ESTIMATION OF A DISTRIBUTION FUNCTION DOMINATING STOCHASTICALLY A KNOWN DISTRIBUTION FUNCTION

This paper considers the problem of estimating a cumulative distribution function (cdf), when it is known a priori to dominate a known cdf. The estimator considered is obtained by adjusting the empirical cdf using the prior information. This adjusted estimator is shown to be consistent, its limiting distribution is found, and its mean squared error (MSE) is shown to be smaller than the MSE of the empirical cdf. Its asymptotic efficiency (compared to the empirical cdf) is also found.     

Local Smoothing for Kernel Distribution Function Estimation

The problem of bandwidth selection for kernel-based estimation of the distribution function (cdf) at a given point is considered. With appropriate bandwidth, a kernel-based estimator (kdf) is known to outperform the empirical distribution function. However, such a bandwidth is unknown in practice. In pointwise estimation, the appropriate bandwidth depends on the point where the function is estimated. The existing smoothing methods use one common bandwidth to estimate the cdf. The accuracy of the resulting estimates varies substantially depending on the cdf and the point where it is estimated. We propose to select bandwidth by minimizing a bootstrap estimator of the MSE of the kdf. The resulting estimator performs reliably, irrespective of where the cdf is estimated. It is shown to be consistent under i.i.d. as well as strongly mixing dependence assumption. Two applications of the proposed estimator are shown in finance and seismology. We report a dataset on the S & P Nifty index values.     

Testing Censoring Point Independence

Identification in censored regression analysis and hazard models of duration outcomes relies on the condition that censoring points are conditionally independent of latent outcomes, an assumption which may be questionable in many settings. This article proposes a test for this assumption based on a Cramer–von-Mises-like test statistic comparing two different nonparametric estimators for the latent outcome cdf: the Kaplan–Meier estimator, and the empirical cdf conditional on the censoring point exceeding (for right-censored data) the cdf evaluation point. The test is consistent and has power against a wide variety of alternatives. Applying the test to unemployment duration data from the NLSY, the SIPP, and the PSID suggests the assumption is frequently suspect.     

A new formula for the computation of multivariate factorized moments by using the joint cumulative distribution function

This work presents a closed formula to compute any muitivariate factorized expected value from the knowledge of the joint cumulative distribution function (cdf) of any random variable. Additionally, a new nonparametric estimator alternative to the sample average is presented for the univariate case.     

Testing Exponentiality Based on Type II Censored Data and a New cdf Estimator

In this article, we use a new cdf estimator to obtain a nanparametric entropy estimate and use it for testing exponentiality and normality. We also use the new cdf estimator to estimate the joint entropy of the Type II censored data which we use for some goodness-of-fit tests based on Kullback–Leibler information and show, by simulation, that it compares favorably with the leading competitor.     

Stable estimation of location parameter -nonasymptotic approach

Let X₁:, X₂:, …, X_n be iidrv's with cdf F?, F?(x)=F (x-θ), R. Let T be an equivariant median-unbiased estimator of θ. Let πε(F)={G = (1 -ε) F+εH, H any cdf} and let M(G, T) be a median of T if X₁ has cdf G. The oscillation of the bias of T, defined as

Bε(T)=sup (M(G₁ T) :G₁,G₂:∈π_σ:(F)} ,is considered and the estimator with the smallest B$epsi;(T) is explicitly constructed     

Estimation of the error distribution in a varying coefficient regression model

This paper deals with the estimation of the error distribution function in a varying coefficient regression model. We propose two estimators and study their asymptotic properties by obtaining uniform stochastic expansions. The first estimator is a residual-based empirical distribution function. We study this estimator when the varying coefficients are estimated by under-smoothed local quadratic smoothers. Our second estimator which exploits the fact that the error distribution has mean zero is a weighted residual-based empirical distribution whose weights are chosen to achieve the mean zero property using empirical likelihood methods. The second estimator improves on the first estimator. Bootstrap confidence bands based on the two estimators are also discussed.     

A simple algorithm for calculating values for folded normal distribution

Folded normal distribution originates from the modulus of normal distribution. In the present article, we have formulated the cumulative distribution function (cdf) of a folded normal distribution in terms of standard normal cdf and the parameters of the mother normal distribution. Although cdf values of folded normal distribution were earlier tabulated in the literature, we have shown that those values are valid for very particular situations. We have also provided a simple approach to obtain values of the parameters of the mother normal distribution from those of the folded normal distribution. These results find ample application in practice, for example, in obtaining the so-called upper and lower α-points of folded normal distribution, which, in turn, is useful in testing of the hypothesis relating to folded normal distribution and in designing process capability control chart of some process capability indices. A thorough study has been made to compare the performance of the newly developed theory to the existing ones. Some simulated as well as real-life examples have been discussed to supplement the theory developed in this article. Codes (generated by R software) for the theory developed in this article are also presented for the ease of application.     

On the sampling distribution of clement's capability index

Most of the research effort concerning the development and statistical study of capability indices has been devoted to normal processes. In this paper a statistical study of a capability index for non-normal processes proposed by Clements (1989) is developed. An approximate distribution for the natural estimator of the index is obtained from a distribution free point of view and a simulation study is used to compare it with its empirical distribution. An approximate conservative lower confidence limit for the index is also constructed.     

Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response

In this paper, bias-adjustment in the jackknife estimator of variance accredited to Rao and Sitter (1995) has been considered. Then the bias-adjusted Rao and Sitter (1995) estimator has been calibrated such that its expected value under the imputing superpopulation model remains the same as the expected value of the mean squared error of the ratio estimator in the presence of non-response. A simulation study has been performed to compare the six different estimators of variance: out of them four estimators belong to Rao and Sitter (1995) and the other two proposed estimators are named as bias-adjusted and bias-adjusted-cum-calibrated estimators. The empirical relative bias and empirical relative efficiency of the two proposed estimators with respect to the four existing estimators accredited to Rao and Sitter (1995) have been investigated through simulations. The bias-adjusted-cum-calibrated estimator has been found to be an efficient estimator in the case of heteroscadastic populations. The present paper considers the situation of simple random and without replacement sampling. The possibility of obtaining a negative estimate of variance by the estimator due to Kim et al. (2006) has been pointed out.     

Estimating the turning point of a bathtub-shaped failure distribution

The turning point of a hazard rate function is useful in assessing the hazard in the useful life phase and helps to determine and plan appropriate burn-in, maintenance, and repair policies and strategies. For many bathtub-shaped distributions, the turning point is unique, and the hazard varies little in the useful life phase. We investigate the performance of an empirical estimator for the turning point in the case of the modified Weibull distribution, a bathtub-shaped generalization of the Weibull distribution, that has been found to be useful in reliability engineering and other areas concerned with life-time data. We illustrate the theory by means of an example, and also conduct a simulation study to assess the performance of the estimator in practice.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

Estimating the distribution function of symmetric pairs

We discuss three classes of bivariate symmetry models and study the estimation of their distribution functions (DFs). Under radial symmetry, an estimator based on the mean of the empirical and survival DFs is considered. For exchangeable symmetry, an estimator based on the mean of the empirical DF and its exchangeable image is presented. At their intersection, we define radial exchangeability and study estimation of its DF. The symmetrized estimators coincide with the non parametric maximum likelihood estimators of the DF under each model. We obtain their mean and variance and state their asymptotic normality. The relative efficiency of the estimators for the bivariate normal distribution is obtained.     

The distribution of a linear predictor after model selection: conditional finite-sample distributions and asymptotic approximations

This paper studies the distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. The finite-sample cumulative distribution function (cdf) of the linear predictor is derived and a detailed analysis of the effects of the model selection step is given. Moreover, a simple approximation to the (complicated) finite-sample cdf is proposed. This approximation facilitates the study of the large-sample limit behavior of the linear predictor and its cdf, in the fixed-parameter case and under local alternatives. The focus of this paper is on the conditional distribution of a linear predictor, conditional on the event that a fixed (possibly incorrect) model has been selected. The unconditional distribution of a linear predictor is studied in the companion paper Leeb (The distribution of a linear predictor after model selection: unconditional finite-sample distributions and asymptotic approximations, Technical Report, Department of Statistics, University of Vienna, 2002).     

An alternative to ratio estimator in post-stratification

This article proposes an alternative to usual ratio estimator of population mean in post-stratified sampling procedure and its properties are analyzed. Both theoretical and empirical findings are encouraging and support the soundness of the proposed procedure for mean estimation over an alternative to ratio estimator in simple random sampling without replacement suggested by Srivenkataramana and Tracy (1980), usual combined ratio estimators suggested by Ige and Tripathi (1989), and usual unbiased estimator in post-stratified sampling scheme. Both theoretical and empirical findings are encouraging and support the soundness of the present study. At the end, a simulation study has been carried out to verify the superiority of the proposed estimator.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.     

Nonparametric estimation of bivariate distribution under right truncation with application to panic disorder

In several studies, investigators are interested in estimating the bivariate distribution of the onset ages of a generic disorder in successive generations. The empirical distribution is inappropriate for this purpose due to truncation: only parent–child pairs with onset ages prior to the ages at interview were included in the sample. In this paper, we propose a simple nonparametric estimator for the underlying bivariate distribution of the onset ages. Compared with the existing estimators, the proposed estimator has a closed form and smaller biases when estimating marginal distributions. A real example is used to illustrate this estimator.     

Empirical distribution function under heteroscedasticity

Neglecting heteroscedasticity of error terms may imply the wrong identification of a regression model (see appendix). Employment of (heteroscedasticity resistent) White's estimator of covariance matrix of estimates of regression coefficients may lead to the correct decision about the significance of individual explanatory variables under heteroscedasticity. However, White's estimator of covariance matrix was established for least squares (LS)-regression analysis (in the case when error terms are normally distributed, LS- and maximum likelihood (ML)-analysis coincide and hence then White's estimate of covariance matrix is available for ML-regression analysis, tool). To establish White's-type estimate for another estimator of regression coefficients requires Bahadur representation of the estimator in question, under heteroscedasticity of error terms. The derivation of Bahadur representation for other (robust) estimators requires some tools. As the key too proved to be a tight approximation of the empirical distribution function (d.f.) of residuals by the theoretical d.f. of the error terms of the regression model. We need the approximation to be uniform in the argument of d.f. as well as in regression coefficients. The present paper offers this approximation for the situation when the error terms are heteroscedastic.