Optimal Sequential Design in a Controlled Non-parametric Regression

Abstract. In a non‐parametric regression, the heteroscedasticity (dependence of the variance of the regression error on the predictor) can be a serious complication in estimation or visualization of an underlying regression function. If a controlled sampling is permitted, then the statistician can choose the design of predictors which attenuates the effect of heteroscedasticity. It is proposed to use a design which minimizes the mean integrated squared error of the regression function estimation. Then the corresponding optimal design density is proportional to the standard deviation of the regression error (the so‐called scale function). Because in general the statistician does not know an underlying scale function, the natural question is as follows: is it possible to suggest a sequential design which performs as well as an oracle that knows the underlying scale function? The answer is ‘yes’, and a corresponding sequential procedure is developed. It is proved, for the first time in the literature, that a data‐driven sequential design, together with an adaptive regression estimator, can mimic the oracle and be sharp minimax. Further, it is shown that the suggested method is feasible for small samples.     

Nonparametric density estimation based on beta prime kernel

Abstract

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.     

Wavelet-Based Density Estimation in a Heteroscedastic Convolution Model

We consider a heteroscedastic convolution density model under the “ordinary smooth assumption.” We introduce a new adaptive wavelet estimator based on term-by-term hard thresholding rule. Its asymptotic properties are explored via the minimax approach under the mean integrated squared error over Besov balls. We prove that our estimator attains near optimal rates of convergence (lower bounds are determined). Simulation results are reported to support our theoretical findings.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.     

Variable kernel density estimation

This paper considers the problem of selecting optimal bandwidths for variable (sample‐point adaptive) kernel density estimation. A data‐driven variable bandwidth selector is proposed, based on the idea of approximating the log‐bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross‐validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.     

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

Abstract

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.     

Nonparametric estimation of a two dimensional continuous–discrete density function by wavelets

We consider the estimation of a two dimensional continuous–discrete density function. A new methodology based on wavelets is proposed. We construct a linear wavelet estimator and a non-linear wavelet estimator based on a term-by-term thresholding. Their rates of convergence are established under the mean integrated squared error over Besov balls. In particular, we prove that our adaptive wavelet estimator attains a fast rate of convergence. A simulation study illustrates the usefulness of the proposed estimators.     

CORRECTING FOR KURTOSIS IN DENSITY ESTIMATION

Using a global window width kernel estimator to estimate an approximately symmetric probability density with high kurtosis usually leads to poor estimation because good estimation of the peak of the distribution leads to unsatisfactory estimation of the tails and vice versa. The technique proposed corrects for kurtosis via a transformation of the data before using a global window width kernel estimator. The transformation depends on a “generalised smoothing parameter” consisting of two real-valued parameters and a window width parameter which can be selected either by a simple graphical method or, for a completely data-driven implementation, by minimising an estimate of mean integrated squared error. Examples of real and simulated data demonstrate the effectiveness of this approach, which appears suitable for a wide range of symmetric, unimodal densities. Its performance is similar to ordinary kernel estimation in situations where the latter is effective, e.g. Gaussian densities. For densities like the Cauchy where ordinary kernel estimation is not satisfactory, our methodology offers a substantial improvement.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

Adaptive Estimation of the Integral of Squared Regression Derivatives

A problem of estimating the integral of a squared regression function and of its squared derivatives has been addressed in a number of papers. For the case of a heteroscedastic model where smoothness of the underlying regression function, the design density, and the variance of errors are known, the asymptotically sharp minimax lower bound and a sharp estimator were found in Pastuchova & Khasminski (1989). However, there are apparently no results on the either rate optimal or sharp optimal adaptive, or data-driven, estimation when neither the degree of regression function smoothness nor design density, scale function and distribution of errors are known. After a brief review of main developments in non-parametric estimation of non-linear functionals, we suggest a simple adaptive estimator for the integral of a squared regression function and its derivatives and prove that it is sharp-optimal whenever the estimated derivative is sufficiently smooth.     

On MISE of a Non linear Wavelet Estimator of the Regression Function Based on Biased Data under Strong Mixing

In this paper, we consider the adaptation of the non linear wavelet-based estimator of the regression function for the biased data setup under strong mixing. We provide an asymptotic sharp bound for the mean integrated squared error (MISE) of the estimator, that is nearly optimal in the minimax sense over a large range of Besov function classes.     

One-sided trimming in small samples with asymmetric contamination

An estimator for location, given a sample of only four or five observations, is proposed. The underlying distribution on of the sample may (with probability p) be contaminated by an outlier from a rightly-skewed distribution. The estimator minimizes the maximum mean squared error over all values of p. In fact, there exists an estimator which is unbiased in both the outlier - free and extreme-outlier cases, but its mean square error is substantially higher than the mean squared error for the minimax estimator. Mean squared errors for various underlying distributional situations are calculated and compared with those of other location estimators such as the mean and the median.     

Adaptive wavelet estimation of a density from mixtures under multiplicative censoring

In this paper, a mixture model under multiplicative censoring is considered. We investigate the estimation of a component of the mixture (a density) from the observations. A new adaptive estimator based on wavelets and a hard thresholding rule is constructed for this problem. Under mild assumptions on the model, we study its asymptotic properties by determining an upper bound of the mean integrated squared error over a wide range of Besov balls. We prove that the obtained upper bound is sharp.     

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.     

Nonparametric Wavelet Regression Based on Biased Data

The estimation of the regression function in the biased nonparametric regression model is investigated. We propose and develop a new wavelet-based methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls.     

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.     

Adaptive Estimation Using Weighted Least Squares

This paper proposes an adaptive estimator that is more precise than the ordinary least squares estimator if the distribution of random errors is skewed or has long tails. The adaptive estimates are computed using a weighted least squares approach with weights based on the lengths of the tails of the distribution of residuals. Smaller weights are assigned to those observations that have residuals in the tails of long-tailed distributions and larger weights are assigned to observations having residuals in the tails of short-tailed distributions. Monte Carlo methods are used to compare the performance of the proposed estimator and the performance of the ordinary least squares estimator. The estimates that were studied in this simulation include the difference between the means of two populations, the mean of a symmetric distribution, and the slope of a regression line. The adaptive estimators are shown to have lower mean squared errors than those for the ordinary least squares estimators for short-tailed, long-tailed, and skewed distributions, provided the sample size is at least 20. The ordinary least squares estimator has slightly lower mean squared error for normally distributed errors. The adaptive estimator is recommended for general use for studies having sample sizes of at least 20 observations unless the random errors are known to be normally distributed.     

Bayesian adaptive bandwidth selector for multivariate discrete kernel estimator

We treat a non parametric estimator for joint probability mass function, based on multivariate discrete associated kernels which are appropriated for multivariate count data of small and moderate sample sizes. Bayesian adaptive estimation of the vector of bandwidths using the quadratic and entropy loss functions is considered. Exact formulas for the posterior distribution and the vector of bandwidths are obtained. Numerical studies indicate that the performance of our approach is better, comparing with other bandwidth selection techniques using integrated squared error as criterion. Some applications are made on real data sets.