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.     

An Improved Class of Estimators in Two-Phase Sampling Using Two Auxiliary Variables

This article suggests an improved class of estimators under the general framework of two-phase sampling scheme in presence of two auxiliary variables. This class includes a large number of estimators (Chand, 1975 Chand , L. ( 1975 ). Some Ratio-Type Estimator Based on Two or More Auxiliary Variables. Unpublished Ph.D. dissertation, Iowa State University, Iowa . [Google Scholar]; Kiregyera, 1980 Kiregyera , B. ( 1980 ). A chain ratio-type estimator in finite population double sampling using two auxiliary variables . Metrika 27 : 217 – 223 .[Crossref] , [Google Scholar], 3; Mukharjee et al., 1987 Mukharjee , R. , Rao , T. J. , Vijayan , K. ( 1987 ). Regression-type estimators using multiple auxiliary information . Aust. J. Statist. 29 : 244 – 254 . [Google Scholar]) and also the class of estimators suggested by Sahoo et al. (1993 Sahoo , J. , Sahoo , L. N. , Mohanty , S. ( 1993 ). A regression approach to estimation in two phase sampling using two auxiliary variables . Curr. Sci. 65 ( 1 ): 73 – 75 . [Google Scholar]).     

The Restricted and Unrestricted Two-Parameter Estimators

In this article, we introduce a new two-parameter estimator by grafting the contraction estimator into the modified ridge estimator proposed by Swindel (1976 Swindel , B. F. ( 1976 ). Good ridge estimators based on prior information . Commun. Statist. Theor. Meth. A5 : 1065 – 1075 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This new two-parameter estimator is a general estimator which includes the ordinary least squares, the ridge, the Liu, and the contraction estimators as special cases. Furthermore, by setting restrictions Rβ = r on the parameter values we introduce a new restricted two-parameter estimator which includes the well-known restricted least squares, the restricted ridge proposed by Groß (2003 Groß , J. ( 2003 ). Restricted ridge estimation . Statist. Probab. Lett. 65 : 57 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]), the restricted contraction estimators, and a new restricted Liu estimator which we call the modified restricted Liu estimator different from the restricted Liu estimator proposed by Kaç?ranlar et al. (1999 Kaç?ranlar , S. , Sakall?o?lu , S. , Akdeniz , F. , Styan , G. P. H. , Werner , H. J. ( 1999 ). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement . Sankhya Ser. B., Ind. J. Statist. 61 : 443 – 459 . [Google Scholar]). We also obtain necessary and sufficient condition for the superiority of the new two-parameter estimator over the ordinary least squares estimator and the comparison of the new restricted two-parameter estimator to the new two-parameter estimator is done by the criterion of matrix mean square error. The estimators of the biasing parameters are given and a simulation study is done for the comparison as well as the determination of the biasing parameters.     

Efficient separate class of estimators of population mean in stratified random sampling

In this paper, efficient class of estimators for population mean using two auxiliary variates is suggested. It has been shown that the suggested estimator is more efficient than usual unbiased estimator in stratified random sampling, usual ratio and product-type estimators, Tailor and Lone (2012 Tailor, R. and Lone, H. A. (2012). Separate ratio-cum- product estimators of finite population mean using auxiliary information. J. Rajasthan Stat. Assoc. 1(2):94–102. [Google Scholar], 2014) estimators, and other considered estimators. The bias and mean-squared error of the suggested estimator are obtained up to the first degree of approximation. Conditions under which the suggested estimator is more efficient than other considered estimators are obtained. An empirical study has been carried out to demonstrate the performances of the suggested estimator.     

Statistical Analysis of the Log-Normal Distribution under Type-II Progressive Hybrid Censoring Schemes

In this article, we obtain the maximum likelihood estimators (MLEs) and approximate maximum likelihood estimators (AMLEs) of the parameters, from a two-parameter log-normal distribution based on the adaptive Type-II progressive hybrid censoring scheme, which was introduced by Ng et al. (2009 Ng , H. K. T. , Kundu , D. , Chan , P. S. ( 2009 ). Statistical analysis of exponential lifetimes under an adaptive Type-II progressively censoring scheme . Naval Research Logistics 56 : 687 – 698 .[Crossref], [Web of Science ®] , [Google Scholar]) for life testing or reliability experiment. In order to compare the results, we calculate corresponding estimators of the Type-II progressive hybrid censoring scheme. In particular, we provide computational formulas of the expected total test time and the expected number of failures for each scheme. We also compute the observed Fisher information matrix and use them to obtain the asymptotic confidence intervals. A simulation study carries out to evaluate the bias and mean square error of the MLEs and AMLEs from the two above-mentioned schemes. Finally, we present a numerical example to illustrate the methods of inference discussed here.     

Some Classes of Estimators for the Population Median Using Auxiliary Information

This article considers some classes of estimators of the population median of the study variable using information on an auxiliary variable with their properties under large sample approximation. Asymptotic optimum estimator (AOE) in each class of estimators has been investigated along with the approximate mean square error formulae. It has been shown that the proposed classes of estimators are better than these considered by Gross (1980 Gross , T. S. ( 1980 ). Median estimation in sample surveys. Proc. Surv. Res. Meth. Sect. Amer. Statist. Assoc. 181–184 . [Google Scholar]), Kuk and Mak (1989 Kuk , A. Y. C. , Mak , T. K. ( 1989 ). Median estimation in the presence of auxiliary information . J. Roy. Statist. Soc. Ser. B51 : 261 – 269 . [Google Scholar]), Singh et al. (2003a Singh , H. P. , Singh , S. , Joarder , A. H. ( 2003a ). Estimation of population median when mode of an auxiliary variable is known . J. Statist. Res. 37 ( 1 ): 57 – 63 . [Google Scholar]), and Al and Cingi (2009 Al , S. , Cingi , H. ( 2009 ). New estimators for the population median in simple random sampling. Tenth Islamic Countries Conference on Statistical Sciences, held in New Cairo, Egypt . [Google Scholar]). An empirical study is carried out to judge the merits of the suggested class of estimators over other existing estimators.     

Efficient Ratio and Product Estimators in Stratified Random Sampling

This paper suggests an efficient class of ratio and product estimators for estimating the population mean in stratified random sampling using auxiliary information. It is interesting to mention that, in addition to many, Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2003 Kadilar , C. , Cingi , H. ( 2003 ). Ratio estimator in stratified sampling . Biometr. J. 45 : 218 – 225 .[Crossref], [Web of Science ®] , [Google Scholar], 2005 Kadilar , C. , Cingi , H. ( 2005 ). A new estimator in stratified random sampling . Commun. Statist. Theor. Meth. 34 : 597 – 602 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Vishwakarma (2007 Singh , H. P. , Vishwakarma , G. K. ( 2007 ). Modified exponential ratio and product estimators for finite population mean in double sampling . Austr. J. Statist. 36 ( 3 ): 217 – 225 . [Google Scholar]) estimators are identified as members of the proposed class of estimators. The expressions of bias and mean square error (MSE) of the proposed estimators are derived under large sample approximation in general form. Asymptotically optimum estimator (AOE) in the class is identified alongwith its MSE formula. It has been shown that the proposed class of estimators is more efficient than combined regression estimator and Koyuncu and Kadilar (2009 Koyuncu , N. , Kadilar , C. ( 2009 ). Ratio and product estimators in stratified random sampling . J. Statist. Plann. Infer. 139 : 2552 – 2558 .[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Moreover, theoretical findings are supported through a numerical example.     

Asymptotic normality of the local linear estimation of the conditional density for functional time-series data

This article focuses on the conditional density of a scalar response variable given a random variable taking values in a semimetric space. The local linear estimators of the conditional density and its derivative are considered. It is assumed that the observations form a stationary α-mixing sequence. Under some regularity conditions, the joint asymptotic normality of the estimators of the conditional density and its derivative is established. The result confirms the prospect in Rachdi et al. (2014 Rachdi, M., A. Laksaci, J. Demongeot, A. Abdali, and F. Madani. 2014. Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data. Computational Statistics and Data Analysis 73 :53–68.[Crossref], [Web of Science ®] , [Google Scholar]) and can be applied in time-series analysis to make predictions and build confidence intervals. The finite-sample behavior of the estimator is investigated by simulations as well.     

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.     

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.     

Improved Exponential Type Estimators of the Mean of a Sensitive Variable in the Presence of Nonsensitive Auxiliary Information

Recently, Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) proposed an exponential type estimator to improve the efficiency of mean estimator based on randomized response technique. In this article, we propose an improved exponential type estimator which is more efficient than the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator, which in turn was shown to be more efficient than the usual mean estimator, ratio estimator, regression estimator, and the Gupta et al. (2012 Gupta, S., Shabbir, J., Sousa, R., Corte-Real, P. (2012). Regression estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics – Theory and Methods 41:2394–2404.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator. Under simple random sampling without replacement (SRSWOR) scheme, bias and mean square error expressions for the proposed estimator are obtained up to first order of approximation and comparisons are made with the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator. A simulation study is used to observe the performances of these two estimators. Theoretical findings are also supported by a numerical example with real data. We also show how to, extend the proposed estimator to the case when more than one auxiliary variable is available.     

Generalized ratio-cum-product type exponential estimator in stratified random sampling

ABSTRACT

In this article, we propose a generalized ratio-cum-product type exponential estimator for estimating population mean in stratified random sampling. Asymptotic expression of the bias and mean squared error of the proposed estimator are obtained. Asymptotic optimum estimator in the proposed estimator has been obtained with its mean squared error formula. Conditions under which the proposed estimator is more efficient than usual unbiased estimator, combined ratio and product type estimators, 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., Dec. 2008, 29–31, Aligarh, India. [Google Scholar]) estimators and Tailor and Chouhan (2014 Tailor, R., Chouhan, S. (2014). Ratio-cum-product type exponential estimator of finite population mean in stratified random sampling. Commun. Statist. Theor. Meth. 43:343–354.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator are obtained. An empirical study has also been carried out.     

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.     

Symmetric quantiles and their applications

To develop estimators with stronger efficiencies than the trimmed means which use the empirical quantile, Kim (1992) Kim, S. J. 1992. The metrically trimmed means as a robust estimator of location. Annals of Statistics, 20: 1534–1547. [Crossref], [Web of Science ®] [Google Scholar] and Chen & Chiang (1996) Chen, L. A. and Chiang, Y. C. 1996. Symmetric type quantile and trimmed means for location and linear regression model. Journal of Nonparametric Statistics, 7: 171–185. [Taylor & Francis Online] [Google Scholar], implicitly or explicitly used the symmetric quantile, and thus introduced new trimmed means for location and linear regression models, respectively. This study further investigates the properties of the symmetric quantile and extends its application in several aspects. (a) The symmetric quantile is more efficient than the empirical quantiles in asymptotic variances when quantile percentage α is either small or large. This reveals that for any proposal involving the α th quantile of small or large α s, the symmetric quantile is the right choice; (b) a trimmed mean based on it has asymptotic variance achieving a Cramer-Rao lower bound in one heavy tail distribution; (c) an improvement of the quantiles-based control chart by Grimshaw & Alt (1997) Grimshaw, S. D. and Alt, F. B. 1997. Control charts for quantile function values. Journal of Quality Technology, 29: 1–7. [Taylor & Francis Online] [Google Scholar] is discussed; (d) Monte Carlo simulations of two new scale estimators based on symmetric quantiles also support this new quantile.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.     

Estimation of Parameters of Mixed Failure Time Distribution Based on an Extended Modified Sampling Scheme

Abstract

The problem of estimation of parameters of a mixture of degenerate (at zero) and exponential distribution is considered by Dixit and Prasad [Dixit, V. U. (Nee: Jayade, V. D.), Prasad, M. S. (1990 Dixit, V. U. and Prasad, M. S. 1990. Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth, 19(12): 4667–4678. (Nee: Jayade, V. D.) [Google Scholar]). Estimation of parameters of mixed failure time distribution. Commun.in Statist.-Theory Meth., 19(12):4667–4678]. The sampling scheme proposed in it is extended to k positive observations in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] and moment estimator, MLE and UMVUE based on it are obtained and their finite sample and asymptotic properties are studied. These results are presented in this paper. It is interesting to mention that the sampling scheme proposed by Shinde and Shanubhogue [Shinde, R. L., Shanubhogue, A. (2000 Shinde, R. L. and Shanubhogue, A. 2000. Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth, 29(11): 2621–2642. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Estimation of parameters and the mean life of a mixed failure time distribution. Commun. Statist.-Theory Meth. 29(11):2621–2642] is a particular case of the sampling scheme proposed in Dixit [Dixit, V. U. (1993 Dixit, V. U. 1993. “Statistical Inference for AR (1) Process with Mixed Errors”. Kolhapur, , India: Shivaji University. Unpublished Ph.D. thesis [Google Scholar]). Statistical Inference for AR (1) Process with Mixed Errors. Unpublished Ph.D. thesis, Shivaji University Kolhapur, India] for n = k.     

Ridge estimator with correlated errors and two-stage ridge estimator under inequality restrictions

Liew (1976a Liew, C.K. (1976a). A two-stage least-squares estimation with inequality restrictions on parameters. Rev. Econ. Stat. LVIII(2):234–238.[Crossref], [Web of Science ®] , [Google Scholar]) introduced generalized inequality constrained least squares (GICLS) estimator and inequality constrained two-stage and three-stage least squares estimators by reducing primal–dual relation to problem of Dantzig and Cottle (1967 Dantzig, G.B., Cottle, R.W. (1967). Positive (semi-) definite matrices and mathematical programming. In: Abadie, J., ed. Nonlinear Programming (pp. 55–73). Amsterdam: North Holland Publishing Co. [Google Scholar]), Cottle and Dantzig (1974 Cottle, R.W., Dantzig, G.B. (1974). Complementary pivot of mathematical programming. In: Dantzig, G.B., Eaves, B.C., eds. Studies in OptimizationVol. 10. Washington: Mathematical Association of America. [Google Scholar]) and solving with Lemke (1962 Lemke, C.E. (1962). A method of solution for quadratic programs. Manage. Sci. 8(4):442–453.[Crossref], [Web of Science ®] , [Google Scholar]) algorithm. The purpose of this article is to present inequality constrained ridge regression (ICRR) estimator with correlated errors and inequality constrained two-stage and three-stage ridge regression estimators in the presence of multicollinearity. Untruncated variance–covariance matrix and mean square error are derived for the ICRR estimator with correlated errors, and its superiority over the GICLS estimator is examined via Monte Carlo simulation.     

On generalized exponential chain ratio estimators under stratified two-phase random sampling

Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) have suggested generalized exponential chain ratio estimators under stratified two-phase sampling scheme for estimating the finite population mean. However, the bias and mean square error (MSE) expressions presented in that work need some corrections, and consequently the study based on efficiency comparison also requires corrections. In this article, we revisit Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) estimator and provide the correct bias and MSE expressions of their estimator. We also propose an estimator which is more efficient than several competing estimators including the classes of estimators in Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]). Three real datasets are used for efficiency comparisons.     

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]).