A Note on the Best Linear Unbiased Estimation Based on Order Statistics

The appropriateness of replacing the term regression to the mean with regression to the mode is discussed.     

A Note on the Predictors of Differenced Sequences

In this note we consider the problems of optimal linear prediction (o.l.p.) and the minimum mean squared error prediction (m.m.s.e.p.) of a sequence X_t, which fits to a stationary and invertible ARMA model through the filter (1 - B^s)^d X_t= Y_t. It is shown that these two predictors are not identical in general from the theoretical point of view. Permitting the degree of differencing d to take any real value, a set of conditions for these commonly applied prediction formulas to be identical is given.     

A Note on Model-Based Stratification

A slight improvement on the stratification methodology in Godfrey et al. (1984) is proposed and an example is advanced to demonstrate its usefulness.     

Invariance principles and almost sure central limit theorem for the error variance estimator in linear models

A number of score statistics are derived for a heterogeneous spatial Poisson process which has a composite intensity. The intensity consists of a 'background' process which is estimated

from a control point process by kernel density estimation. The parametric form of the composite intensity yields score tests for particular spatial effects. A numerical example concerning respiratory cancer mortality is given.     

Matching Using Sufficient Dimension Reduction for Causal Inference

ABSTRACT

To estimate causal treatment effects, we propose a new matching approach based on the reduced covariates obtained from sufficient dimension reduction. Compared with the original covariates and the propensity score, which are commonly used for matching in the literature, the reduced covariates are nonparametrically estimable and are effective in imputing the missing potential outcomes, under a mild assumption on the low-dimensional structure of the data. Under the ignorability assumption, the consistency of the proposed approach requires a weaker common support condition. In addition, researchers are allowed to employ different reduced covariates to find matched subjects for different treatment groups. We develop relevant asymptotic results and conduct simulation studies as well as real data analysis to illustrate the usefulness of the proposed approach.     

A Note on Superiority Comparisons of Homogeneous Linear Estimators

This note extends some results on homogeneous linear estimators to the general, even nonlinear case.A Sufficient condition for the difference of mean square error matrices of minimum conditional mean square error estimator and minimum average risk linear estimator to be postive definite is derived.     

A Note on the Hyperbolic-Secant Distribution

This note discusses briefly some basic properties of the hyperbolic-secant distribution (e.g., Johnson and Kotz 1970; Manoukian 1986; Perks 1932; Talacko 1956), which has not received sufficient attention in the published literature, and may be useful for students and practitioners.     

A Shrinkage Estimation of Central Subspace in Sufficient Dimension Reduction

Sliced regression is an effective dimension reduction method by replacing the original high-dimensional predictors with its appropriate low-dimensional projection. It is free from any probabilistic assumption and can exhaustively estimate the central subspace. In this article, we propose to incorporate shrinkage estimation into sliced regression so that variable selection can be achieved simultaneously with dimension reduction. The new method can improve the estimation accuracy and achieve better interpretability for the reduced variables. The efficacy of proposed method is shown through both simulation and real data analysis.     

A Note on a Commonly Used Ridge Regression Monte Carlo Design

Ridge estimators are usually examined through Monte Carlo simulations since their properties are difficult to obtain analytically. In this paper we argue that a simulation design commonly used in the literature will give biased results of Monte Carlo simulations in favor of ridge regression over ordinary least square estimators. Specifically, it is argued that the properties of ridge estimators that are functions of p distinct regressor eigenvalues should not be evaluated through Monte Carlo designs using only two distinct eigenvalues.     

Model Reduction for Prediction in Regression Models

We look at prediction in regression models under squared loss for the random x case with many explanatory variables. Model reduction is done by conditioning upon only a small number of linear combinations of the original variables. The corresponding reduced model will then essentially be the population model for the chemometricians' partial least squares algorithm. Estimation of the selection matrix under this model is briefly discussed, and analoguous results for the case with multivariate response are formulated. Finally, it is shown that an assumption of multinormality may be weakened to assuming elliptically symmetric distribution, and that some of the results are valid without any distributional assumption at all.     

A Note on Window Length Selection in Singular Spectrum Analysis

In singular spectrum analysis (SSA) window length is a critical tuning parameter that must be assigned by the practitioner. This paper provides a theoretical analysis of signal–noise separation and time series reconstruction in SSA that can serve as a guide to optimal window choice. We establish numerical bounds on the mean squared reconstruction error and present their almost sure limits under very general regularity conditions on the underlying data generating mechanism. We also provide asymptotic bounds for the mean squared separation error. Evidence obtained using simulation experiments and real data sets indicates that the theoretical properties are reflected in observed behaviour, even in relatively small samples, and the results indicate how, in practice, an optimal assignment for the window length can be made.     

A new nonparametric estimator for the mean of the selected population

A modified bootstrap estimator of the mean of the population selected from two populations is proposed which is a convex combination of the two sample means, where the weights are random quantities. The estimator is shown to be strongly consistent. The small sample behavior of the estimator is investigated and compared with some competitors by means of Monte Carlo studies. It is found that the newly proposed estimator has smaller mean squared error for a wide range of parameter values.     

A Note on Prediction Error with Collinearity

Abstract

Least squares (LS) estimator is the best linear unbiased estimator for linear models. It is well known that LS performs poorly in estimation when collinearity is present among regressors. However, it is not fully understood and is even controversial whether LS performs well in prediction. To address this controversy, we study the mean and variance of the prediction squared error (PSE) of LS estimator, and conclude theoretically that although the mean PSE remains invariant regardless of the collinearity, the variance of PSE increases with the collinearity. Thus the prediction error is sensitive to the location in the feature space.     

A Note on Generalized Functional Linear Model and Its Application

Motivated by a biomarker study for colorectal neoplasia, we consider generalized functional linear models where the functional predictors are measured with errors at discrete design points. Assuming that the true functional predictor and the slope function are smooth, we investigate a two-step estimating procedure where both the true functional predictor and the slope function are estimated through spline smoothing. The operating characteristics of the proposed method are derived; the usefulness of the proposed method is illustrated by a simulation study as well as data analysis for the motivating colorectal neoplasia study.     

A New PC-Based Test for Varian's Weak Separability Conditions

This article develops a new method to evaluate revealed preference separability conditions. In contrast to previous studies, our results generally find weak separability, even when datasets have some measurement error. In addition, revealed preference and weak separability appear robust to measurement error, different price distributions, and alternative preference settings. Measurement error generally results in relatively few violations of revealed preference or weak separability.     

Note on the minimum mean integrated squared error of kernel estimates of a distribution function and its derivatives

An exact expiession for the minimum integrated squared error associated with the kernel distribution function and its derivatives is given. Furthermore, the virtual optimality of the Fourier integral estimate in density estimation, shown by Davis (1977), is extended to estimation of a distibution function and its derivatives.     

A Better Alternative to Non-parametric Approaches for Adjusting for Covariate Measurement Errors in Logistic Regression

In this article, we propose a flexible parametric (FP) approach for adjusting for covariate measurement errors in regression that can accommodate replicated measurements on the surrogate (mismeasured) version of the unobserved true covariate on all the study subjects or on a sub-sample of the study subjects as error assessment data. We utilize the general framework of the FP approach proposed by Hossain and Gustafson in 2009 for adjusting for covariate measurement errors in regression. The FP approach is then compared with the existing non-parametric approaches when error assessment data are available on the entire sample of the study subjects (complete error assessment data) considering covariate measurement error in a multiple logistic regression model. We also developed the FP approach when error assessment data are available on a sub-sample of the study subjects (partial error assessment data) and investigated its performance using both simulated and real life data. Simulation results reveal that, in comparable situations, the FP approach performs as good as or better than the competing non-parametric approaches in eliminating the bias that arises in the estimated regression parameters due to covariate measurement errors. Also, it results in better efficiency of the estimated parameters. Finally, the FP approach is found to perform adequately well in terms of bias correction, confidence coverage, and in achieving appropriate statistical power under partial error assessment data.     

A Note on Derived Means of Scaled Data

Details are given of the procedures for calculating derived means of square-root and log transformed variables from un-subdivided and layered samples. A method for determining approximate confidence limits of the derived means is also described. The temporal distribution of nematodes in grass herbage is given as an example to illustrate the procedures used.     

A Note on Estimation in Hilbertian Linear Models

We study estimation and prediction in linear models where the response and the regressor variable both take values in some Hilbert space. Our main objective is to obtain consistency of a principal component‐based estimator for the regression operator under minimal assumptions. In particular, we avoid some inconvenient technical restrictions that have been used throughout the literature. We develop our theory in a time‐dependent setup that comprises as important special case the autoregressive Hilbertian model.     

A Note on Capacity Utilization and Measurement of Scale Economies

This article investigates the measurement of returns to scale (RTS) via a cost function in the presence of quasi-fixed inputs. Caves, Christensen, and Swanson (CCS) proposed estimation of a variable-cost function and derived a formula for computing RTS from it. This article extends the CCS results by (a) investigating the bias that would result from erroneously using a total-cost function for measuring RTS, (b) indicating that the CCS procedure for measuring scale economies in the presence of quasi-fixed inputs is inconsistent for the case of nonhomothetic production technology, and (c) proposing an alternative procedure that allows one to evaluate RTS at long-run equilibrium points, even when firms in the sample are in disequilibrium. The similarity between this proposed method and the “temporary” equilibrium approach for measuring productivity growth is noted as well. The empirical results on RTS are compared among the three alternative methods (total-cost, CCS variable-cost, and proposed long-run equilibrium methods).