Optimal and Robust Designs for Estimating the Concentration Curve and the AUC

The problem of interest is to estimate the concentration curve and the area under the curve (AUC) by estimating the parameters of a linear regression model with an autocorrelated error process. We introduce a simple linear unbiased estimator of the concentration curve and the AUC. We show that this estimator constructed from a sampling design generated by an appropriate density is asymptotically optimal in the sense that it has exactly the same asymptotic performance as the best linear unbiased estimator. Moreover, we prove that the optimal design is robust with respect to a minimax criterion. When repeated observations are available, this estimator is consistent and has an asymptotic normal distribution. Finally, a simulated annealing algorithm is applied to a pharmacokinetic model with correlated errors.     

Empirical bayes estimation of binomial parameter with symmetric priors

This paper deals with the problem of estimating the binomial parameter via the nonparametric empirical Bayes approach. This estimation problem has the feature that estimators which are asymptotically optimal in the usual empirical Bayes sense do not exist (Robbins (1958, 1964)), However, as pointed out by Liang (1934) and Gupta and Liang (1988), it is possible to construct asymptotically optimal empirical Bayes estimators if the unknown prior is symmetric about the point 1/2, In this paper, assuming symmetric priors a monotone empirical Bayes estimator is constructed by using the isotonic regression method. This estimator is asymptotically optimal in the usual empirical Bayes sense. The corresponding rate of convergence is investigated and shown to be of order n^-1, where n is the number of past observations at hand.     

SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS

Härdle & Marron (1990) treated the problem of semiparametric comparison of nonparametric regression curves by proposing a kernel-based estimator derived by minimizing a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are n-consistent, which prompts a consideration of issues. of optimality. We show that when the unknown mean function is periodic, an optimal nonparametric estimator may be motivated by an elegantly simple argument based on maximum likelihood estimation in a parametric model with normal errors. Strikingly, the asymptotic variance of an optimal estimator of θ does not depend at all on the manner of estimating error variances, provided they are estimated n-consistently. The optimal kernel-based estimator derived via these considerations is asymptotically equivalent to a periodic version of that suggested by Härdle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case.     

Optimal sequential estimation procedures of a function of a probability of success under LINEX loss

In this paper, we investigate the problem of estimating a function g(p), where p is the probability of success in a sequential sample of independent identically Bernoulli distributed random variables. As a loss associated with estimation we introduce a generalized LINEX loss function. We construct a sequential procedure possessing some asymptotically optimal properties in the case when p tends to zero. In this approach to the problem, the conditions are given, under which the stopping time is asymptotically efficient and normal, and the corresponding sequential estimator is asymptotically normal. The procedure constructed guarantees that its sequential risk is asymptotically equal to a prescribed constant.     

Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions

Abstract.  We consider an asymptotically efficient estimator of the drift parameter for a multi-dimensional diffusion process with small dispersion parameter ɛ . In the situation where the sample path is observed at equidistant times k / n , k  = 0, 1, …,  n , we study asymptotic properties of an M -estimator derived from an approximate martingale estimating function as ɛ tends to 0 and n tends to ∞ simultaneously.     

Estimating partially linear panel data models with one-way error components

We consider the problem of estimating a partially linear panel data model whenthe error follows an one-way error components structure. We propose a feasiblesemiparametric generalized least squares (GLS) type estimator for estimating the coefficient of the linear component and show that it is asymptotically more efficient than a semiparametric ordinary least squares (OLS) type estimator. We also discussed the case when the regressor of the parametric component is correlated with the error, and propose an instrumental variable GLS-type semiparametric estimator.     

New generalized regression estimator in the presence of non response under unequal probability sampling

In this paper, we propose a new generalized regression estimator for the problem of estimating the population total using unequal probability sampling without replacement. A modified automated linearization approach is applied in order to transform the proposed estimator to estimate variance of population total. The variance and estimated value of the variance of the proposed estimator is investigated under a reverse framework assuming that the sampling fraction is negligible and there are equal response probabilities for all units. We prove that the proposed estimator is an asymptotically unbiased estimator and that it does not require a known or estimated response probability to function.     

An Extended Single‐index Model with Missing Response at Random

An extended single‐index model is considered when responses are missing at random. A three‐step estimation procedure is developed to define an estimator for the single‐index parameter vector by a joint estimating equation. The proposed estimator is shown to be asymptotically normal. An algorithm for computing this estimator is proposed. This algorithm only involves one‐dimensional nonparametric smoothers, thereby avoiding the data sparsity problem caused by high model dimensionality. Some simulation studies are conducted to investigate the finite sample performances of the proposed estimators.     

Ratios of linear combinations in the general linear model

We consider the problem of estimating. the ratio of two linear combinations of thevector of parameters in the general linear model. The nonexistence of an unbiased estimator under normal errorsisdiscussed. Properties of an often used estimator, the maximum likelihood estimator under normal errors, are presented, This is done both for fixed sample size and asymptotically, in the

presence of normal and non-normal errors.     

An emprical bayes approach to two-sample problems with censored data

Empirical Bayes methods are used in estimating the probability based on randomly right-censored samples. The estimator is shown to be asymptotically optimal. Thus, in a way, this uork is similar to the results of Hollander and Korwar (1976) who used a similar approach in estimating A in the case of non-censored data. We also give hero a shorter proof to their rate result. In addition, a. resting procedure is obtained to test the hypothesis against on the basis of censored data. It is shown that this procedure is asymptotically optimal with rate of convergence n . Tnis result is analogous to our earlier result for the uncensored case (1970) The empirical Hayes procedure has been illustrated by means of a practical example.     

Estimating a uniform distribution when data are measured with a normal additive error with unknown variance

The problem of estimating the width of a symmetric uniform distribution on the line together with the error variance, when data are measured with normal additive error, is considered. The main purpose is to analyse the maximum-likelihood (ML) estimator and to compare it with the moment-method estimator. It is shown that this two-parameter model is regular so that the ML estimator is asymptotically efficient. Necessary and sufficient conditions are given for the existence of the ML estimator. As numerical problems are known to frequently occur while computing the ML estimator in this model, useful suggestions for computing the ML estimator are also given.     

Failure Inference and Optimization for Step Stress Model Based on Bivariate Wiener Model

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distribution.     

Interpolation,outliers mid inverse autocorrelations

The paper addresses the problem of estimating missing observations in an infinite realization of a linear, possibly nonstationary, stochastic processes when the model is known. The general case of any possible distribution of missing observations in the time series is considered, and analytical expressions for the optimal estimators and their associated mean squared errors are obtained. These expressions involve solely the elements of the inverse or dual autocorrelation function of the series.

This optimal estimator -the conditional expectation of the missing observations given the available ones- is equal to the estimator that results from filling the missing values in the series with arbitrary numbers, treating these numbers as additive outliers, and removing with intervention analysis the outlier effects from the invented numbers.     

Econometric Reviews

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Statistical inference for a single-index varying-coefficient model

We investigate the estimators of parameters of interest for a single-index varying-coefficient model. To estimate the unknown parameter efficiently, we first estimate the nonparametric component using local linear smoothing, then construct an estimator of parametric component by using estimating equations. Our estimator for the parametric component is asymptotically efficient, and the estimator of nonparametric component has asymptotic normality and optimal uniform convergence rate. Our results provide ways to construct confidence regions for the involved unknown parameters. The finite-sample behavior of the new estimators is evaluated through simulation studies, and applications to two real data are illustrated.     

Minimax Linear Smoothers

We consider the problem of estimating the mean of a multivariate distribution. As a general alternative to penalized least squares estimators, we consider minimax estimators for squared error over a restricted parameter space where the restriction is determined by the penalization term. For a quadratic penalty term, the minimax estimator among linear estimators can be found explicitly. It is shown that all symmetric linear smoothers with eigenvalues in the unit interval can be characterized as minimax linear estimators over a certain parameter space where the bias is bounded. The minimax linear estimator depends on smoothing parameters that must be estimated in practice. Using results in Kneip (1994), this can be done using Mallows' C _L -statistic and the resulting adaptive estimator is now asymptotically minimax linear. The minimax estimator is compared to the penalized least squares estimator both in finite samples and asymptotically.     

A smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes

In this paper, we propose a smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes. In contrast to the Q‐learning algorithm in which nonregular inference is involved, we show that, under assumptions adopted in this paper, the proposed smoothed Q‐learning estimator is asymptotically normally distributed even when the Q‐learning estimator is not and its asymptotic variance can be consistently estimated. As a result, inference based on the smoothed Q‐learning estimator is standard. We derive the optimal smoothing parameter and propose a data‐driven method for estimating it. The finite sample properties of the smoothed Q‐learning estimator are studied and compared with several existing estimators including the Q‐learning estimator via an extensive simulation study. We illustrate the new method by analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness–Alzheimer's Disease (CATIE‐AD) study.     

Generalized mixed estimator for nonlinear models: a maximum likelihood approach

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Linear-representation Based Estimation of Stochastic Volatility Models

Abstract.  A new way of estimating stochastic volatility models is developed. The method is based on the existence of autoregressive moving average (ARMA) representations for powers of the log-squared observations. These representations allow to build a criterion obtained by weighting the sums of squared innovations corresponding to the different ARMA models. The estimator obtained by minimizing the criterion with respect to the parameters of interest is shown to be consistent and asymptotically normal. Monte-Carlo experiments illustrate the finite sample properties of the estimator. The method has potential applications to other non-linear time-series models.     

Multistage plug—in bandwidth selection for kernel distribution function estimates

The use of a kernel estimator as a smooth estimator for a distribution function has been suggested by many authors An expression for the bandwidth that minimizes the mean integrated square error asymptotically has been available for some time. However, few practical data based methods ior estimating this bandwidth have been investigated. In this paper we propose multisstage plug-in type estimater for this optimal bandwith and derive its asymptotic properties. In particular we show that two stages are required for good asymptotic properties. This behavior is verified for finite samples using a simulation study.