Metrika最新文献

The paper considers the nonparametric Poisson regression problem with a regular equidistant design on the unit interval. The nonparametric estimation of the Poisson regression function is studied. The uniform consistency conditions are established and the limit theorems are proved for continuous functionals on (C[a,1-a]), (0<a<frac{1}{2}) .

Abstract

In this work, we construct a stable and fairly fast estimator for solving multidimensional non-parametric regression problems. The proposed estimator is based on the use of a novel and special system of multivariate Jacobi polynomials that generate a basis for a reduced size of (d-) variate finite dimensional polynomials space. An ANOVA decomposition trick has been used for building this space. Also, by using some results from the theory of positive definite random matrices, we show that the proposed estimator is stable under the condition that the i.i.d. (d-) dimensional random sampling training points follow a (d-) dimensional Beta distribution. In addition, we provide the reader with an estimate for the (L^2-) risk error of the estimator. This risk error depends on the (L^2-) error of the orthogonal projection error of the regression function over the considered polynomials space. An involved study of this orthogonal projection error is done under the condition that the regression function belongs to a given weighted Sobolev space. Thanks to this novel estimate of the orthogonal projection error, we give the optimal convergence rate of our estimator. Furthermore, we give a regularized extension version of our estimator, that is capable of handling random sampling training vectors drawn according to an unknown multivariate pdf. Moreover, we derive an upper bound for the empirical risk error of this regularized estimator. Finally, we give some numerical simulations that illustrate the various theoretical results of this work. In particular, we provide simulations on a real data that compares the performance of our estimator with some existing and popular NP regression estimators.

This paper presents a score-based weighted likelihood estimator (SWLE) for robust estimations of the generalized linear model (GLM) for insurance loss data. The SWLE exhibits a limited sensitivity to the outliers, theoretically justifying its robustness against model contaminations. Also, with the specially designed weight function to effectively diminish the contributions of extreme losses to the GLM parameter estimations, most statistical quantities can still be derived analytically, minimizing the computational burden for parameter calibrations. Apart from robust estimations, the SWLE can also act as a quantitative diagnostic tool to detect outliers and systematic model misspecifications. Motivated by the coverage modifications which make insurance losses often random censored and truncated, the SWLE is extended to accommodate censored and truncated data. We exemplify the SWLE on three simulation studies and two real insurance datasets. Empirical results suggest that the SWLE produces more reliable parameter estimates than the MLE if outliers contaminate the dataset. The SWLE diagnostic tool also successfully detects any systematic model misspecifications with high power, accompanying some potential model improvements.

The minimum aberration criterion is popular for selecting good designs with qualitative factors under an ANOVA model, and the minimum (beta )-aberration criterion is more suitable for selecting designs with quantitative factors under a polynomial model. However, numerous computer experiments involve both qualitative and quantitative factors, while there is a lack of a reasonable criterion to assess the effectiveness of such designs. This paper proposes some important properties of the (beta )-wordlength pattern for mixed-level designs, and introduces the minimum (theta )-aberration criterion for comparing and selecting designs with qualitative and quantitative factors based on a full model involving all interactions of the factors. The computation of the (theta )-wordlength pattern is optimized by the generalized wordlength enumerator. Then we provide some construction methods for designs with less (theta )-aberration, and apply this criterion to screen the marginally coupled designs and the doubly coupled designs.

Abstract

(L^p) -quantiles are a class of generalized quantiles defined as minimizers of an asymmetric power function. They include both quantiles, (p=1) , and expectiles, (p=2) , as special cases. This paper studies composite (L^p) -quantile regression, simultaneously extending single (L^p) -quantile regression and composite quantile regression. A Bayesian approach is considered, where a novel parameterization of the skewed exponential power distribution is utilized. Further, a Laplace prior on the regression coefficients allows for variable selection. Through a Monte Carlo study and applications to empirical data, the proposed method is shown to outperform Bayesian composite quantile regression in most aspects.

Beta regression models are employed to model continuous response variables in the unit interval, like rates, percentages, or proportions. Their applications rise in several areas, such as medicine, environment research, finance, and natural sciences. The maximum likelihood estimation is widely used to make inferences for the parameters. Nonetheless, it is well-known that the maximum likelihood-based inference suffers from the lack of robustness in the presence of outliers. Such a case can bring severe bias and misleading conclusions. Recently, robust estimators for beta regression models were presented in the literature. However, these estimators require non-trivial restrictions in the parameter space, which limit their application. This paper develops new robust estimators that overcome this drawback. Their asymptotic and robustness properties are studied, and robust Wald-type tests are introduced. Simulation results evidence the merits of the new robust estimators. Inference and diagnostics using the new estimators are illustrated in an application to health insurance coverage data. The new R package robustbetareg is introduced.

This paper is concerned with prediction for skew-normal models, and more specifically the Bayes estimation of a predictive density for (Y left. right| mu sim {mathcal {S}} {mathcal {N}}_p (mu , v_y I_p, lambda )) under Kullback–Leibler loss, based on (X left. right| mu sim {mathcal {S}} {mathcal {N}}_p (mu , v_x I_p, lambda )) with known dependence and skewness parameters. We obtain representations for Bayes predictive densities, including the minimum risk equivariant predictive density (hat{p}_{pi _{o}}) which is a Bayes predictive density with respect to the noninformative prior (pi _0equiv 1). George et al. (Ann Stat 34:78–91, 2006) used the parallel between the problem of point estimation and the problem of estimation of predictive densities to establish a connection between the difference of risks of the two problems. The development of similar connection, allows us to determine sufficient conditions of dominance over (hat{p}_{pi _{o}}) and of minimaxity. First, we show that (hat{p}_{pi _{o}}) is a minimax predictive density under KL risk for the skew-normal model. After this, for dimensions (pge 3), we obtain classes of Bayesian minimax densities that improve (hat{p}_{pi _{o}}) under KL loss, for the subclass of skew-normal distributions with small value of skewness parameter. Moreover, for dimensions (pge 4), we obtain classes of Bayesian minimax densities that improve (hat{p}_{pi _{o}}) under KL loss, for the whole class of skew-normal distributions. Examples of proper priors, including generalized student priors, generating Bayesian minimax densities that improve (hat{p}_{pi _{o}}) under KL loss, were constructed when (pge 5). This findings represent an extension of Liang and Barron (IEEE Trans Inf Theory 50(11):2708–2726, 2004), George et al. (Ann Stat 34:78–91, 2006) and Komaki (Biometrika 88(3):859–864, 2001) results to a subclass of asymmetrical distributions.

We are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general Hölder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.

Abstract

Stimulated by the need of describing useful notions related to information measures, we introduce the ‘pdf-related distributions’. These are defined in terms of transformation of absolutely continuous random variables through their own probability density functions. We investigate their main characteristics, with reference to the general form of the distribution, the quantiles, and some related notions of reliability theory. This allows us to obtain a characterization of the pdf-related distribution being uniform for distributions of exponential and Laplace type as well. We also face the problem of stochastic comparing the pdf-related distributions by resorting to suitable stochastic orders. Finally, the given results are used to analyse properties and to compare some useful information measures, such as the differential entropy and the varentropy.

The frequency polygon estimation, which is based on histogram technique, has similar convergence rate as those of non-negative kernel estimators and the advantages of computational simplicity. This work will study the Berry–Esséen bound of frequency polygon estimation with (rho )-mixing samples under some general conditions. The rates are shown to be (O(n^{-1/9})) if the mixing coefficients decay polynomially and (O(n^{-1/6}log ^{1/3}n)) if the mixing coefficients decay geometrically. These results improve and extend the corresponding ones in the literature and reveal that the frequency polygon estimator also has similar Berry–Esséen bound as those of kernel estimators. Moreover, some numerical analysis is also presented to assess the finite sample performance of the theoretical results.