Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(_β1,…,_βp)$\beta =({\beta }_1,\dots ,{\beta }_p)$ and independent and identically N(0,σ²)$N(0,{\sigma }^2)$ distributed errors. Suppose that the parameter of interest is θ=a^Tβ$\theta =a^T\beta$ where a$a$ is a specified vector. Define the parameter τ=c^Tβ-t$\tau =c^T\beta -t$ where the vector c$c$ and the number t$t$ are specified and a$a$ and c$c$ are linearly independent. Also suppose that we have uncertain prior information that τ=0$\tau =0$. We present a new frequentist 1-α$1-\alpha$ confidence interval for θ$\theta$ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α$1-\alpha$ confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0$\tau =0$. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ$\tau$ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2$2\times 2$ factorial experiment with 20 replicates. Suppose that the parameter of interest θ$\theta$ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ$\theta$ that utilizes this prior information.     

Goodness-of-fit tests for parametric regression with selection biased data

Consider the nonparametric location-scale regression model Y=m(X)+σ(X)ε$Y=m(X)+\sigma (X)\epsilon$, where the error ε$\epsilon$ is independent of the covariate X$X$, and m$m$ and σ$\sigma$ are smooth but unknown functions. The pair (X,Y)$(X,Y)$ is allowed to be subject to selection bias. We construct tests for the hypothesis that m(·)$m(·)$ belongs to some parametric family of regression functions. The proposed tests compare the nonparametric maximum likelihood estimator (NPMLE) based on the residuals obtained under the assumed parametric model, with the NPMLE based on the residuals obtained without using the parametric model assumption. The asymptotic distribution of the test statistics is obtained. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, the finite sample performance of the proposed tests is studied in a simulation study, and the developed tests are applied on environmental data.     

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y$y$ on x$x$ given by a pair of mean and variance functions with a parameter vector θ$\theta$ to be estimated that also appears in the distribution of the regressor variable x$x$. The estimation of θ$\theta$ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y$y$ unbiased estimating functions. Of special interest is the case where the distribution of x$x$ depends only on a subvector α$\alpha$ of θ$\theta$, which may be considered a nuisance parameter. In general, α$\alpha$ must be estimated simultaneously together with the rest of θ$\theta$, but there are cases where α$\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.     

Quasi-minimax estimation in the general linear regression model

This paper is mainly concerned with minimax estimation in the general linear regression model y=Xβ+ε$y=X\beta +\epsilon$ under ellipsoidal restrictions on the parameter space and quadratic loss function. We confine ourselves to estimators that are linear in the response vector y  . The minimax estimators of the regression coefficient β$\beta$ are derived under homogeneous condition and heterogeneous condition, respectively. Furthermore, these obtained estimators are the ridge-type estimators and mean dispersion error (MDE) superior to the best linear unbiased estimator b=(X^′W^-1X)^-1X^′W^-1y$b=(X^{\prime }{W}^{-1}X{)}^{-1}{X}^{\prime }{W}^{-1}y$ under some conditions.     

Estimation of smooth regression functions in monotone response models

We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z,X)$(Z,X)$, where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z   is distributed according to the density p(·,ψ(Z))$p(·,\psi (Z))$, where p(·,θ)$p(·,\theta )$ is a one-parameter exponential family. The function ψ$\psi$ is a smooth monotone function. Under this formulation, the regression function E(X|Z)$E(X|Z)$ is monotone in the co-variate Z   (and can be expressed as a one–one function of ψ$\psi$); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ$\psi$ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ$\psi$ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.     

A surprising property of uniformly best linear affine estimation in linear regression when prior information is fuzzy

It is already shown in Arnold and Stahlecker (2009) that, in linear regression, a uniformly best estimator exists in the class of all Γ-compatible$\Gamma \text{-compatible}$ linear affine estimators. Here, prior information is given by a fuzzy set Γ$\Gamma$ defined by its ellipsoidal α-cuts$\alpha \text{-cuts}$. Surprisingly, such a uniformly best linear affine estimator is uniformly best not only in the class of all Γ-compatible$\Gamma \text{-compatible}$ linear affine estimators but also in the class of all estimators satisfying a very weak and sensible condition. This property of a uniformly best linear affine estimator is shown in the present paper. Furthermore, two illustrative special cases are mentioned, where a generalized least squares estimator on the one hand and a general ridge or Kuks–Olman estimator on the other hand turn out to be uniformly best.     

Moderate deviations for LS estimator in simple linear EV regression model

In this paper, we consider the following simple linear Errors-in-Variables (EV) regression model _ηi=θ+β_xi+_?i${\eta }_i=\theta +\beta x_i+{?}_i$, _ξi=_xi+_δi${\xi }_i=x_i+{\delta }_i$, 1?i?n$1?i?n$. The moderate deviation principle for the least squares (LS) estimators of the unknown parameters θ$\theta$, β$\beta$ in the model are obtained.     

Estimation of extreme quantiles from heavy and light tailed distributions

In Gardes et al. (2011), a new family of distributions is introduced, depending on two parameters τ$\tau$ and θ$\theta$, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for θ$\theta$ and extreme quantiles are also proposed, but they both depend on the unknown parameter τ$\tau$, making them useless in practical situations. In this paper, we propose an estimator of τ$\tau$ which is independent of θ$\theta$. Plugging our estimator of τ$\tau$ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.     

A sequential design for a clinical trial with a linear prognostic factor

We study a randomized adaptive design to assign one of the L$L$ treatments to patients who arrive sequentially by means of an urn model. At each stage n$n$, a reward is distributed between treatments. The treatment applied is rewarded according to its response, 0?_Yn?1$0?Y_n?1$, and 1-_Yn$1-Y_n$ is distributed among the other treatments according to their performance until stage n-1$n-1$. Patients can be classified in K+1$K+1$ levels and we assume that the effect of this level in the response to the treatments is linear. We study the asymptotic behavior of the design when the ordinary least square estimators are used as a measure of performance until stage n-1$n-1$.     

The use of an identity in Anderson for multivariate multiple testing

Consider the model where there are I$I$ independent multivariate normal treatment populations with p×1$p\times 1$ mean vectors _μi${\mathit{\mu }}_i$, i=1,…,I$i=1,\dots ,I$, and covariance matrix Σ$\Sigma$. Independently the (I+1)$(I+1)$st population corresponds to a control and it too is multivariate normal with mean vector _μI+1${\mathit{\mu }}_{I+1}$ and covariance matrix Σ$\Sigma$. Now consider the following two multiple testing problems.     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Semiparametric partially linear regression models for functional data

In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time point. For discarding this restricted condition of observation number of each subject, we consider the semiparametric partially linear regression models with mean function x^?β$x^{?}\beta$ + g(z), where x and z   are functional data. The estimations of β$\beta$ and g(z) are presented and some asymptotic results are given. It is shown that the estimator of the parametric component is asymptotically normal. The convergence rate of the estimator of the nonparametric component is also obtained. Here, the observation number of each subject is completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

Optimally robust credible sets with a class of priors

We determine a credible set A   that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ$\Gamma$ of the starting prior _π0(θ)${\pi }_0(\theta )$. Among the class of sets with credibility γ$\gamma$ under _π0${\pi }_0$, the “optimally robust” set will be the one which maximizes the minimum probability of including θ$\theta$ as the prior varies over Γ$\Gamma$. This procedure is also Γ-minimax$\Gamma \text{-minimax}$ with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ$\Gamma$, the ε-contamination$\epsilon \text{-contamination}$ class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

Optimal global rates of convergence for nonparametric regression with unbounded data

Estimation of regression functions from independent and identically distributed data is considered. The _L2$L_2$ error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates a boundedness assumption on the explanatory variable X$X$ is made besides smoothness assumptions on the regression function and moment conditions on the response variable Y$Y$. In this article we consider the kernel estimate and show that by replacing the boundedness assumption on X$X$ by a proper moment condition the same (optimal) rate of convergence can be shown as for bounded data. This answers Question 1 in Stone [1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040–1053].     

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X^(?)AX${\mathbf{X}}^{(?)}\mathbf{AX}$, when A=A^(?)$\mathbf{A}={\mathbf{A}}^{(?)}$ is a positive semidefinite matrix, X$\mathbf{X}$ has a matrix multivariate elliptical distribution and X^(?)${\mathbf{X}}^{(?)}$ denotes the usual conjugate transpose of X$\mathbf{X}$. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.