fully nonparametric models, when the target is to predict a scalar response using a function-valued covariate. However, the performance of these models may be adversely affected by extremely large values or skewness in the response. In addition, they are not able to offer a full picture of the conditional distribution of the response. Motivated by using trajectories of $$\hbox {PM}_{{10}}$$ concentrations of last day to predict the maximum $$\hbox {PM}_{{10}}$$ concentration of the current day, a functional single-index quantile regression model is proposed to address those issues. A generalized profiling method is employed to estimate the model. Simulation studies are conducted to investigate the finite sample performance of the proposed estimator. We apply the proposed framework to predict the maximal value of $$\hbox {PM}_{{10}}$$ concentrations based on the intraday $$\hbox {PM}_{{10}}$$ concentrations of the previous day.     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X ₁and X ₂ are identified in terms of relations between the conditional expectation of ( X_2:2² -X_1:2²)^r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X _1:2 (or X_2:2^2k{X_{2:2}^{2k}} given X_1:2,"k ￡ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X _1:2 and X _2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.     

Best quadratic unbiased estimation in variance-covariance component models 2

This paper deals with the existence of best quadratic unbiased estimators in variance covariance component models. It extends and unifies results previously obtained by Seely, Zyskind, Klonecki, Zmy?lony, Gnot, Kleffe and Pincus. The author considers a quasinormally distributed random vector y such that Ey = Xβ, Cov y∈L, where L is a linear space of symmetric square matrices. Conditions for the existence of a BLUE of Ey and a BQUE of Cov y (Eyy′) are investigated. A BLUE exists iff symmetry conditions for certain matrices are met while a BQUE exists iff some modified quadratic subspace conditions are met. At the end of the paper three examples are studied in which all these conditions are met: The Random Coefficient Regression Model, the multivariate linear model and the Behrens-Fisher model. The proofs of the theorems are obtained by considering linear model in y and yy′, respectively.     

Estimation of a normal mean relative to balanced loss functions

LetX ₁,…,X _nbe a random sample from a normal distribution with mean θ and variance σ². The problem is to estimate θ with Zellner's (1994) balanced loss function, % MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean % MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form % MathType!End!2!1! under % MathType!End!2!1!. We also consider the weighted balanced loss function, % MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e ^θ .     

Let us do the twist again

Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.     

On the natural restrictions in the singular Gauss–Markov model

A Gauss–Markov model is said to be singular if the covariance matrix of the observable random vector in the model is singular. In such a case, there exist some natural restrictions associated with the observable random vector and the unknown parameter vector in the model. In this paper, we derive through the matrix rank method a necessary and sufficient condition for a vector of parametric functions to be estimable, and necessary and sufficient conditions for a linear estimator to be unbiased in the singular Gauss–Markov model. In addition, we give some necessary and sufficient conditions for the ordinary least-square estimator (OLSE) and the best linear unbiased estimator (BLUE) under the model to satisfy the natural restrictions.      

Simultaneous marginal survival estimators when doubly censored data is present

A doubly censoring scheme occurs when the lifetimes T being measured, from a well-known time origin, are exactly observed within a window [L, R] of observational time and are otherwise censored either from above (right-censored observations) or below (left-censored observations). Sample data consists on the pairs (U, δ) where U = min{R, max{T, L}} and δ indicates whether T is exactly observed (δ = 0), right-censored (δ = 1) or left-censored (δ = −1). We are interested in the estimation of the marginal behaviour of the three random variables T, L and R based on the observed pairs (U, δ). We propose new nonparametric simultaneous marginal estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} for the survival functions of T, L and R, respectively, by means of an inverse-probability-of-censoring approach. The proposed estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} are not computationally intensive, generalize the empirical survival estimator and reduce to the Kaplan-Meier estimator in the absence of left-censored data. Furthermore, [^(S)]_T{\hat S_{T}} is equivalent to a self-consistent estimator, is uniformly strongly consistent and asymptotically normal. The method is illustrated with data from a cohort of drug users recruited in a detoxification program in Badalona (Spain). For these data we estimate the survival function for the elapsed time from starting IV-drugs to AIDS diagnosis, as well as the potential follow-up time. A simulation study is discussed to assess the performance of the three survival estimators for moderate sample sizes and different censoring levels.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Minimax prediction in the linear model with a relative squared error

Arnold and Stahlecker (Stat Pap 44:107–115, 2003) considered the prediction of future values of the dependent variable in the linear regression model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution d* of that problem. In the paper we generalize this result proving that the decision rule d* is also minimax when the class D{\mathcal{D}} of possible predictors of the dependent variable is unrestricted. Then we show that d* remains minimax in D{\mathcal{D}} when the disturbances are random with the mean vector zero and the known positive definite covariance matrix.     

Biased estimation of a variance

Summary Let , whereX _i are i.i.d. random variables with a finite variance σ² and is the usual estimate of the mean ofX _i. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S ²(σ)−σ²|^k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian cases we present Monte Carlo results regarding Pitman's criteron. This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade.     

Computations of predictors/estimators under a linear random-effects model with parameter restrictions

This paper considers the problem of simultaneously predicting/estimating unknown parameter spaces in a linear random-effects model with both parameter restrictions and missing observations. We shall establish explicit formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameters in such a model, and derive a variety of mathematical and statistical properties of the BLUPs under general assumptions. We also discuss some matrix expressions related to the covariance matrix of the BLUP, and present various necessary and sufficient conditions for several equalities and inequalities of the covariance matrix of the BLUP to hold.