On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

A stochastic restricted k–d class estimator

In this paper, we introduce a stochastic restricted k–d class estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. The stochastic restricted k–d class estimator is a generalization of the ordinary mixed estimator and the k–d class estimator. We show that our new biased estimator is superior in the mean squared error matrix sense to the k–d class estimator [S. Sakall?o?lu and S. Kaçiranlar, A new biased estimator based on ridge estimation, Statist. Papers 49 (2008), pp. 669–689] and the stochastic restricted Liu estimator [H. Yang and J.W. Xu, An alternative stochastic restricted Liu estimator in linear regression, Statist. Papers 50 (2009), pp. 639–647]. Finally, a numerical example is given to show the theoretical results.     

On an exponential bound for the Kaplan–Meier estimator

We review limit theory and inequalities for the Kaplan–Meier Kaplan and Meier (J Am Stat Assoc 53:457–481, 1958) product limit estimator of a survival function on the whole line . Along the way we provide bounds for the constant in an interesting inequality due to Biotouzé et al. (Ann Inst H Poincaré Probab Stat 35:735–763, 1999), and provide some numerical evidence in support of one of their conjectures. Supported in part by NSF grant DMS-0503822 and by NI-AID grant 2R01 AI291968-04.     

On the Kaplan–Meier estimator based on ranked set samples

When quantification of all sampling units is expensive but a set of units can be ranked, without formal measurement, ranked set sampling (RSS) is a cost-efficient alternate to simple random sampling (SRS). In this paper, we study the Kaplan–Meier estimator of survival probability based on RSS under random censoring time setup, and propose nonparametric estimators of the population mean. We present a simulation study to compare the performance of the suggested estimators. It turns out that RSS design can yield a substantial improvement in efficiency over the SRS design. Additionally, we apply the proposed methods to a real data set from an environmental study.     

Heteroskedasticity-consistent covariance matrix estimation:white's estimator and the bootstrap ∗

This paper considers the issue of estimating the covariance matrix of ordinary least squares estimates in a linear regression model when heteroskedasticity is suspected. We perform Monte Carlo simulation on the White estimator, which is commonly used in.

empirical research, and also on some alternatives based on different bootstrapping schemes. Our results reveal that the White estimator can be considerably biased when the sample size is not very large, that bias correction via bootstrap does not work well, and that the weighted bootstrap estimators tend to display smaller biases than the White estimator and its variants, under both homoskedasticity and heteroskedasticity. Our results also reveal that the presence of (potentially) influential observations in the design matrix plays an important role in the finite-sample performance of the heteroskedasticity-consistent estimators.     

Rate of convergence for asymptotic variance of the Horvitz–Thompson estimator

Drawing distinct units without replacement and with unequal probabilities from a population is a problem often considered in the literature (e.g. Hanif and Brewer, 1980, Int. Statist. Rev. 48, 317–355). In such a case, the sample mean is a biased estimator of the population mean. For this reason, we use the unbiased Horvitz–Thompson estimator (1951). In this work, we focus our interest on the variance of this estimator. The variance is cumbersome to compute because it requires the calculation of a large number of second-order inclusion probabilities. It would be helpful to use an approximation that does not need heavy calculations. The Hájek (1964) variance approximation provides this advantage as it is free of second-order inclusion probabilities. Hájek (1964) proved that this approximation is valid under restrictive conditions that are usually not fulfilled in practice. In this paper, we give more general conditions and we show that this approximation remains acceptable for most practical problems.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

A stable estimator of the information matrix under EM for dependent data

This article develops a new and stable estimator for information matrix when the EM algorithm is used in maximum likelihood estimation. This estimator is constructed using the smoothed individual complete-data scores that are readily available from running the EM algorithm. The method works for dependent data sets and when the expectation step is an irregular function of the conditioning parameters. In comparison to the approach of Louis (J. R. Stat. Soc., Ser. B 44:226–233, 1982), this new estimator is more stable and easier to implement. Both real and simulated data are used to demonstrate the use of this new estimator.     

r − k Class estimator in the linear regression model with correlated errors

Autocorrelation in errors and multicollinearity among the regressors are serious problems in regression analysis. The aim of this paper is to examine multicollinearity and autocorrelation problems concurrently and to compare the r ? k class estimator to the generalized least squares estimator, the principal components regression estimator and the ridge regression estimator by the scalar and matrix mean square error criteria in the linear regression model with correlated errors.     

The flood algorithm—a multivariate, self-organizing-map-based, robust location and covariance estimator

Self-organizing maps (SOMs) introduced by Kohonen (Biol. Cybern. 43(1):59–69, 1982) are well-known in the field of artificial neural networks. The way SOMs are performing is very intuitive, leading to great popularity and numerous applications (related to statistics: classification, clustering). The result of the unsupervised learning process performed by SOMs is a non-linear, low-dimensional projection of the high-dimensional input data, that preserves certain features of the underlying data, e.g. the topology and probability distribution (Lee and Verleysen in Nonlinear Dimensionality Reduction, Springer, 2007; Kohonen in Self-organizing Maps, 3rd edn., Springer, 2001).     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

Nonparametric estimation of the multivariate survivor function: the multivariate Kaplan–Meier estimator

The Dabrowska (Ann Stat 16:1475–1489, 1988) product integral representation of the multivariate survivor function is extended, leading to a nonparametric survivor function estimator for an arbitrary number of failure time variates that has a simple recursive formula for its calculation. Empirical process methods are used to sketch proofs for this estimator’s strong consistency and weak convergence properties. Summary measures of pairwise and higher-order dependencies are also defined and nonparametrically estimated. Simulation evaluation is given for the special case of three failure time variates.     

Asymptotic expansion of the risk of maximum likelihood estimator with respect to α-divergence

For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to α-divergence, which includes the special cases of Kullback–Leibler divergence, the Hellinger distance, and essentially χ²-divergence. The asymptotic expansion of the risk is given with respect to sample sizes up to order n^{? 2}. Each term in the expansion is expressed with the geometrical properties of the Riemannian manifold formed by the parametric probability model.     

An extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe

In this paper, an extension of Horvitz–Thompson estimator used in adaptive cluster sampling to continuous universe is developed. Main new results are presented in theorems. The primary notions of discrete population are transferred to continuous population. First and second order inclusion probabilities for networks are delivered. Horvitz–Thompson estimator for adaptive cluster sampling in continuous universe is constructed. The unbiasedness of the estimator is proven. Variance and unbiased variance estimator are delivered. Finally, the theory is illustrated with an example.     

A consistent estimator of the smoothing operator in the functional Hodrick–Prescott filter

In this article, we consider a version of the functional Hodrick–Prescott filter for functional time series. We show that the associated optimal smoothing operator preserves the “noise-to-signal ratio” structure. Moreover, as the main result, we propose a consistent estimator of this optimal smoothing operator.     

Nadaraya–Watson estimator for I.I.D. paths of diffusion processes

This paper deals with a nonparametric Nadaraya–Watson (NW) estimator of the drift function computed from independent continuous observations of a diffusion process. Risk bounds on the estimator and its discrete-time approximation are established. The paper also deals with extensions of the PCO and leave-one-out cross-validation bandwidth selection methods for our NW estimator. Finally, some numerical experiments are provided.     

Comments on “Improvement of generalized difference-based mixed Liu estimator in partially linear model”

Abstract

We make some comments about the article of Wu (2018 Wu, J. B. 2018. Improvement of generalized difference-based mixed Liu estimator in partially linear model. Communications in Statistics: Theory and Methods 47 (18):4433–42.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and correct the theorems in that article.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.