Strong uniform consistency rates of kernel estimators of cumulative distribution functions

In this paper we consider the uniform strong consistency, along with a rate, of the cumulative distribution function (CDF) estimator. We extend the extended Glivenko–Cantelli lemma (for empirical distribution function) in Fabian and Hannan (1985 Fabian, V., Hannan, J. (1985). Introduction to Probability and Mathematical Statistics. New York: Wiley, ISBN-13:978-0471250234. [Google Scholar], pp. 80–83) to the kernel estimator of the CDF.     

The smooth colonel and the reverend find common ground

A semiparametric regression estimator that exploits categorical (i.e., discrete-support) kernel functions is developed for a broad class of hierarchical models including the pooled regression estimator, the fixed-effects estimator familiar from panel data, and the varying coefficient estimator, among others. Separate shrinking is allowed for each coefficient. Regressors may be continuous or discrete. The estimator is motivated as an intuitive and appealing generalization of existing methods. It is then supported by demonstrating that it can be realized as a posterior mean in the Lindley and Smith (1972 Lindley, D. V., Smith, A. F. M. (1972). Bayes estimates for the linear model. Journal of the Royal Statistical Society 34:1–41. [Google Scholar]) framework. As a demonstration of the flexibility of the proposed approach, the model is extended to nonparametric hierarchical regression based on B-splines.     

An Asymmetric Kernel Estimator of Density Function for Stationary Associated Sequences

Here, we apply the smoothing technique proposed by Chaubey et al. (2007 Chaubey , Y. P. , Sen , A. , Sen , P. K. ( 2007 ). A new smooth density estimator for non-negative random variables. Technical Report No. 1/07. Department of Mathematics and Statistics, Concordia University, Montreal, Canada . [Google Scholar]) for the empirical survival function studied in Bagai and Prakasa Rao (1991 Bagai , I. , Prakasa Rao , B. L. S. ( 1991 ). Estimation of the survival function for stationary associated processes . Statist. Probab. Lett. 12 : 385 – 391 .[Crossref], [Web of Science ®] , [Google Scholar]) for a sequence of stationary non-negative associated random variables.The derivative of this estimator in turn is used to propose a nonparametric density estimator. The asymptotic properties of the resulting estimators are studied and contrasted with some other competing estimators. A simulation study is carried out comparing the recent estimator based on the Poisson weights (Chaubey et al., 2011 Chaubey , Y. P. , Dewan , I. , Li , J. ( 2011 ). Smooth estimation of survival and density functions for a stationary associated process using poisson weights . Statist. Probab. Lett. 81 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) showing that the two estimators have comparable finite sample global as well as local behavior.     

Weighted log-normal kernel density estimation

ABSTRACT

The log-normal (LN) kernel estimator of a density with support [0, ∞) was discussed by Jin and Kawczak (2003 Jin, X., Kawczak, J. (2003). Birnbaum–Saunders and lognormal kernel estimators for modelling durations in high frequency financial data. Ann. Econ. Finance 4:103–124. [Google Scholar]). The contribution of this paper is to suggest a new class of LN kernel estimators using the idea of weighted distribution. The asymptotic properties of the new class of estimators are studied. Also, numerical studies based on both simulated and real data set are presented.     

Inversion Theorem Based Kernel Density Estimation for the Ordinary Least Squares Estimator of a Regression Coefficient

The traditional confidence interval associated with the ordinary least squares estimator of linear regression coefficient is sensitive to non-normality of the underlying distribution. In this article, we develop a novel kernel density estimator for the ordinary least squares estimator via utilizing well-defined inversion based kernel smoothing techniques in order to estimate the conditional probability density distribution of the dependent random variable. Simulation results show that given a small sample size, our method significantly increases the power as compared with Wald-type CIs. The proposed approach is illustrated via an application to a classic small data set originally from Graybill (1961 Graybill, F.A. (1961). Introduction to Linear Statistical Models. Vol. 1. New York: McGraw-Hill Book Company. [Google Scholar]).     

Skewing and Generalized Jackknifing in Kernel Density Estimation

Abstract

Kernel methods are very popular in nonparametric density estimation. In this article we suggest a simple estimator which reduces the bias to the fourth power of the bandwidth, while the variance of the estimator increases only by at most a moderate constant factor. Our proposal turns out to be a fourth order kernel estimator and may be regarded as a new version of the generalized jackknifing approach (Schucany W. R., Sommers, J. P. (1977 Schucany, W. R. and Sommers, J. P. 1977. Improvement of kernel type estimators. Journal of the American Statistical Association, 72: 420–423. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Improvement of Kernal type estimators. Journal of the American Statistical Association 72:420–423.) applied to kernel density estimation.     

Asymptotics of the Lp-Norms of Density Estimators in the Nonlinear Autoregressive Models

This article investigates the asymptotic behavior of the error density function in nonlinear autoregressive stationary time series regression models. For any 1 ? p < ∞, the kernel density estimator of residuals is shown to be consistent for the error estimator concerning the L_p-distance, which extends the result developed by Cheng and Sun (2008 Cheng, F. X. (2005). Asymptotic distributions of error density estimators in first-order autoregression models. Sankhy Ind. J. Statist. 67:553–567. [Google Scholar]) in L₂-norm. Moreover, the result developed in this article is extended the results of Horváth and Zitikis (2003 Horváth, L., Zitikis, R. (2003). Asymptotics of the L_p-norms of density estimators in the first-order autoregressive models. Statist. Probab. Lett. 65:331342.[Crossref], [Web of Science ®] , [Google Scholar]) to nonlinear autoregressive models.     

Estimation of the Mean of a Sensitive Variable in the Presence of Auxiliary Information

Sousa et al. (2010 Sousa , R. , Shabbir , J. , Real , P. C. , Gupta , S. ( 2010 ). Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information . J. Statist. Theor. Prac. 4 ( 3 ): 495 – 507 .[Taylor & Francis Online] , [Google Scholar]) introduced a ratio estimator for the mean of a sensitive variable and showed that this estimator performs better than the ordinary mean estimator based on a randomized response technique (RRT). In this article, we introduce a regression estimator that performs better than the ratio estimator even for modest correlation between the primary and the auxiliary variables. The underlying assumption is that the primary variable is sensitive in nature but a non sensitive auxiliary variable exists that is positively correlated with the primary variable. Expressions for the Bias and MSE (Mean Square Error) are derived based on the first order of approximation. It is shown that the proposed regression estimator performs better than the ratio estimator and the ordinary RRT mean estimator (that does not utilize the auxiliary information). We also consider a generalized regression-cum-ratio estimator that has even smaller MSE. An extensive simulation study is presented to evaluate the performances of the proposed estimators in relation to other estimators in the study. The procedure is also applied to some financial data: purchase orders (a sensitive variable) and gross turnover (a non sensitive variable) in 2009 for a population of 5,336 companies in Portugal from a survey on Information and Communication Technologies (ICT) usage.     

A class of exponential-type ratio estimators of a general parameter

This paper addresses the problem of estimating a general parameter using information on an auxiliary variable X. We have suggested a class of exponential-type ratio estimators for the parameter and its properties are studied. It is identified that the estimators due to Upadhyaya et al. [Journal of Statistical Theory and Practice (2011), 5(2), 285–302] and Yadav and Kadilar [Revista Columbiana de Estadistica, (2013), 36(1), 145–152] are members of the proposed estimator. We have also shown that the suggested estimator is more efficient than the estimators of Upadhyaya et al. (2011 Upadhyaya, L.N., Singh, H.P., Chatterjee, S., Yadav, R. (2011). Improved ratio and product exponential type estimators. J. Stat. Theo. Pract. 5 (2): 285–302.[Taylor &; Francis Online] , [Google Scholar]) and Yadav and Kadilar (2013 Yadav, S.K., Kadilar, C. (2013). Improved exponential type ratio estimator of population variance. Revis. Colum. de Estadist. 36(1): 145–152. [Google Scholar]). Numerical illustration is provided in support of the present study.     

Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density

We investigate the issue of bandwidth estimation in a functional nonparametric regression model with function-valued, continuous real-valued and discrete-valued regressors under the framework of unknown error density. Extending from the recent work of Shang (2013 Shang, H.L. (2013), ‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198. doi: 10.1016/j.csda.2013.05.006[Crossref], [Web of Science ®] , [Google Scholar]) [‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198], we approximate the unknown error density by a kernel density estimator of residuals, where the regression function is estimated by the functional Nadaraya–Watson estimator that admits mixed types of regressors. We derive a likelihood and posterior density for the bandwidth parameters under the kernel-form error density, and put forward a Bayesian bandwidth estimation approach that can simultaneously estimate the bandwidths. Simulation studies demonstrated the estimation accuracy of the regression function and error density for the proposed Bayesian approach. Illustrated by a spectroscopy data set in the food quality control, we applied the proposed Bayesian approach to select the optimal bandwidths in a functional nonparametric regression model with mixed types of regressors.     

On Wavelet Estimation of the Derivatives of a Density Based on Biased Data

Here, we consider wavelet based estimation of the derivatives of a probability density function under random sampling from a weighted distribution and extend the results regarding the asymptotic convergence rates under the i.i.d. setup studied in Prakasa Rao (1996 Rao, B. L.S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernat. 28:91–100. [Google Scholar]) to the biased-data setup. We compare the performance of the wavelet based estimator with that of the kernel based estimator obtained by differentiating the Efromovich (2004 Efromovich, S. (2004). Density estimation for biased data. Ann. Statist. 32:1137–1161.[Crossref], [Web of Science ®] , [Google Scholar]) kernel density estimator through a simulation study.     

Kernel Density Estimator From Ranked Set Samples

We study kernel density estimator from the ranked set samples (RSS). In the kernel density estimator, the selection of the bandwidth gives strong influence on the resulting estimate. In this article, we consider several different choices of the bandwidth and compare their asymptotic mean integrated square errors (MISE). We also propose a plug-in estimator of the bandwidth to minimize the asymptotic MISE. We numerically compare the MISE of the proposed kernel estimator (having the plug-in bandwidth estimator) to its simple random sampling counterpart. We further propose two estimators for a symmetric distribution, and show that they outperform in MISE all other estimators not considering symmetry. We finally apply the methods in this article to analyzing the tree height data from Platt et al. (1988 Platt, W.J., Evans, G.M., Rathbun, S.L. (1988). The population dynamics of long-lived conifer (Pinus plaustris) (1988). Amer. Natrualist 131:491–525.[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2003 Chen, Z., Bai, Z., Sinha, B.K. (2003). Ranked Set Sampling: Theory and Applications. New York: Springer. [Google Scholar]).     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Ratio-Cum-Product Type Exponential Estimator of Finite Population Mean in Stratified Random Sampling

This article addresses the problem of estimating the finite population mean in stratified random sampling using auxiliary information. Motivated by Singh (1967 Singh , M. P. ( 1967 ). Ratio cum product method of estimation . Metrika 12 : 34 – 42 .[Crossref] , [Google Scholar]) and Bahl and Tuteja (1991 Bahl , S. , Tuteja , R. K. ( 1991 ). Ratio and product type exponential estimator . Inform. Optimiz. Sci. 12 ( 1 ): 159 – 163 .[Taylor &; Francis Online] , [Google Scholar]) a ratio-cum-product type exponential estimator has been suggested and its bias and mean squared error have been derived under large sample approximation. Suggested estimator has been compared with usual unbiased estimator of population mean in stratified random sampling, combined ratio estimator, combined product estimator, ratio and product type exponential estimator of Singh et al. (2008 Singh , R. , Kumar , M. , Singh , R. D. , Chaudhary , M. K. ( 2008 ). Exponential ratio type estimators in stratified random sampling. Presented in International Symposium on Optimisation and Statistics (I.S.O.S) at A.M.U., Aligarh, India, during 29–31 Dec . [Google Scholar]). Conditions under which suggested estimator is more efficient than other considered estimators have been obtained. A numerical illustration is given in support of the theoretical findings.     

Convergence rate of wavelet density estimator with data missing randomly when covariables are present

In this article, we study global L₂ error of non linear wavelet estimator of density in the Besov space B^s_pq for missing data model when covariables are present and prove that the estimator can achieve the optimal rate of convergence, which is similar to the result studied by Donoho et al. (1996) Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Stat. 24:508–539.[Crossref], [Web of Science ®] , [Google Scholar] in complete independent data case with term-by-term thresholding of the empirical wavelet coefficients. Finite-sample behavior of the proposed estimator is explored via simulations.     

Some new methods to solve multicollinearity in logistic regression

The binary logistic regression is a widely used statistical method when the dependent variable is binary or dichotomous. In some of the situations of logistic regression, independent variables are collinear which leads to the problem of multicollinearity. It is known that multicollinearity affects the variance of maximum likelihood estimator (MLE) negatively. Thus, this article introduces new methods to estimate the shrinkage parameters of Liu-type logistic estimator proposed by Inan and Erdogan (2013 Inan, D., Erdogan, B. E. (2013). Liu-type logistic estimator. Communications in Statistics-Simulation and Computation 42(7):1578–1586. [Google Scholar]) which is a generalization of the Liu-type estimator defined by Liu (2003 Liu, K. (2003). Using Liu-type estimator to combat collinearity. Communications in Statistics: Theory and Methods 32(5):1009–1020. [Google Scholar]) for the linear model. A Monte Carlo study is used to show the effectiveness of the proposed methods over MLE using the mean squared error (MSE) and mean absolute error (MAE) criteria. A real data application is illustrated to show the benefits of new methods. According to the results of the simulation and application proposed methods have better performance than MLE.     

Quantile Regression with Left-Truncated and Right-Censored Data in a Reproducing Kernel Hilbert Space

Li et al. (2007 Li, Y., Liu, Y., Zhu, J. (2007). Quantile regression in reproducing kernel Hilbert spaces. J. Amer. Statist. Assoc. 102:255–268.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) developed an estimation method for quantile functions in a reproducing kernel Hilbert space for complete data, and Park and Kim (2011 Park, J., Kim, J. (2011). Quantile regression with an epsilon-insensitive loss in a reproducing kernel Hilbert space. Statist. Probab. Lett. 81:62–70.[Crossref], [Web of Science ®] , [Google Scholar]) proposed an estimation method using the ε-insensitive loss. This article extends these estimation methods to left-truncated and right-censored data. As a measure of goodness of fit, the check loss and the ε-insensitive loss were used to estimate the quantile function. The ε-insensitive loss can shrink the estimated coefficients toward zero; hence, it can reduce the variability of the estimates. Simulation studies show that the estimated quantile functions based on the ε-insensitive loss perform slightly better when ε is adequately chosen.     

Joint influence of measurement error and non response on estimation of population mean

In this paper, we propose a method to jointly incorporate measurement error and non response in the estimators of population mean using auxiliary information in simple random sampling. We have not only studied some available estimators but also suggested three new estimators in the presence of two types of non sampling errors occurring jointly: the measurement error and the non response. The expressions for the bias and mean square errors of proposed estimator have been derived. A comparative study is made among the proposed estimators, the Hansen and Hurwitz (1946 Hansen, M.H., Hurwitz, W.N. (1946). The problem of non-response in sample surveys. J. Am. Stat. Assoc. 41:517–529.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator, the Cochran's (1977 Cochran, W.G. (1977). Sampling Techniques, 3rd Edn. New York: John Wiley &; Sons, Inc. [Google Scholar]) estimator, and the Singh and Kumar (2008 Singh, H.P., Karpe, N. (2008). Estimation of population variance using auxiliary information in the presence of measurement errors. Stat. Trans. New Ser. 9(3):443–470. [Google Scholar]) estimator.     

A new estimator of finite population mean based on the dual use of the auxiliary information

When a sufficient correlation between the study variable and the auxiliary variable exists, the ranks of the auxiliary variable are also correlated with the study variable, and thus, these ranks can be used as an effective tool in increasing the precision of an estimator. In this paper, we propose a new improved estimator of the finite population mean that incorporates the supplementary information in forms of: (i) the auxiliary variable and (ii) ranks of the auxiliary variable. Mathematical expressions for the bias and the mean-squared error of the proposed estimator are derived under the first order of approximation. The theoretical and empirical studies reveal that the proposed estimator always performs better than the usual mean, ratio, product, exponential-ratio and -product, classical regression estimators, and Rao (1991 Rao, T.J. (1991). On certail methods of improving ration and regression estimators. Commun. Stat. Theory Methods 20(10):3325–3340.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (2009 Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2009). Improvement in estimating the population mean using exponential estimator in simple random sampling. Int. J. Stat. Econ. 3(A09):13–18. [Google Scholar]), Shabbir and Gupta (2010 Shabbir, J., Gupta, S. (2010). On estimating finite population mean in simple and stratified random sampling. Commun. Stat. Theory Methods 40(2):199–212.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Grover and Kaur (2011 Grover, L.K., Kaur, P. (2011). An improved estimator of the finite population mean in simple random sampling. Model Assisted Stat. Appl. 6(1):47–55. [Google Scholar], 2014) estimators.