Non-parametric estimation of reciprocal coordinate subtangent for right censored dependent scheme

The concept of reciprocal coordinate subtangent (RCST) has been used as a useful tool to study the monotone behavior of a continuous density function and for characterizing probability distributions. In this paper, we propose a non-parametric estimator for RCST based on the censored dependent data. Asymptotic properties of the estimator are established under suitable regularity conditions. A simulation study is carried out to examine the performance of the estimator. The usefulness of the estimator is also examined through a real data.     

Bayesian Prediction of the Median of the Burr Distribution with Fixed and Random Sample Sizes

sLingappaiah (1986) was the first to introduce the idea of Bayesian prediction in life testing when the size of the future sample is a random variable. In this paper we discuss the Bayesion prediction of the sample median when the parent distribution is a generalized Burr distribution (GBD), the old sample is censored type II and the size of the future sample is a random variable. A numerical illustration is provided.     

Estimation of inaccuracy measure for censored dependent data

In the present paper, we propose non parametric estimators for the inaccuracy measure for the lifetime distribution based on censored data. This measure plays important roles in reliability and survival analysis in connection with modeling and analysis of life time data. Asymptotic properties of the estimators are established under suitable regularity conditions. Monte Carlo simulation studies are carried out to compare the performance of the estimators using the mean-squared error. The methods are illustrated using a real data set.     

Sample moments of the autocorrelations of moving average processes and a modification to bartlett'sasymptotic variance formula

In this paper we express the sample autocorrelations for a moving average process of order q as a function of its own theoretical autocorrelations and the sample autocorrelations for the generating white noise series. Approximate analytic expressions are then obtained forthe moments of the sample autocorrelations of the moving average process.

Using these expressions, together with numerical evidence, we show that Bartlett's asymptotic formula for the variance of the sample autocorrelations of moving average processes, which is used widely in identifying these processes, is a large overestimate when considering finitesample sizes.

Our approach is for motivational purposes and so is purely formal, the amount of mathematics presented being kept to a minimum.     

Matrix density estimation

The accuracy of orthogonal series types of density estimators can be conveniently measured in terms of their Mean Integrated Squared Error, or MISE. Further reduction In MISE is achieved by introducing certain weighting factors into the estimators. In this paper we consider optimal weighting matrices, and the result is a new class of density estimators, the collection of matrix density estimators.     

Invalidity of average squared error criterion in density estimation

The average squared error has been suggested earlier as an appropriate estimate of the integrated squared error, but an example is given which shows their ratio can tend to infinity. The results of a Monte Carlo study are also presented which suggest the average squared error can seriously underestimate the errors inherent in even the simplest density estimations.     

Exact small sample properties of an operational variant of the minimum mean squared error estimator

In this paper, we show a sufficient condition for an operational variant of the minimum mean squared error estimator (simply, the minimum MSE estimator) to dominate the ordinary least squares (OLS) estimator. It is also shown numerically that the minimum MSE estimator dominates the OLS estimator if the number of regression coefficients is larger than or equal to three, even if the sufficient condition is not satisfied. When the number of regression coefficients is smaller than three, our numerical results show that the gain in MSE of using the minimum MSE estimator is larger than the loss.     

On boundary correction in kernel density estimation

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. In this paper, we propose a new general method of boundary correction for univariate kernel density estimation. The proposed method generates a class of boundary corrected estimators. They all possess desirable properties such as local adaptivity and non-negativity. In simulation, it is observed that the proposed method perform quite well when compared with other existing methods available in the literature for most shapes of densities, showing a very important robustness property of the method. The theory behind the new approach and the bias and variance of the proposed estimators are given. Results of a data analysis are also given.     

The influence function of the optimal bandwidth for kernel density estimation

Theories about the bandwidth of kernel density estimation have been well established by many statisticians. However, the influence function of the bandwidth has not been well investigated. The influence function of the optimal bandwidth that minimizes the mean integrated square error is derived and the asymptotic property of the bandwidth selectors based on the influence function is provided.     

Smoothed bootstrap bandwidth selection for nonparametric hazard rate estimation

A smoothed bootstrap method is presented for the purpose of bandwidth selection in nonparametric hazard rate estimation for iid data. In this context, two new bootstrap bandwidth selectors are established based on the exact expression of the bootstrap version of the mean integrated squared error of some approximations of the kernel hazard rate estimator. This is very useful since Monte Carlo approximation is no longer needed for the implementation of the two bootstrap selectors. A simulation study is carried out in order to show the empirical performance of the new bootstrap bandwidths and to compare them with other existing selectors. The methods are illustrated by applying them to a diabetes data set.     

Residual variance estimation in moving average models

We consider time series models of the MA (moving average) family, and deal with the estimation of the residual variance. Results are known for maximum likelihood estimates under normality, both for known or unknown mean, in which case the asymptotic biases depend on the number of parameters (including the mean), and do not depend on the values of the parameters. For moment estimates the situation is different, because we find that the asymptotic biases depend on the values of the parameters, and become large as they approach the boundary of the region of invertibility. Our approach is to use Taylor series expansions, and the objective is to obtain asymptotic biases with error of o(l/T), where T is the sample size. Simulation results are presented, and corrections for bias suggested.     

Some finite-sample-size properties of rosenblatt density estimates

The results of a Monte Carlo study of the sensitivity of Rosenblatt density estimates to the scale factor are presented.     

Local orthogonal polynomial expansion for density estimation

A local orthogonal polynomial expansion (LOrPE) of the empirical density function is proposed as a novel method to estimate the underlying density. The estimate is constructed by matching localised expectation values of orthogonal polynomials to the values observed in the sample. LOrPE is related to several existing methods, and generalises straightforwardly to multivariate settings. By manner of construction, it is similar to local likelihood density estimation (LLDE). In the limit of small bandwidths, LOrPE functions as kernel density estimation (KDE) with high-order (effective) kernels inherently free of boundary bias, a natural consequence of kernel reshaping to accommodate endpoints. Consistency and faster asymptotic convergence rates follow. In the limit of large bandwidths LOrPE is equivalent to orthogonal series density estimation (OSDE) with Legendre polynomials, thereby inheriting its consistency. We compare the performance of LOrPE to KDE, LLDE, and OSDE, in a number of simulation studies. In terms of mean integrated squared error, the results suggest that with a proper balance of the two tuning parameters, bandwidth and degree, LOrPE generally outperforms these competitors when estimating densities with sharply truncated supports.     

Adaptive normal reference bandwidth based on quantile for kernel density estimation

Bandwidth selection is an important problem of kernel density estimation. Traditional simple and quick bandwidth selectors usually oversmooth the density estimate. Existing sophisticated selectors usually have computational difficulties and occasionally do not exist. Besides, they may not be robust against outliers in the sample data, and some are highly variable, tending to undersmooth the density. In this paper, a highly robust simple and quick bandwidth selector is proposed, which adapts to different types of densities.     

Orthogonal series density estimation for complex surveys

We propose an orthogonal series density estimator for complex surveys, where samples are neither independent nor identically distributed. The proposed estimator is proved to be design-unbiased and asymptotically design-consistent. The asymptotic normality is proved under both design and combined spaces. Two data driven estimators are proposed based on the proposed oracle estimator. We show the efficiency of the proposed estimators in simulation studies. A real survey data example is provided for an illustration.     

On pitman nearness and variance of estimators

In this article, some conditions on variances are presented under which the (Generalized) Pitman Nearness Criterion Would Prefer one estimator over another. Results for univariate as well as multivariate cases are derived. An exact expression for a result of Rao, Keating and Mason (1985) is provided.     

A note on maximum likelihood estimation for the first-order autoregressive process

The maximum likelihood estimator of the parameters of a zero-mean normal stationary first-order autoregressive process is in-vestigated. it is shown that the likelihood function is uniquely maximized at a point in the interior of the parameter space. A closed-form expression is obtained for the estimator.     

A generalized reflection method of boundary correction in kernel density estimation

The kernel method of estimation of curves is now popular and widely used in statistical applications. Kernel estimators suffer from boundary effects, however, when the support of the function to be estimated has finite endpoints. Several solutions to this problem have already been proposed. Here the authors develop a new method of boundary correction for kernel density estimation. Their technique is a kind of generalized reflection involving transformed data. It generates a class of boundary corrected estimators having desirable properties such as local smoothness and nonnegativity. Simulations show that the proposed method performs quite well when compared with the existing methods for almost all shapes of densities. The authors present the theory behind this new methodology, and they determine the bias and variance of their estimators.     

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.