BETWEEN OLSE AND BLUE

Several estimators of X β under the general Gauss–Markov model are considered. Particular attention is paid to those estimators whose efficiency lies between that of the ordinary least squares estimator and that of the best linear unbiased estimator.     

On Projection‐type Estimators of Multivariate Isotonic Functions

Abstract. Let M be an isotonic real‐valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection‐type estimators are r_n‐equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability‐weighted moment, quantile) and biometric (mean remaining lifetime) functions.     

Negative Dependence in Sampling

Abstract. The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0‐1 random variables that satisfy the strong Rayleigh property and let . We show that {Z_i} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Brändén (Invent. Math., 177, 2009, 521–569). This entails that a number of important π ps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.     

A REGRESSION APPROACH TO THE ESTIMATION OF THE FINITE POPULATION MEAN IN THE PRESENCE OF NON‐RESPONSE

This paper presents various estimators for estimating the population mean of the study variable y using information on the auxiliary variable x in the presence of non‐response. Properties of the suggested estimators are studied and compared with those of existing estimators. It is shown that the estimators suggested in this paper are among the best of all the estimators considered. An empirical study is carried out to demonstrate the performance of the suggested estimators and of others, and it is found that the empirical results support the theoretical study.     

A Class of Normalized Random Measures with an Exact Predictive Sampling Scheme

Abstract. In this article, we define and investigate a novel class of non‐parametric prior distributions, termed the class . Such class of priors is dense with respect to the homogeneous normalized random measures with independent increments and it is characterized by a richer predictive structure than those arising from other widely used priors. Our interest in the class is mainly motivated by Bayesian non‐parametric analysis of some species sampling problems concerning the evaluation of the species relative abundances in a population. We study both the probability distribution of the number of species present in a sample and the probability of discovering a new species conditionally on an observed sample. Finally, by using the coupling from the past method, we provide an exact sampling scheme for the system of predictive distributions characterizing the class .     

CONFIDENCE INTERVALS UTILIZING PRIOR INFORMATION IN THE BEHRENS–FISHER PROBLEM

Consider two independent random samples of size f + 1 , one from an N (μ₁, σ²₁) distribution and the other from an N (μ₂, σ²₂) distribution, where σ²₁/σ²₂∈ (0, ∞) . The Welch ‘approximate degrees of freedom’ (‘approximate t‐solution’) confidence interval for μ₁?μ₂ is commonly used when it cannot be guaranteed that σ²₁/σ²₂= 1 . Kabaila (2005, Comm. Statist. Theory and Methods 34 , 291–302) multiplied the half‐width of this interval by a positive constant so that the resulting interval, denoted by J₀, has minimum coverage probability 1 ?α. Now suppose that we have uncertain prior information that σ²₁/σ²₂= 1. We consider a broad class of confidence intervals for μ₁?μ₂ with minimum coverage probability 1 ?α. This class includes the interval J₀, which we use as the standard against which other members of will be judged. A confidence interval utilizes the prior information substantially better than J₀ if (expected length of J)/(expected length of J₀) is (a) substantially less than 1 (less than 0.96, say) for σ²₁/σ²₂= 1 , and (b) not too much larger than 1 for all other values of σ²₁/σ²₂ . For a given f, does there exist a confidence interval that satisfies these conditions? We focus on the question of whether condition (a) can be satisfied. For each given f, we compute a lower bound to the minimum over of (expected length of J)/(expected length of J₀) when σ²₁/σ²₂= 1 . For 1 ?α= 0.95 , this lower bound is not substantially less than 1. Thus, there does not exist any confidence interval belonging to that utilizes the prior information substantially better than J₀.     

Geometrically Corrected Second Order Analysis of Events on a Linear Network,with Applications to Ecology and Criminology

Abstract. We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. Okabe and Yamada developed a ‘network K function’, analogous to Ripley's K function, for analysis of such data. However, values of the network K‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of K, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network K‐function is the identity. The corrected estimator is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of under complete randomness. We extend these results to an ‘inhomogeneous’ network K‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).     

On the Validity of the Bootstrap in Non‐Parametric Functional Regression

Abstract. We consider the functional non‐parametric regression model Y= r( χ )+?, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ? satisfies E(? | χ ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.     

Estimating prediction error in microarray classification: Modifications of the 0.632+ bootstrap when ${\bf n} < {\bf p}$

We are interested in estimating prediction error for a classification model built on high dimensional genomic data when the number of genes (p) greatly exceeds the number of subjects (n). We examine a distance argument supporting the conventional 0.632+ bootstrap proposed for the $n > p$ scenario, modify it for the $n < p$ situation and develop learning curves to describe how the true prediction error varies with the number of subjects in the training set. The curves are then applied to define adjusted resampling estimates for the prediction error in order to achieve a balance in terms of bias and variability. The adjusted resampling methods are proposed as counterparts of the 0.632+ bootstrap when $n < p$ , and are found to improve on the 0.632+ bootstrap and other existing methods in the microarray study scenario when the sample size is small and there is some level of differential expression. The Canadian Journal of Statistics 41: 133–150; 2013 © 2012 Statistical Society of Canada     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Central Limit Theorems for the Non‐Parametric Estimation of Time‐Changed Lévy Models

Abstract. Let {Z_t}_{t 0} be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β(?):=∫?(x)ν(dx), valid when both the sampling frequency and the observation time‐horizon of the process get larger. Our results combine the long‐run ergodic properties of the diffusion process with the short‐term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox–Ingersoll–Ross process .     

TESTING THE EQUALITY OF PARAMETERS IN THE DISTRIBUTIONS WHEN THE RANGE DEPENDS UPON THE PARAMETER

The authors derive the null and non-null distributions of the test statistic v=y_min/y_max (where y_min= min x_ij, y_max= max x_ij, J=1,2, …, k) connected with testing the equality of scale parameters θ₁, θ₂, …θ_k in certain, class of density functions given by      

How Many Iterations are Sufficient for Efficient Semiparametric Estimation?

Abstract. A common practice in obtaining an efficient semiparametric estimate is through iteratively maximizing the (penalized) full log‐likelihood w.r.t. its Euclidean parameter and functional nuisance parameter. A rigorous theoretical study of this semiparametric iterative estimation approach is the main purpose of this study. We first show that the grid search algorithm produces an initial estimate with the proper convergence rate. Our second contribution is to provide a formula in calculating the minimal number of iterations k ^* needed to produce an efficient estimate . We discover that (i) k ^* depends on the convergence rates of the initial estimate and the nuisance functional estimate, and (ii) k ^* iterations are also sufficient for recovering the estimation sparsity in high dimensional data. The last contribution is the novel construction of which does not require knowing the explicit expression of the efficient score function. The above general conclusions apply to semiparametric models estimated under various regularizations, for example, kernel or penalized estimation. As far as we are aware, this study provides a first general theoretical justification for the ‘one‐/two‐step iteration’ phenomena observed in the semiparametric literature.     

Constrained nonparametric maximum likelihood estimation of stochastically ordered survivor functions

This paper considers estimators of survivor functions subject to a stochastic ordering constraint based on right censored data. We present the constrained nonparametric maximum likelihood estimator (C‐NPMLE) of the survivor functions in one‐and two‐sample settings where the survivor distributions could be discrete or continuous and discuss the non‐uniqueness of the estimators. We also present a computationally efficient algorithm to obtain the C‐NPMLE. To address the possibility of non‐uniqueness of the C‐NPMLE of $S_1(t)$ when $S_1(t)\le S_2(t)$ , we consider the maximum C‐NPMLE (MC‐NPMLE) of $S_1(t)$ . In the one‐sample case with arbitrary upper bound survivor function $S_2(t)$ , we present a novel and efficient algorithm for finding the MC‐NPMLE of $S_1(t)$ . Dykstra ( 1982 ) also considered constrained nonparametric maximum likelihood estimation for such problems, however, as we show, Dykstra's method has an error and does not always give the C‐NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed and consistency of the estimators is proved. Data from a study on larynx cancer are analysed to illustrate the method. The Canadian Journal of Statistics 40: 22–39; 2012 © 2012 Statistical Society of Canada     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada     

New estimation and feature selection methods in mixture‐of‐experts models

We study estimation and feature selection problems in mixture‐of‐experts models. An $l_2$ ‐penalized maximum likelihood estimator is proposed as an alternative to the ordinary maximum likelihood estimator. The estimator is particularly advantageous when fitting a mixture‐of‐experts model to data with many correlated features. It is shown that the proposed estimator is root‐$n$ consistent, and simulations show its superior finite sample behaviour compared to that of the maximum likelihood estimator. For feature selection, two extra penalty functions are applied to the $l_2$ ‐penalized log‐likelihood function. The proposed feature selection method is computationally much more efficient than the popular all‐subset selection methods. Theoretically it is shown that the method is consistent in feature selection, and simulations support our theoretical results. A real‐data example is presented to demonstrate the method. The Canadian Journal of Statistics 38: 519–539; 2010 © 2010 Statistical Society of Canada     

Further Properties of Frequentist Confidence Intervals in Regression that Utilize Uncertain Prior Information

Consider a linear regression model with n‐dimensional response vector, regression parameter and independent and identically distributed errors. Suppose that the parameter of interest is where a is a specified vector. Define the parameter where c and t are specified. Also suppose that we have uncertain prior information that . Part of our evaluation of a frequentist confidence interval for is the ratio (expected length of this confidence interval)/(expected length of standard confidence interval), which we call the scaled expected length of this interval. We say that a confidence interval for utilizes this uncertain prior information if: (i) the scaled expected length of this interval is substantially less than 1 when ; (ii) the maximum value of the scaled expected length is not too much larger than 1; and (iii) this confidence interval reverts to the standard confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri (2009) present a new method for finding such a confidence interval. Let denote the least squares estimator of . Also let and . Using computations and new theoretical results, we show that the performance of this confidence interval improves as increases and decreases.     

Measuring the Discrepancy of a Parametric Model via Local Polynomial Smoothing

Abstract. In the context of multivariate mean regression, we propose a new method to measure and estimate the inadequacy of a given parametric model. The measure is basically the missed fraction of variation after adjusting the best possible parametric model from a given family. The proposed approach is based on the minimum L ² ‐distance between the true but unknown regression curve and a given model. The estimation method is based on local polynomial averaging of residuals with a polynomial degree that increases with the dimension d of the covariate. For any d ≥ 1 and under some weak assumptions we give a Bahadur‐type representation of the estimator from which ‐consistency and asymptotic normality are derived for strongly mixing variables. We report the outcomes of a simulation study that aims at checking the finite sample properties of these techniques. We present the analysis of a dataset on ultrasonic calibration for illustration.     

Contaminated normal modeling with application to microarray data analysis

A contaminated beta model $(1-\gamma) B(1,1) + \gamma B(\alpha,\beta)$ is often used to describe the distribution of $P$ ‐values arising from a microarray experiment. The authors propose and examine a different approach: namely, using a contaminated normal model $(1-\gamma) N(0,\sigma^2) + \gamma N(\mu,\sigma^2)$ to describe the distribution of $Z$ statistics or suitably transformed $T$ statistics. The authors then address whether a researcher who has $Z$ statistics should analyze them using the contaminated normal model or whether the $Z$ statistics should be converted to $P$ ‐values to be analyzed using the contaminated beta model. The authors also provide a decision‐theoretic perspective on the analysis of $Z$ statistics. The Canadian Journal of Statistics 38: 315–332; 2010 © 2010 Statistical Society of Canada     

COMBINING INDIVIDUAL PARTICIPANT DATA AND SUMMARY STATISTICS FROM BOTH CONTINUOUSLY VALUED AND BINARY VARIABLES TO ESTIMATE REGRESSION PARAMETERS

Recent research has extended standard methods for meta‐analysis to more general forms of evidence synthesis, where the aim is to combine different data types or data summaries that contain information about functions of multiple parameters to make inferences about the parameters of interest. We consider one such scenario in which the goal is to make inferences about the association between a primary binary exposure and continuously valued outcome in the context of several confounding exposures, and where the data are available in various different forms: individual participant data (IPD) with repeated measures, sample means that have been aggregated over strata, and binary data generated by thresholding the underlying continuously valued outcome measure. We show that an estimator of the population mean of a continuously valued outcome can be constructed using binary threshold data provided that a separate estimate of the outcome standard deviation is available. The results of a simulation study show that this estimator has negligible bias but is less efficient than the sample mean – the minimum variance ratio is based on a Taylor series expansion. Combining this estimator with sample means and IPD from different sources (such as a series of published studies) using both linear and probit regression does, however, improve the precision of estimation considerably by incorporating data that would otherwise have been excluded for being in the wrong format. We apply these methods to investigate the association between the G277S mutation in the transferrin gene and serum ferritin (iron) levels separately in pre‐ and post‐menopausal women based on data from three published studies.