A note on bias reduction in variable-kernel density estimates

It is well known that the inverse-square-root rule of Abramson (1982) for the bandwidth h of a variable-kernel density estimator achieves a reduction in bias from the fixed-bandwidth estimator, even when a nonnegative kernel is used. Without some form of “clipping” device similar to that of Abramson, the asymptotic bias can be much greater than O(h⁴) for target densities like the normal (Terrell and Scott 1992) or even compactly supported densities. However, Abramson used a nonsmooth clipping procedure intended for pointwise estimation. Instead, we propose a smoothly clipped estimator and establish a globally valid, uniformly convergent bias expansion for densities with uniformly continuous fourth derivatives. The main result extends Hall's (1990) formula (see also Terrell and Scott 1992) to several dimensions, and actually to a very general class of estimators. By allowing a clipping parameter to vary with the bandwidth, the usual O(h⁴) bias expression holds uniformly on any set where the target density is bounded away from zero.     

L 1-estimation for varying coefficient models

The varying coefficient model is a useful extension of linear models and has many advantages in practical use. To estimate the unknown functions in the model, the kernel type with local linear least-squares (L ₂) estimation methods has been proposed by several authors. When the data contain outliers or come from population with heavy-tailed distributions, L ₁-estimation should yield better estimators. In this article, we present the local linear L ₁-estimation method and derive the asymptotic distributions of the L ₁-estimators. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L ₁-estimators outperform the L ₂-estimators.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

Weighted L1-estimates for the First-order Bifurcating Autoregressive Model

We developed robust estimators that minimize a weighted L₁ norm for the first-order bifurcating autoregressive model. When all of the weights are fixed, our estimate is an L₁ estimate that is robust against outlying points in the response space and more efficient than the least squares estimate for heavy-tailed error distributions. When the weights are random and depend on the points in the factor space, the weighted L₁ estimate is robust against outlying points in the factor space. Simulated and artificial examples are presented. The behavior of the proposed estimate is modeled through a Monte Carlo study.     

Minimum local distance density estimation

We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sample distance to x over all such subsets is used to define the density estimate at x. The tuning parameter is thus the number of subsets instead of the typical bandwidth of kernel or histogram-based density estimators. The proposed method is similar to nearest-neighbor density estimators but it provides smoother estimates. We derive the asymptotic distribution of this minimum sample distance statistic to study globally optimal values for the number and size of the subsets. Simulations are used to illustrate and compare the convergence properties of the estimator. The results show that the method provides good estimates of a wide variety of densities without changes of the tuning parameter, and that it offers competitive convergence performance.     

Density Estimation by Total Variation Penalized Likelihood Driven by the Sparsity ℓ1 Information Criterion

Abstract. We propose a non‐linear density estimator, which is locally adaptive, like wavelet estimators, and positive everywhere, without a log‐ or root‐transform. This estimator is based on maximizing a non‐parametric log‐likelihood function regularized by a total variation penalty. The smoothness is driven by a single penalty parameter, and to avoid cross‐validation, we derive an information criterion based on the idea of universal penalty. The penalized log‐likelihood maximization is reformulated as an ?₁‐penalized strictly convex programme whose unique solution is the density estimate. A Newton‐type method cannot be applied to calculate the estimate because the ?₁‐penalty is non‐differentiable. Instead, we use a dual block coordinate relaxation method that exploits the problem structure. By comparing with kernel, spline and taut string estimators on a Monte Carlo simulation, and by investigating the sensitivity to ties on two real data sets, we observe that the new estimator achieves good L ₁ and L ₂ risk for densities with sharp features, and behaves well with ties.     

On the Optimality of Prediction-based Selection Criteria and the Convergence Rates of Estimators

Several estimators of squared prediction error have been suggested for use in model and bandwidth selection problems. Among these are cross-validation, generalized cross-validation and a number of related techniques based on the residual sum of squares. For many situations with squared error loss, e.g. nonparametric smoothing, these estimators have been shown to be asymptotically optimal in the sense that in large samples the estimator minimizing the selection criterion also minimizes squared error loss. However, cross-validation is known not to be asymptotically optimal for some `easy' location problems. We consider selection criteria based on estimators of squared prediction risk for choosing between location estimators. We show that criteria based on adjusted residual sum of squares are not asymptotically optimal for choosing between asymptotically normal location estimators that converge at rate n ^1/2but are when the rate of convergence is slower. We also show that leave-one-out cross-validation is not asymptotically optimal for choosing between √ n -differentiable statistics but leave- d -out cross-validation is optimal when d ∞ at the appropriate rate.     

Local Linear Estimation for Spatiotemporal Models Based on Least Absolute Deviation

When the data contain outliers or come from population with heavy-tailed distributions, which appear very often in spatiotemporal data, the estimation methods based on least-squares (L₂) method will not perform well. More robust estimation methods are required. In this article, we propose the local linear estimation for spatiotemporal models based on least absolute deviation (L₁) and drive the asymptotic distributions of the L₁-estimators under some mild conditions imposed on the spatiotemporal process. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L₁-estimators perform better than the L₂-estimators.     

On the L1‐consistency of wavelet density estimates

The authors analyze the L₁ performance of wavelet density estimators. They prove that under mild conditions on the family of wavelets, such estimates are universally consistent in the L₁ sense.     

A New Family of Nonparametric Quantile Estimators

A new core methodology for creating nonparametric L-quantile estimators is introduced and three new quantile L-estimators (SV1 _p , SV2 _p , and SV3 _p ) are constructed using the new methodology. Monte Carlo simulation was used in order to investigate the performance of the new estimators for small and large samples under normal distribution and a variety of light and heavy-tailed symmetric and asymmetric distributions. The new estimators outperform, in most of the cases studied, the Harrell–Davis quantile estimator and the weighted average at X _([np]) quantile estimator.     

Kernel density estimation on the torus

Kernel density estimation for multivariate, circular data has been formulated only when the sample space is the sphere, but theory for the torus would also be useful. For data lying on a d-dimensional torus (d?1), we discuss kernel estimation of a density, its mixed partial derivatives, and their squared functionals. We introduce a specific class of product kernels whose order is suitably defined in such a way to obtain L₂-risk formulas whose structure can be compared to their Euclidean counterparts. Our kernels are based on circular densities; however, we also discuss smaller bias estimation involving negative kernels which are functions of circular densities. Practical rules for selecting the smoothing degree, based on cross-validation, bootstrap and plug-in ideas are derived. Moreover, we provide specific results on the use of kernels based on the von Mises density. Finally, real-data examples and simulation studies illustrate the findings.     

The Consistency and Robustness of Modified Cramér–Von Mises and Kolmogorov–Cramér Estimators

This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér–von Mises distance. The generalized power form of Cramér–von Mises distance is defined together with the so-called Kolmogorov–Cramér distance which includes both standard Kolmogorov and Cramér–von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L₁-norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L₁-norm on ?-contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov–Cramér, generalized Cramér–von Mises, standard Kolmogorov, and Cramér–von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the contaminated and uncontaminated pseudo-random samples.     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.     

On the performance of L2E estimation in modelling heterogeneous count responses with extreme values

In healthcare studies, count data sets measured with covariates often exhibit heterogeneity and contain extreme values. To analyse such count data sets, we use a finite mixture of regression model framework and investigate a robust estimation approach, called the L₂E [D.W. Scott, On fitting and adapting of density estimates, Comput. Sci. Stat. 30 (1998), pp. 124–133], to estimate the parameters. The L₂E is based on an integrated L₂ distance between parametric conditional and true conditional mass functions. In addition to studying the theoretical properties of the L₂E estimator, we compare the performance of L₂E with the maximum likelihood (ML) estimator and a minimum Hellinger distance (MHD) estimator via Monte Carlo simulations for correctly specified and gross-error contaminated mixture of Poisson regression models. These show that the L₂E is a viable robust alternative to the ML and MHD estimators. More importantly, we use the L₂E to perform a comprehensive analysis of a Western Australia hospital inpatient obstetrical length of stay (LOS) (in days) data that contains extreme values. It is shown that the L₂E provides a two-component Poisson mixture regression fit to the LOS data which is better than those based on the ML and MHD estimators. The L₂E fit identifies admission type as a significant covariate that profiles the predominant subpopulation of normal-stayers as planned patients and the small subpopulation of long-stayers as emergency patients.     

Density estimation under the koziol-green model of censoring by penalized likelihood methods

We propose two density estimators of the survival distribution in the setting of the Koziol-Green random-censoring model. The estimators are obtained by maximum-penalized-likelihood methods, and we provide an algorithm for their numerical evaluation. We establish the strong consistency of the estimators in the Hellinger metric, the L_p-norms, p= 1,2, ∞, and a Sobolev norm, under mild conditions on the underlying survival density and the censoring distribution.     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

Choosing a robustness tuning parameter

A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L ₂ distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.     

Asymptotics of cross-validated risk estimation in estimator selection and performance assessment

Risk estimation is an important statistical question for the purposes of selecting a good estimator (i.e., model selection) and assessing its performance (i.e., estimating generalization error). This article introduces a general framework for cross-validation and derives distributional properties of cross-validated risk estimators in the context of estimator selection and performance assessment. Arbitrary classes of estimators are considered, including density estimators and predictors for both continuous and polychotomous outcomes. Results are provided for general full data loss functions (e.g., absolute and squared error, indicator, negative log density). A broad definition of cross-validation is used in order to cover leave-one-out cross-validation, V-fold cross-validation, Monte Carlo cross-validation, and bootstrap procedures. For estimator selection, finite sample risk bounds are derived and applied to establish the asymptotic optimality of cross-validation, in the sense that a selector based on a cross-validated risk estimator performs asymptotically as well as an optimal oracle selector based on the risk under the true, unknown data generating distribution. The asymptotic results are derived under the assumption that the size of the validation sets converges to infinity and hence do not cover leave-one-out cross-validation. For performance assessment, cross-validated risk estimators are shown to be consistent and asymptotically linear for the risk under the true data generating distribution and confidence intervals are derived for this unknown risk. Unlike previously published results, the theorems derived in this and our related articles apply to general data generating distributions, loss functions (i.e., parameters), estimators, and cross-validation procedures.