Asymptotic Efficiency Properties Research Articles

Efficient closed-form estimation of large spatial autoregressions

Newton-step approximations to pseudo maximum likelihood estimates of spatial autoregressive models with a large number of parameters are examined, in the sense that the parameter space grows slowly as a function of sample size. These have the same asymptotic efficiency properties as maximum likelihood under Gaussianity but are of closed form. Hence they are computationally simple and free from compactness assumptions, thereby avoiding two notorious pitfalls of implicitly defined estimates of large spatial autoregressions. When commencing from an initial least squares estimate, the Newton step can also lead to weaker regularity conditions for a central limit theorem than some extant in the literature. A simulation study demonstrates excellent finite sample gains from Newton iterations, especially in large multiparameter models for which grid search is costly. A small empirical illustration shows improvements in estimation precision with real data.

Linear regression with many controls of limited explanatory power

We consider inference about a scalar coefficient in a linear regression model. One previously considered approach to dealing with many controls imposes sparsity, that is, it is assumed known that nearly all control coefficients are (very nearly) zero. We instead impose a bound on the quadratic mean of the controls' effect on the dependent variable, which also has an interpretation as anR2‐type bound on the explanatory power of the controls. We develop a simple inference procedure that exploits this additional information in general heteroskedastic models. We study its asymptotic efficiency properties and compare it to a sparsity‐based approach in a Monte Carlo study. The method is illustrated in three empirical applications.

Model selection for the robust efficient signal processing observed with small Lévy noise

We develop a new model selection method for an adaptive robust efficient nonparametric signal estimation observed with impulse noise which is defined by a general non-Gaussian Levy process. On the basis of the developed method, we construct estimation procedures which are analyzed in two settings: in non-asymptotic and in asymptotic ones. For the first time for such models, we show non-asymptotic sharp oracle inequalities for quadratic and robust risks, i.e., we show that the constructed procedures are optimal in the sense of sharp oracle inequalities. Next, by making use of the obtained oracle inequalities, we provide an asymptotic efficiency property for the developed estimation methods in an adaptive setting when the signal/noise ratio goes to infinity. We apply the developed model selection methods for the signal number detection problem in multi-path information transmission.

Local asymptotic normality for Student-Lévy processes under high-frequency sampling

ABSTRACTThere is considerable interest in parameter estimation in Lévy models. The maximum likelihood estimator is widely used because under certain conditions it enjoys asymptotic efficiency properties. The toolkit for Lévy processes is the local asymptotic normality which guarantees these conditions. Although the likelihood function is not known explicitly, we prove local asymptotic normality for the location and scale parameters of the Student-Lévy process assuming high-frequency data. In addition, we propose a numerical method to make maximum likelihood estimates feasible based on the Monte Carlo expectation-maximization algorithm. A simulation study verifies the theoretical results.

Negative Binomial Quasi‐Likelihood Inference for General Integer‐Valued Time Series Models

Two negative binomial quasi‐maximum likelihood estimates (NB‐QMLEs) for a general class of count time series models are proposed. The first one is the profile NB‐QMLE calculated while arbitrarily fixing the dispersion parameter of the negative binomial likelihood. The second one, termed two‐stage NB‐QMLE, consists of four stages estimating both conditional mean and dispersion parameters. It is shown that the two estimates are consistent and asymptotically Gaussian under mild conditions. Moreover, the two‐stage NB‐QMLE enjoys a certain asymptotic efficiency property provided that a negative binomial link function relating the conditional mean and conditional variance is specified. The proposed NB‐QMLEs are compared with the Poisson QMLE asymptotically and in finite samples for various well‐known particular classes of count time series models such as the Poisson and negative binomial integer‐valued GARCH model and the INAR(1) model. Application to a real dataset is given.

Purely Sequential and Two-Stage Fixed-Accuracy Confidence Interval Estimation Methods for Count Data from Negative Binomial Distributions in Statistical Ecology: One-Sample and Two-Sample Problems

Our main focus is on count data arising from statistical ecology. Anscombe (1949) emphasized negative binomial (NB) modeling for overdispersed count data. A large majority of existing methodologies, both sequential and nonsequential, were reviewed by Mukhopadhyay and Banerjee (2012). We assume that the thatch parameter remains known and revisit a sequential confidence interval estimation method for an NB mean proposed by Willson and Folks (1983) that may not always guarantee a positive lower confidence limit. Moreover, any postsampling adjustment of their lower confidence limit would compromise the preset confidence coefficient. In this article, we provide an appropriate resolution by estimating the NB mean with a fixed-accuracy confidence interval such that both confidence limits are positive. Our proposed purely sequential and two-stage estimation methodologies enjoy asymptotic consistency and asymptotic efficiency properties. Next, we consider estimating the ratio of two NB means under a two-sample setup assuming that the thatch parameters remain known with equal sample sizes and provide both purely sequential and two-stage methodologies. These are shown to enjoy asymptotic consistency and asymptotic efficiency properties. Extensive sets of analyses based on both simulated data and real data from potato beetle infestation are presented to highlight some of the exciting small- and moderate-sample features of the proposed estimation methodologies.

Consistent estimation of regression parameters under replicated ultrastructural model with non-normal errors

This article discusses the construction and efficiency properties of consistent estimators of regression parameters under replicated ultrastructural model with not necessarily normally distributed measurement errors. The variances of measurement errors associated with the study and explanatory variables are estimated from the replicated sample observations and are used for the consistent estimation of regression parameters. The asymptotic efficiency properties of the estimators are derived and analysed. The finite sample performance of the estimators is empirically studied through a Monte Carlo simulation.

Variable selection strategies in survival models with multiple imputations

In this paper, the variable selection strategies (criteria) are thoroughly discussed and their use in various survival models is investigated. The asymptotic efficiency property, in the sense of Shibata Ann Stat 8: 147-164, 1980, of a class of variable selection strategies which includes the AIC and all criteria equivalent to it, is established for a general class of survival models, such as parametric frailty or transformation models and accelerated failure time models, under minimum conditions. Furthermore, a multiple imputations method is proposed which is found to successfully handle censored observations and constitutes a competitor to existing methods in the literature. A number of real and simulated data are used for illustrative purposes.

Finite sample nonparametric inference and large sample efficiency

Given a sample $X_1,\dots, X_n$ from a distribution $F$, the problem of constructing nonparametric confidence intervals for the mean $\mu(F)$ is considered. Unlike bootstrap procedures or those based on normal approximations, we insist on any procedure being truly nonparametric in the sense that the probability that the confidence interval contains $\mu(F)$ based on a sample of size $n$ from $F$ be at least $1 - \alpha$ for all $F$ and all $n$. Bahadur and Savage proved it is impossible to find an effective (or bounded) confidence interval for $\mu(F)$ without some restrictions. Thus,we assume that $F$ is supported on a known compact set, which we take to be $[0, 1]$. In this setting, an asymptotic efficiency result is obtained that gives a lower bound on the size of any conservative interval. We then provide a construction of an interval that meets our finite sample requirement on level, yet has an asymptotic efficiency property. Thus, the price to be paid for using fully nonparametric procedures when considering the trade-off between exact inference statements and asymptotic efficiency is negligible. Much of what is accomplished for the mean generalizes to other settings as well.

Estimation in large and disaggregated demand systems: an estimator for conditionally linear systems

Empirical demand systems that do not impose unreasonable restrictions on preferences are typically non-linear. We show, however, that all popular systems possess the property of conditional linearity. A computationally attractive iterated linear least squares estimator (ILLE) is proposed for large non-linear simultaneous equation systems which are conditionally linear in unknown parameters. The estimator is shown to be consistent and its asymptotic efficiency properties are derived. An application is given for a 22-commodity quadratic demand system using household-level data from a time series of repeated cross-sections. Copyright © 1999 John Wiley & Sons, Ltd.

Estimation in large and disaggregated demand systems: an estimator for conditionally linear systems

Empirical demand systems that do not impose unreasonable restrictions on preferences are typically non-linear. We show, however, that all popular systems possess the property of conditional linearity. A computationally attractive iterated linear least squares estimator (ILLE) is proposed for large non-linear simultaneous equation systems which are conditionally linear in unknown parameters. The estimator is shown to be consistent and its asymptotic efficiency properties are derived. An application is given for a 22-commodity quadratic demand system using household-level data from a time series of repeated cross-sections.

Fixed-Width Simultaneous Confidence Intervals for Multinormal Means in Several Intraclass Correlation Models

The problem of constructing fixed-width simultaneous confidence intervals for comparing mean vectors ofk(⩾2) independent multivariate normal distributions is considered when those covariance matrices have the intraclass correlation structures. Two-stage procedures are developed for which the simultaneous confidence levels are shown to beat least1−α, the preassigned nominal value, 0<α<1. Asymptotic efficiency properties are addressed. In the case of fixed and finite initial sample size, efficiency related issues are also discussed.

Asymptotic efficiency properties of least squares in an ultrastructural model

This paper considers the estimation of both the intercept term and the slope parameter in a linear ultrastructural model using direct regression and reverse regression methods. Without assuming the errors to be normally distributed, asymptotic expressions for the bias and the variance of the estimators are derived and analyzed. The effect of departure of the errors from normality is also studied.

Analysis of an importance sampling estimator for tandem queues

We analyze the performance of an importance sampling estimator for a rare-event probability in tandem Jackson networks. The rare event we consider corresponds to the network population reaching K before returning to ø, starting from ø, with K large. The estimator we study is based on interchanging the arrival rate and the smallest service rate and is therefore a generalization of the asymptotically optimal estimator for an M/M/1 queue. We examine its asymptotic performance for large K , showing that in certain parameter regions the estimator has an asymptotic efficiency property, but that in other regions it does not. The setting we consider is perhaps the simplest case of a rare-event simulation problem in which boundaries on the state space play a significant role.

Asymptotic study of the multivariate functional model in the case of a random number of observations for each mean

We consider the multivariate linear and affine functional models for which several observations for each mean are available (replications of observations). In the case of a simple random sampling, which is the assumption made in this study, the number of observations for each mean is a random variable. Let be the sampling covariance matrix Explained by the partition (also called the between covariance matrix) and M a symmetric positive definite p × p matrix that defines a quadratic metric on . The least squares estimation of the parameters of the model in ( , M)amounts to the diagonalization of M The estimators are consistent for any M, but we show that they satisfy an asymptotic efficiency property if and only if we choose for M the inverse of the errors covariance matrix Γ-1. When Γ is unknown and estimated by the sampling Residual covariance matrix (also called the within covariance matrix), we are led to the diagonalization of or (with . A study of the asymptotic properties of the estimators is then feasible in the framework of Discriminant Factorial Analysis, in the case where the population between covariance matrix V E is assumed to be of rank q.

Three-Stage Procedures for Selecting the Largest Normal Mean

SYNOPTIC ABSTRACTWe select, from k(≥2) independent normal populations, the one with the largest mean, where the common variance is unknown. Under the “indifference zone” approach of Bechhofer (1954), we consider three–stage selection procedures. Asymptotic second order expansions for the average sample size and the probability of correct selection are provided here for arbitrary k. The second order expansions show that our three–stage procedure enjoys the second–order asymptotic efficiency property, and gives us the asymptotic number of extra samples necessary over the sequential procedure of Robbins, Sobel and Starr (1968), and over the fixed sample approach, had the variance been known, of Bechhofer (1954). Computer simulations are used to explore moderate sample size performance.

Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics

Abstract Considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics. The most commonly used generalizations are the log-rank test statistic, which extends the Savage exponential scores statistic (Cox 1972; Mantel 1966) and the generalized Wilcoxon statistics (Gehan 1965; Peto and Peto 1972). Gill (1980) showed that in the two-sample problem the asymptotic efficiency properties of the Savage and Wilcoxon statistics are maintained in censored data by the log-rank and Peto and Peto Wilcoxon statistics, respectively, when censoring is the same in both samples. Heuristics and simulations have implied that some recently proposed supremum statistics, closely related to these two popular linear rank statistics, can be used to construct tests that provide good power against a much wider range of nonlocal alternatives (see, e.g., Fleming and Harrington 1981; Fleming, O'Brien, O'Fallon, and Harrington 1980; Gill 1980; Schumacher 1984). In this article we consider n...

Brown-Mood Type Median Estimators for Simple Regression Models

In the context of simple linear regression we investigate a class of estimators for the parameters which contains the Brown-Mood estimators. We derive the asymptotic efficiency properties of the class of slope estimators and find the most efficient slope estimator. This estimator is shown to have efficiency properties analogous to those of other median estimators for slope found in the literature. The equations which define the optimal estimators are median analogues of the least squares normal equation.

Distribution-free comparison of two probability distributions with reference to their hazard rates

S-UMMARY Let F(x) and G(x) be the absolutely continuous distribution functions of two life distributions with rF(x) and rG(x) as their respective hazard rates. In the class of increasing failure rate distributions, one-sided scale as well as one-sided location alternatives imply that one hazard rate is uniformly smaller than the other. A new distribution-free test for testing Ho: rF(x) = rG(x) against HA: rF(x) < rG(x) has been proposed. The test is seen to possess robust asymptotic efficiency properties.

Robust estimates of linear trend in multivariate time series

Median and weighted median estimates are obtained for the linear trend parameters of a univariate time series by applying the Hodges-Lehmann method to some well known nonparametric tests for trend. The estimation procedure is extended to the multivariate trend model and asymptotic efficiency properties relative to the classical estimates are studied.