Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Polynomial Histograms for Multivariate Density and Mode Estimation

Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n^?6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.     

Bias reduction in kernel density estimation

In this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.     

Kernel Method Starting with Half-Normal Detection Function for Line Transect Density Estimation

In this article, we introduce the nonparametric kernel method starting with half-normal detection function using line transect sampling. The new method improves bias from O(h ²), as the smoothing parameter h → 0, to O(h ³) and in some cases to O(h ⁴). Properties of the proposed estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the estimator are investigated and compared with the traditional kernel estimator by using simulation technique. A numerical results show that improvements over the traditional kernel estimator often can be realized even when the true detection function is far from the half-normal detection function.     

An efficient and fast algorithm for estimating the frequencies of superimposed exponential signals in zero-mean multiplicative and additive noise

In this paper, we consider the superimposed exponential signals in zero-mean multiplicative and additive noise when all the noise are independently and identically distributed. We use a three-step iterative procedure to estimate the frequencies of the considered model. It is observed that the estimators are consistent and work quite well in terms of biases and mean square errors. Moreover, the convergence rate of the estimators attains O _p (N ^?3/2), which is the best convergence rate in the case of only additive noise and constant amplitude.     

An Asymptotic Characterization of Finite Degree U-statistics With Sample Size-Dependent Kernels: Applications to Nonparametric Estimators and Test Statistics

We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic’s kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric R² test for overall significance of a nonparametric regression model.     

Realized Volatility: A Review

This article reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time model is presented in order to motivate the main results. A continuous time specification provides the theoretical foundation for the main results in this literature. Cases with and without microstructure noise are considered, and it is shown how microstructure noise can cause severe problems in terms of consistent estimation of the daily realized volatility. Independent and dependent noise processes are examined. The most important methods for providing consistent estimators are presented, and a critical exposition of different techniques is given. The finite sample properties are discussed in comparison with their asymptotic properties. A multivariate model is presented to discuss estimation of the realized covariances. Various issues relating to modelling and forecasting realized volatilities are considered. The main empirical findings using univariate and multivariate methods are summarized.     

Robust kernel estimators for additive models with dependent observations

Robust nonparametric estimators for additive regression or autoregression models under an α-mixing condition are proposed. They are based on local M-estimators or local medians with kernel weights, and their asymptotic behaviour is studied. Moreover, diese local M-estimators achieve the same univariate rate of convergence as their linear relatives.     

Realized Volatility: A Review

This article reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time model is presented in order to motivate the main results. A continuous time specification provides the theoretical foundation for the main results in this literature. Cases with and without microstructure noise are considered, and it is shown how microstructure noise can cause severe problems in terms of consistent estimation of the daily realized volatility. Independent and dependent noise processes are examined. The most important methods for providing consistent estimators are presented, and a critical exposition of different techniques is given. The finite sample properties are discussed in comparison with their asymptotic properties. A multivariate model is presented to discuss estimation of the realized covariances. Various issues relating to modelling and forecasting realized volatilities are considered. The main empirical findings using univariate and multivariate methods are summarized.     

Power Variations and Testing for Co‐Jumps: The Small Noise Approach

In this paper, we study the effects of noise on bipower variation, realized volatility (RV) and testing for co‐jumps in high‐frequency data under the small noise framework. We first establish asymptotic properties of bipower variation in this framework. In the presence of the small noise, RV is asymptotically biased, and the additional asymptotic conditional variance term appears in its limit distribution. We also propose consistent estimators for the asymptotic variances of RV. Second, we derive the asymptotic distribution of the test statistic proposed in (Ann. Stat. 37, 1792‐1838) under the presence of small noise for testing the presence of co‐jumps in a two‐dimensional Itô semimartingale. In contrast to the setting in (Ann. Stat. 37, 1792‐1838), we show that the additional asymptotic variance terms appear and propose consistent estimators for the asymptotic variances in order to make the test feasible. Simulation experiments show that our asymptotic results give reasonable approximations in the finite sample cases.     

ON ASYMPTOTIC PROPERTIES OF A TWO DIMENSIONAL FREQUENCY ESTIMATOR

Estimating parameters of a two dimensional frequency model is an important problem in statistical signal processing. In this paper, we consider the two-dimensional frequency model in presence of an additive stationary noise. We consider two different estimators and obtain their asymptotic properties. The asymptotic properties can be used to construct confidence intervals of the unknown parameters and for testing purposes also. The small sample performances of these estimators are observed using numerical simulations.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

Simultaneous confidence intervals on partial means of classes in the two-stage stratified sampling

Let’s consider a finite population of P units, each of them assumes a specific amount of the quantitative variable X. Moreover we assume that the range of values of X is subdivided into k classes and the sampling data come out from a two stage stratified sampling. The main purpose of the work is to determine the estimators, as well as their asymptotic distribution, of the partial means of classes, each of them is defined as a non linear function of the other parameters. Particularly, we are interested in determining the linear approximation estimators and, under convergence theorems, the asymptotic distribution. Afterwards we define the estimator of the vector of the partial means of classes and its asymptotic convergence to multivariate normal distribution is determined. These results are useful to develop simultaneous inferential procedures.     

Boundary and Bias Correction in Kernel Hazard Estimation

A new class of local linear hazard estimators based on weighted least square kernel estimation is considered. The class includes the kernel hazard estimator of Ramlau-Hansen (1983), which has the same boundary correction property as the local linear regression estimator (see Fan & Gijbels, 1996). It is shown that all the local linear estimators in the class have the same pointwise asymptotic properties. We derive the multiplicative bias correction of the local linear estimator. In addition we propose a new bias correction technique based on bootstrap estimation of additive bias. This latter method has excellent theoretical properties. Based on an extensive simulation study where we compare the performance of competing estimators, we also recommend the use of the additive bias correction in applied work.     

Inference for Multi‐dimensional High‐frequency Data with an Application to Conditional Independence Testing

We find the asymptotic distribution of the multi‐dimensional multi‐scale and kernel estimators for high‐frequency financial data with microstructure. Sampling times are allowed to be asynchronous and endogenous. In the process, we show that the classes of multi‐scale and kernel estimators for smoothing noise perturbation are asymptotically equivalent in the sense of having the same asymptotic distribution for corresponding kernel and weight functions. The theory leads to multi‐dimensional stable central limit theorems and feasible versions. Hence, they allow to draw statistical inference for a broad class of multivariate models, which paves the way to tests and confidence intervals in risk measurement for arbitrary portfolios composed of high‐frequently observed assets. As an application, we enhance the approach to construct a test for investigating hypotheses that correlated assets are independent conditional on a common factor.     

Histogram Model with Plausible Parametric Detection Function for Line Transect Data

This article develops a new model that combines between the histogram and plausible parametric detection function to estimate the population density (abundance) by using line transects technique. A parametric detection function is introduced to improve the properties of the classical histogram estimator. Asymptotic properties of the resulting estimator are derived and an expression for the asymptotic mean square error (AMSE) is given. A general formula for the optimal choice of the histogram bin width based on AMSE is derived. Moreover, other possible alternative procedures to select the bin width are suggested and studied via simulation technique. The results show the superiority of the proposed estimators over both the classical histogram and the usual kernel estimators in most reasonable cases. In addition, the simulation results indicate that the choice of a plausible detection function is less sensitive than the choice of a bin width on the performance of the proposed estimator.     

An estimating equation for parametric shared frailty models with marginal additive hazards

Summary.  Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In some contexts, however, it may be more reasonable to use the marginal additive hazards model. We derive asymptotic properties of the Lin and Ying estimators for the marginal additive hazards model for multivariate failure time data. Furthermore we suggest estimating equations for the regression parameters and association parameters in parametric shared frailty models with marginal additive hazards by using the Lin and Ying estimators. We give the large sample properties of the estimators arising from these estimating equations and investigate their small sample properties by Monte Carlo simulation. A real example is provided for illustration.     

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.     

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

In this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T² chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.     

Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields

We investigate the asymptotic behaviour of binned kernel density estimators for dependent and locally non-stationary random fields converging to stationary random fields. We focus on the study of the bias and the asymptotic normality of the estimators. A simulation experiment conducted shows that both the kernel density estimator and the binned kernel density estimator have the same behavior and both estimate accurately the true density when the number of fields increases. We apply our results to the 2002 incidence rates of tuberculosis in the departments of France.