Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

On the boundary properties of Bernstein polynomial estimators of density and distribution functions

For density and distribution functions supported on [0,1], Bernstein polynomial estimators are known to have optimal mean integrated squared error (MISE) properties under the usual smoothness conditions on the function to be estimated. These estimators are also known to be well-behaved in terms of bias: they have uniform bias over the entire unit interval. What is less known, however, is that some of these estimators do experience a boundary effect, but of a different nature than what is seen with the usual kernel estimators.     

Wavelet-based estimators of multivariable mean regression function with long-memory data

Wavelet analysis has been proved to be a powerful statistical technique in the non parametric regression. In this paper, we propose non linear wavelet-based estimators for multivariable mean regression function with long-memory data. We also provide an asymptotic expansion for the mean integrated squared error (MISE) of the function estimators. This MISE expansion still works even when the underlying mean regression function is only piecewise smooth. This paper extends the corresponding results in the literature for single variable to multivariable case.     

Nonparametric hazard rate estimation by orthogonal wavelet methods

We propose linear and nonlinear wavelet-based hazard rate estimators where the linear estimator is equivalent to a generalized kernel estimator. An asymptotic formula for the mean integrated squared error (MISE) of the nonlinear wavelet-based hazard rate estimator is provided. It is shown that the MISE formula for the nonlinear estimator is available for hazard rates which are smooth only in a piecewise sense, a feature not available for the kernel estimators.     

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT

We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.     

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.     

Bootstrap MISE Estimators to Obtain Bandwidth for Kernel Density Estimation

Research in the area of bandwidth selection was an active topic in the 1980s and 1990s, however, recently there has been little research in the area. We re-opened this investigation and have found a new method for estimating mean integrated squared error for kernel density estimators. We provide an overview of other methods to obtain optimal bandwidths and offer a comparison of these methods via a simulation study. In certain situations, our method of estimating an optimal bandwidth yields a smaller MISE than competing methods to compute bandwidths. This procedure is illustrated by an application to two data sets.     

Nonparametric kernel estimation of an isotropic variogram

In this paper, we propose nonparametric kernel estimators of the semivariogram, under the assumption of isotropy. At first, a symmetric kernel is considered in order to construct a consistent estimator, so that the selection of the bandwidth parameter is treated via the MSE or the MISE criteria. Next, the use of a boundary kernel will be suggested in order to obtain satisfactory estimates near the semivariogram endpoint. In all cases, an adaptation of Shapiro and Botha's fit is proposed to produce valid semivariogram estimators. Finally, we describe a numerical study carried out to illustrate the performance of the kernel estimators.     

Consistent Smooth Bootstrap Kernel Intensity Estimation for Inhomogeneous Spatial Poisson Point Processes

Non‐parametric estimation and bootstrap techniques play an important role in many areas of Statistics. In the point process context, kernel intensity estimation has been limited to exploratory analysis because of its inconsistency, and some consistent alternatives have been proposed. Furthermore, most authors have considered kernel intensity estimators with scalar bandwidths, which can be very restrictive. This work focuses on a consistent kernel intensity estimator with unconstrained bandwidth matrix. We propose a smooth bootstrap for inhomogeneous spatial point processes. The consistency of the bootstrap mean integrated squared error (MISE) as an estimator of the MISE of the consistent kernel intensity estimator proves the validity of the resampling procedure. Finally, we propose a plug‐in bandwidth selection procedure based on the bootstrap MISE and compare its performance with several methods currently used through both as a simulation study and an application to the spatial pattern of wildfires registered in Galicia (Spain) during 2006.     

Nonparametric estimation for survival data with censoring indicators missing at random

In this paper, we consider the problem of hazard rate estimation in the presence of covariates, for survival data with censoring indicators missing at random. We propose in the context usually denoted by MAR (missing at random, in opposition to MCAR, missing completely at random, which requires an additional independence assumption), nonparametric adaptive strategies based on model selection methods for estimators admitting finite dimensional developments in functional orthonormal bases. Theoretical risk bounds are provided, they prove that the estimators behave well in term of mean square integrated error (MISE). Simulation experiments illustrate the statistical procedure.     

On the effect of misspecifying the error density in a deconvolution problem

The author studies the effect of a misspecification of the error density on the mean integrated squared error (MISE) of the deconvolution estimator. He shows that the MISE converges to a certain functional which he defines. He also illustrates the fact that the limit can sometimes be infinite. Finally, he derives some guidelines for selecting the error density in order to ensure robustness properties of the procedure.     

Alternative GMM estimators for first-order autoregressive panel model: An improving efficiency approach

This article considers first-order autoregressive panel model that is a simple model for dynamic panel data (DPD) models. The generalized method of moments (GMM) gives efficient estimators for these models. This efficiency is affected by the choice of the weighting matrix that has been used in GMM estimation. The non-optimal weighting matrices have been used in the conventional GMM estimators. This led to a loss of efficiency. Therefore, we present new GMM estimators based on optimal or suboptimal weighting matrices. Monte Carlo study indicates that the bias and efficiency of the new estimators are more reliable than the conventional estimators.     

Statistical Inference for Renewal Processes

We consider non‐parametric estimation for interarrival times density of a renewal process. For continuous time observation, a projection estimator in the orthonormal Laguerre basis is built. Nonstandard decompositions lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev–Laguerre spaces are deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation is more difficult to handle. A first strategy consists in neglecting the discretization error. A more precise strategy aims at taking into account the convolution structure of the data. Under a simplifying ‘dead‐zone’ condition, the corresponding MISE is given for any sampling step. In the three cases, an automatic model selection procedure is described and gives the best MISE, up to a logarithmic term. The results are illustrated through a simulation study.     

On consistent testing for serial correlation in seasonal time series models

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel-based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.     

Handling missing data by deleting completely observed records

When data are missing, analyzing records that are completely observed may cause bias or inefficiency. Existing approaches in handling missing data include likelihood, imputation and inverse probability weighting. In this paper, we propose three estimators inspired by deleting some completely observed data in the regression setting. First, we generate artificial observation indicators that are independent of outcome given the observed data and draw inferences conditioning on the artificial observation indicators. Second, we propose a closely related weighting method. The proposed weighting method has more stable weights than those of the inverse probability weighting method (Zhao, L., Lipsitz, S., 1992. Designs and analysis of two-stage studies. Statistics in Medicine 11, 769–782). Third, we improve the efficiency of the proposed weighting estimator by subtracting the projection of the estimating function onto the nuisance tangent space. When data are missing completely at random, we show that the proposed estimators have asymptotic variances smaller than or equal to the variance of the estimator obtained from using completely observed records only. Asymptotic relative efficiency computation and simulation studies indicate that the proposed weighting estimators are more efficient than the inverse probability weighting estimators under wide range of practical situations especially when the missingness proportion is large.     

Interval-censored unimodal kernel density estimation via data sharpening

In this paper, we extend Choi and Hall's [Data sharpening as a prelude to density estimation. Biometrika. 1999;86(4):941–947] data sharpening algorithm for kernel density estimation to interval-censored data. Data sharpening has several advantages, including bias and mean integrated squared error (MISE) reduction as well as increased robustness to bandwidth misspecification. Several interval metrics are explored for use with the kernel function in the data sharpening transformation. A simulation study based on randomly generated data is conducted to assess and compare the performance of each interval metric. It is found that the bias is reduced by sharpening, often with little effect on the variance, thus maintaining or reducing overall MISE. Applications involving time to onset of HIV and running distances subject to measurement error are used for illustration.     

On density and regression estimation with incomplete data

We consider the problem of estimation of a density function in the presence of incomplete data and study the Hellinger distance between our proposed estimators and the true density function. Here, the presence of incomplete data is handled by utilizing a Horvitz–Thompson-type inverse weighting approach, where the weights are the estimates of the unknown selection probabilities. We also address the problem of estimation of a regression function with incomplete data.     

Orthogonal series density estimation in a disaggregation scheme

In this paper we consider long-memory processes obtained by aggregation of independent random parameter AR(1) processes. We propose an estimator of the density of the underlying random parameter. This estimator is based on the expansion of the density function on the basis of Gegenbauer polynomials. Rate of convergence to zero of the mean integrated square error (MISE) and of the uniform error are obtained. The results are illustrated by Monte-Carlo simulations.     

Approximation rates of the error distribution of wavelet estimators of a density function under censorship

In this paper, a wavelet estimator and a random weighted estimator of a probability density function are constructed under right censored data. The normal approximation rates and random weighting approximation rates of the error distribution of wavelet estimators are obtained under suitable conditions. To illustrate the application of the technique, the confidence interval of f(x) is constructed by the results in this paper, and the simulation calculation is studied by the artificially generated data and “real world” data set.     

Some results on random design regression with long memory errors and predictors

This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the mean integrated squared error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that averaged squared error (ASE) is a good approximation of MISE; however, this is not the case for a cross-validation criterion.