Improvement of goodness-of-fit test for normal distribution based on entropy and power comparison

Vasicek's entropy test for normality is based on sample entropy and a parametric entropy estimator. These estimators are known to have bias in small samples. The use of Vasicek's test could affect the capability of detecting non-normality to some extent. This paper presents an improved entropy test, which uses bias-corrected entropy estimators. A Monte Carlo simulation study is performed to compare the power of the proposed test under several alternative distributions with some other tests. The results report that as anticipated, the improved entropy test has consistently higher power than the ordinary entropy test in nearly all sample sizes and alternatives considered, and compares favorably with other tests.     

Goodness-of-fit test based on correcting moments of modified entropy estimator

In this paper, we first propose a new estimator of entropy for continuous random variables. Our estimator is obtained by correcting the coefficients of Vasicek's [A test for normality based on sample entropy, J. R. Statist. Soc. Ser. B 38 (1976), pp. 54–59] entropy estimator. We prove the consistency of our estimator. Monte Carlo studies show that our estimator is better than the entropy estimators proposed by Vasicek, Ebrahimi et al. [Two measures of sample entropy, Stat. Probab. Lett. 20 (1994), pp. 225–234] and Correa [A new estimator of entropy, Commun. Stat. Theory Methods 24 (1995), pp. 2439–2449] in terms of root mean square error. We then derive the non-parametric distribution function corresponding to our proposed entropy estimator as a piece-wise uniform distribution. We also introduce goodness-of-fit tests for testing exponentiality and normality based on the said distribution and compare its performance with their leading competitors.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

A new method of imputation in survey sampling

In this paper, an alternative estimator of population mean in the presence of non-response has been suggested which comes in the form of Walsh's estimator. The estimator of mean obtained from the proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean-squared error (MSE) of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optimum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. A suggestion to form ‘warm deck’ method of imputation has been suggested. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. The work has been extended to the case of multi-auxiliary information to be used for imputation. Numerical illustrations are also provided.     

A comparison of biased regression estimators using a pitman nearness criterion

Biased regression estimators have traditionally benn studied using the Mean Square Error (MSE) criterion. Usually these comparisons have been based on the sum of the MSE's of each of the individual parameters, i.e., a scaler valued measure that is the trace of the MSE matrix. However, since this summed MSE does not consider the covariance structure of the estimators, we propose the use of a Pitman Measure of Closeness (PMC) criterion (Keating and Gupta, 1984; Keating and Mason, 1985). In this paper we consider two versions of PMC. One of these compares the estimates and the other compares the resultant predicted values for 12 different regression estimators. These estimators represent three classes of estimators, namely, ridge, shrunken, and principal component estimators. The comparisons of these estimators using the PMC criteria are contrasted with the usual MSE criteria as well as the prediction mean square error. Included in the estimators is a relatively new estimator termed the generalized principal component estimator proposed by Jolliffe. This estimator has previously received little attention in the literature.     

An evaluation of ridge estimator in linear mixed models: an example from kidney failure data

This paper is concerned with the ridge estimation of fixed and random effects in the context of Henderson's mixed model equations in the linear mixed model. For this purpose, a penalized likelihood method is proposed. A linear combination of ridge estimator for fixed and random effects is compared to a linear combination of best linear unbiased estimator for fixed and random effects under the mean-square error (MSE) matrix criterion. Additionally, for choosing the biasing parameter, a method of MSE under the ridge estimator is given. A real data analysis is provided to illustrate the theoretical results and a simulation study is conducted to characterize the performance of ridge and best linear unbiased estimators approach in the linear mixed model.     

Liu-Type Logistic Estimator

It is known that multicollinearity inflates the variance of the maximum likelihood estimator in logistic regression. Especially, if the primary interest is in the coefficients, the impact of collinearity can be very serious. To deal with collinearity, a ridge estimator was proposed by Schaefer et al. The primary interest of this article is to introduce a Liu-type estimator that had a smaller total mean squared error (MSE) than the Schaefer's ridge estimator under certain conditions. Simulation studies were conducted that evaluated the performance of this estimator. Furthermore, the proposed estimator was applied to a real-life dataset.     

The Statistical Properties of the Equity Estimator

In this article we consider the Equity estimator proposed by Krishnamurthi and Rangaswamy. We show that this estimator is inconsistent and does not necessarily improve on the mean squared error (MSE) of the least squares (LS) estimator. We perform a Monte Carlo experiment based on the price-promotion model used in marketing research, with marketing data, comparing the MSE of the Equity estimator to that of two empirical Bayes estimators and the LS estimator. We find that the empirical Bayes estimators have substantially smaller MSE than the Equity estimator in almost every case.     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

On variable bandwidth selection in local polynomial regression

The performances of data-driven bandwidth selection procedures in local polynomial regression are investigated by using asymptotic methods and simulation. The bandwidth selection procedures considered are based on minimizing 'prelimit' approximations to the (conditional) mean-squared error (MSE) when the MSE is considered as a function of the bandwidth h . We first consider approximations to the MSE that are based on Taylor expansions around h=0 of the bias part of the MSE. These approximations lead to estimators of the MSE that are accurate only for small bandwidths h . We also consider a bias estimator which instead of using small h approximations to bias naïvely estimates bias as the difference of two local polynomial estimators of different order and we show that this estimator performs well only for moderate to large h . We next define a hybrid bias estimator which equals the Taylor-expansion-based estimator for small h and the difference estimator for moderate to large h . We find that the MSE estimator based on this hybrid bias estimator leads to a bandwidth selection procedure with good asymptotic and, for our Monte Carlo examples, finite sample properties.     

Estimation of a distribution function bounded by two known distribution functions

The problem of estimation of a cumulative distribution function (cdf), bounded by two known cdf's, is considered. An estimator satisfying the desired restriction has been obtained by suitably adjusting the empirical cdf. Consistency of the adjusted estimator has been established and its mean square error (MSE) has been shown to be smallerthan that of the empirical cdf. The new estimator has been comparedwith the empirical cdf for some special cases.     

Robust extreme ranked set sampling

In this paper, a robust extreme ranked set sampling (RERSS) procedure for estimating the population mean is introduced. It is shown that the proposed method gives an unbiased estimator with smaller variance, provided the underlying distribution is symmetric. However, for asymmetric distributions a weighted mean is given, where the optimal weights are computed by using Shannon's entropy. The performance of the population mean estimator is discussed along with its properties. Monte Carlo simulations are used to demonstrate the performance of the RERSS estimator relative to the simple random sample (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) estimators. The results indicate that the proposed estimator is more efficient than the estimators based on the traditional sampling methods.     

A bayesian approach to bioassay. technical report m498

We describe the design and analysis for a simulation experiment to compare the mean-squared errors (MSE's) of two quantile estimators defined for random walk designs. The dependence of the easily computed MSE of the first estimator on the levels of five factors is examined via multiple regression. This information is used to plan a simulation to compute the MSE of the second estimator using a fraction of a 3³5²factorial allowing uncorrelated estimates for all main effects and the two-factor interactions of a specified factor. Efficient estimation of the MSE of the second estimator is attempted through antithetic and control variate techniques of variance reduction, with modest success.     

Focused Information Criterion for Capture–Recapture Models for Closed Populations

Abstract. We propose a criterion for selecting a capture–recapture model for closed populations, which follows the basic idea of the focused information criterion (FIC) of Claeskens and Hjort. The proposed criterion aims at selecting the model which, among the available models, leads to the smallest mean‐squared error (MSE) of the resulting estimator of the population size and is based on an index which, up to a constant term, is equal to the asymptotic MSE of the estimator. Two alternative approaches to estimate this FIC index are proposed. We also deal with multimodel inference; in this case, the population size is estimated by using a weighted average of the estimates coming from different models, with weights chosen so as to minimize the MSE of the resulting estimator. The proposed model selection approach is compared with more common approaches through a series of simulations. It is also illustrated by an application based on a dataset coming from a live‐trapping experiment.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

Entropy estimation and goodness-of-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling

ABSTRACT

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.     

A Bayesian approach with generalized ridge estimation for high-dimensional regression and testing

This paper adopts a Bayesian strategy for generalized ridge estimation for high-dimensional regression. We also consider significance testing based on the proposed estimator, which is useful for selecting regressors. Both theoretical and simulation studies show that the proposed estimator can simultaneously outperform the ordinary ridge estimator and the LSE in terms of the mean square error (MSE) criterion. The simulation study also demonstrates the competitive MSE performance of our proposal with the Lasso under sparse models. We demonstrate the method using the lung cancer data involving high-dimensional microarrays.     

Mean squared error estimators of small area means using survey weights

Using survey weights, You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] proposed a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator of a small area mean under a nested error linear regression model. This estimator borrows strength across areas through a linking model, and makes use of survey weights to ensure design consistency and preserve benchmarking property in the sense that the estimators add up to a reliable direct estimator of the mean of a large area covering the small areas. In this article, a second‐order approximation to the mean squared error (MSE) of the pseudo‐EBLUP estimator of a small area mean is derived. Using this approximation, an estimator of MSE that is nearly unbiased is derived; the MSE estimator of You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] ignored cross‐product terms in the MSE and hence it is biased. Empirical results on the performance of the proposed MSE estimator are also presented. The Canadian Journal of Statistics 38: 598–608; 2010 © 2010 Statistical Society of Canada     

A mail survey design for sensitive character without using randomization device

In the present study, we develop theory for Takahasi and Sakasegawa's (1977) suggest ion and propose a new randomized response model suitable for mail surveys. The proposed model based estimator is unbiased and is always more efficient than the estimator based on Takahasi and Sakasegawa model.