Estimating population characteristics by incorporating prior values in stratified random sampling/ranked set sampling

In this paper, we discuss the estimation of population characteristics using stratified random sampling in an infinite population framework, including ranked set sampling as a special case. The use of prior values is considered and the underlying distribution is assumed to be unknown. The estimator considered in each stratum is the weighted mean of the U-statistic and prior value. The optimum weight is obtained by minimizing the mean squared error of the estimator of the population characteristics, but it contains unknown parameters and those parameters are replaced with their estimates. Simulation results show the gains in efficiency of the proposed estimator, yielding gains of at least 1.2 times larger than the usual unbiased estimator under certain condition specified in the text. Guidelines for the usage of the proposed estimator are shown and an application to a real data set is provided.     

ON THE USE OF THE STEIN VARIANCE ESTIMATOR IN THE DOUBLE k-CLASS ESTIMATOR IN REGRESSION

This paper investigates the predictive mean squared error performance of a modified double k-class estimator by incorporating the Stein variance estimator. Recent studies show that the performance of the Stein rule estimator can be improved by using the Stein variance estimator. However, as we demonstrate below, this conclusion does not hold in general for all members of the double k-class estimators. On the other hand, an estimator is found to have smaller predictive mean squared error than the Stein variance-Stein rule estimator, over quite large parts of the parameter space.     

Model averaging by jackknife criterion for varying-coefficient partially linear models

Abstract

This paper is concerned with model averaging procedure for varying-coefficient partially linear models. We proposed a jackknife model averaging method that involves minimizing a leave-one-out cross-validation criterion, and developed a computational shortcut to optimize the cross-validation criterion for weight choice. The resulting model average estimator is shown to be asymptotically optimal in terms of achieving the smallest possible squared error. The simulation studies have provided evidence of the superiority of the proposed procedures. Our approach is further applied to a real data.     

Optimal model averaging estimation for correlation structure in generalized estimating equations

Longitudinal data analysis requires a proper estimation of the within-cluster correlation structure in order to achieve efficient estimates of the regression parameters. When applying likelihood-based methods one may select an optimal correlation structure by the AIC or BIC. However, such information criteria are not applicable for estimating equation based approaches. In this paper we develop a model averaging approach to estimate the correlation matrix by a weighted sum of a group of patterned correlation matrices under the GEE framework. The optimal weight is determined by minimizing the difference between the weighted sum and a consistent yet inefficient estimator of the correlation structure. The computation of our proposed approach only involves a standard quadratic programming on top of the standard GEE procedure and can be easily implemented in practice. We provide theoretical justifications and extensive numerical simulations to support the application of the proposed estimator. A couple of well-known longitudinal data sets are revisited where we implement and illustrate our methodology.     

Regularization through variable selection and conditional MLE with application to classification in high dimensions

It is often the case that high-dimensional data consist of only a few informative components. Standard statistical modeling and estimation in such a situation is prone to inaccuracies due to overfitting, unless regularization methods are practiced. In the context of classification, we propose a class of regularization methods through shrinkage estimators. The shrinkage is based on variable selection coupled with conditional maximum likelihood. Using Stein's unbiased estimator of the risk, we derive an estimator for the optimal shrinkage method within a certain class. A comparison of the optimal shrinkage methods in a classification context, with the optimal shrinkage method when estimating a mean vector under a squared loss, is given. The latter problem is extensively studied, but it seems that the results of those studies are not completely relevant for classification. We demonstrate and examine our method on simulated data and compare it to feature annealed independence rule and Fisher's rule.     

Strong convergence rate of the least median absolute estimator in linear regression models

We propose a least median of absolute (LMA) estimator for a linear regression model, based on minimizing the median absolute deviation of the residuals. Under some regularity conditions on the design points and disturbances, the strong convergence rate of the LMA estimator is established.     

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.     

NONPARAMETRIC ESTIMATORS FOR CENSORED CORRELATED DATA

ABSTRACT

In the context of failure time data, over the long run, dependent observations that might be censored are commonly encountered in practice. The main objective of this paper is to make inference about the common marginal distribution of the failure times. To this end, one nonparametric estimator, namely, the Nelson-Aalen estimator is modified to incorporate the dependence among the observations. The modified estimator is the weighted moving average (WMA) version of the existing estimator used for independent data. It has been shown that the new version is better in the sense of minimizing the one-step ahead forecast errors. Also, the new estimator can be used as a crude measure for checking independence among observations.     

A note on adaptation in garch models

In the framework of the Engle-type (G)ARCH models, I demonstrate that there is a family of symmetric and asymmetric density functions for which the asymptotic efficiency of the semiparametric estimator is equal to the asymptotic efficiency of the maximum likelihood estimator. This family of densities is bimodal (except for the normal). I also chracterize the solution to the problem of minimizing the mean squared distance between the parametric score and the semiparametric score in order to search for unimodal densities for which the semiparametric estimator is likely to perform well. The LaPlace density function emerges as one of these cases.     

Local Smoothing Using the Bootstrap

We consider the problem of data-based choice of the bandwidth of a kernel density estimator, with an aim to estimate the density optimally at a given design point. The existing local bandwidth selectors seem to be quite sensitive to the underlying density and location of the design point. For instance, some bandwidth selectors perform poorly while estimating a density, with bounded support, at the median. Others struggle to estimate a density in the tail region or at the trough between the two modes of a multimodal density. We propose a scale invariant bandwidth selection method such that the resulting density estimator performs reliably irrespective of the density or the design point. We choose bandwidth by minimizing a bootstrap estimate of the mean squared error (MSE) of a density estimator. Our bootstrap MSE estimator is different in the sense that we estimate the variance and squared bias components separately. We provide insight into the asymptotic accuracy of the proposed density estimator.     

Local Smoothing for Kernel Distribution Function Estimation

The problem of bandwidth selection for kernel-based estimation of the distribution function (cdf) at a given point is considered. With appropriate bandwidth, a kernel-based estimator (kdf) is known to outperform the empirical distribution function. However, such a bandwidth is unknown in practice. In pointwise estimation, the appropriate bandwidth depends on the point where the function is estimated. The existing smoothing methods use one common bandwidth to estimate the cdf. The accuracy of the resulting estimates varies substantially depending on the cdf and the point where it is estimated. We propose to select bandwidth by minimizing a bootstrap estimator of the MSE of the kdf. The resulting estimator performs reliably, irrespective of where the cdf is estimated. It is shown to be consistent under i.i.d. as well as strongly mixing dependence assumption. Two applications of the proposed estimator are shown in finance and seismology. We report a dataset on the S & P Nifty index values.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Robust Principal Component Functional Logistic Regression

In this article, we discuss the estimation of the parameter function for a functional logistic regression model in the presence of outliers. We consider ways that allow for the parameter estimator to be resistant to outliers, in addition to minimizing multicollinearity and reducing the high dimensionality, which is inherent with functional data. To achieve this, the functional covariates and functional parameter of the model are approximated in a finite-dimensional space generated by an appropriate basis. This approach reduces the functional model to a standard multiple logistic model with highly collinear covariates and potential high-dimensionality issues. The proposed estimator tackles these issues and also minimizes the effect of functional outliers. Results from a simulation study and a real world example are also presented to illustrate the performance of the proposed estimator.     

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.     

On variable bandwidth selection in local polynomial regression

The performances of data-driven bandwidth selection procedures in local polynomial regression are investigated by using asymptotic methods and simulation. The bandwidth selection procedures considered are based on minimizing 'prelimit' approximations to the (conditional) mean-squared error (MSE) when the MSE is considered as a function of the bandwidth h . We first consider approximations to the MSE that are based on Taylor expansions around h=0 of the bias part of the MSE. These approximations lead to estimators of the MSE that are accurate only for small bandwidths h . We also consider a bias estimator which instead of using small h approximations to bias naïvely estimates bias as the difference of two local polynomial estimators of different order and we show that this estimator performs well only for moderate to large h . We next define a hybrid bias estimator which equals the Taylor-expansion-based estimator for small h and the difference estimator for moderate to large h . We find that the MSE estimator based on this hybrid bias estimator leads to a bandwidth selection procedure with good asymptotic and, for our Monte Carlo examples, finite sample properties.     

On finite-sample properties of adaptive least squares regression estimates

This paper is mainly concerned with deriving finite-sample properties of least squares estimators for the regression function in a nonparametric regression situation under some simplifying assumptions such as normally distributed errors with a common known variance. The selection of basis functions to be used for the construction of an estimator may be regarded as a smoothing problem, and will usually be done in a data-dependent way, A straightforward application of a result by P. J. Kernpthorne yields that, under a squared error loss, all selection procedures are admissible. Furthermore, the minimax approach provides an interpolating estimator, which is often impractical, Therefore, within a certain class of selection procedures an optimal one is determined using the minimax regret principle. It can be seen to behave similarly to the procedure minimizing either an unbiased risk estimator or, equivalently, the C_p-criterion.     

A gm estimation of the location parameters in a spatial linear model

Universal kriging is a form of interpolation that takes into account the local trends in data when minimizing the error associated with the estimator. Under multivariate normality assumptions, the given predictor is the best linear unbiased predictor. but if the underlying distribution is not normal, the estimator will not be unbiased and will be vulnerable to outliers. With spatial data, it is not only the presence of outliers that may spoil the predictions, but also the boundary sites. usually corners, that tend to have high leverage. As an alternative, a weighted one-step generalized M estimator of the location parameters in a spatial linear model is proposed. It is especially recommended in the case of irregularly spaced data.     

Frequentist model averaging for envelope models

The envelope method produces efficient estimation in multivariate linear regression, and is widely applied in biology, psychology, and economics. This paper estimates parameters through a model averaging methodology and promotes the predicting abilities of the envelope models. We propose a frequentist model averaging method by minimizing a cross-validation criterion. When all the candidate models are misspecified, the proposed model averaging estimator is proved to be asymptotically optimal. When correct candidate models exist, the coefficient estimator is proved to be consistent, and the sum of the weights assigned to the correct models, in probability, converges to one. Simulations and an empirical application demonstrate the effectiveness of the proposed method.     

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.     

The optimal group size using inverse binomial group testing considering misclassification

ABSTRACT

Inverse binomial sampling is preferred for quick report. It is also recommended when the population proportion is really small to ensure a positive sample is contained. Group testing has been discussed extensively under binomial model, but not so much under negative binomial model. In this study, we investigate the problem of how to determine the group size using inverse binomial group testing. We propose to choose the optimal group size by minimizing asymptotic variance of the estimator or the cost relative to Fisher information. We show the good performance of our estimator by applying to the data of Chlamydia.