NONNORMAL REGRESSION. II. SYMMETRIC DISTRIBUTIONS

Salient features of a family of short-tailed symmetric distributions, introduced recently by Tiku and Vaughan [1] Tiku, M. L. and Vaughan, D. C. 1999. “A family of short-tailed symmetric distributions”. In Technical Report Canada: McMaster University.  [Google Scholar], are enunciated. Assuming the error distribution to be one of this family, the methodology of modified likelihood is used to derive MML estimators of parameters in a linear regression model. The estimators are shown to be efficient, and robust to inliers. This paper is essentially the first to achieve robustness to inliers. The methodology is extended to long-tailed symmetric distributions and the resulting estimators are shown to be efficient, and robust to outliers. This paper should be read in conjunction with Islam et al. [2] Islam, Q., Tiku, M. L. and Yildirim, F. 2001. Nonnormal regression, Part I: Skew distributions. Commun. Stat.—Theory Meth., to appear [Google Scholar]who develop modified likelihood methodology for skew distributions in the context of linear regression.     

EM-based algorithms for autoregressive models with t-distributed innovations

This paper considers the estimation of parameters of AR(p) models for time series with t-distribution via EM-based algorithms. The paper develops asymptotic properties for the estimation to show that the estimators are efficient. Also testing theory for the estimators is considered. The robustness of the estimators and various tests to deviations from an assumed model is investigated. The study shows that the algorithms have equal estimation efficiency even if the error distribution is miss-specified or perturbed by outliers. Interestingly, the estimators from these algorithms performed better than that of the Modified Maximum Likelihood (MML) considered in Tiku et al. (2000 Tiku, M. L., Wong, W. K., Vaughan, D. C., Bian, G. (2000). Time series models in non-normal situations: Symmetric innovations. Journal of Time Series Analysis, 21: 571–596. [Google Scholar]).     

ESTIMATING PARAMETERS IN AUTOREGRESSIVE MODELS IN NON-NORMAL SITUATIONS: ASYMMETRIC INNOVATIONS

The estimation of coefficients in a simple autoregressive model is considered in a supposedly difficult situation where the innovations have an asymmetric distribution. Two distributions, gamma and generalized logistic, are considered for illustration. Closed form estimators are obtained and shown to be efficient and robust. Efficiencies of least squares estimators are evaluated and shown to be very low. This work is an extension of that of Tiku, Wong and Bian [1] Tiku, M. L., Wong, W. K. and Bian, G. 1999. Time Series Models with Asymmetric Innovations. Commun. Stat.-Theory Meth., 28: 1131–1160.  [Google Scholar] who give solutions for a simple AR(1) model.

     

On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]).     

The Confidence Ellipses in Decomposition Multiple Non Symmetric Correspondence Analysis

Non Symmetric Correspondence Analysis (NSCA) (D'Ambra and Lauro, 1989 D'Ambra , L. , Lauro , N. ( 1989 ). Non symmetrical analysis of three way contingency tables . In: Multiway Data Analysis , Coppi , R. , Bolasco , S. , Eds., North Holland , Amsterdam : pp. 301 – 315 . [Google Scholar]) is a useful technique for analyzing a two-way contingency table.

The key difference between the symmetrical and non symmetrical versions of correspondence analysis rests on the measure of the association used to quantify the relationship between the variables. For a two-way, or multi-way, contingency table, the Pearson chi-squared statistic is commonly used when it can be assumed that the categorical variables are symmetrically related. However, for a two-way table, it may be that one variable can be treated as a predictor variable and the second variable can be considered as a response variable.

Yet, for such a variable structure, the Pearson chi-squared statistic is not an appropriate measure of the association. Instead, one may consider the Goodman-Kruskal tau index. In the case that there are more than two cross-classified variables, multivariate versions of the Goodman-Kruskal tau index can be considered. These include Marcotorchino's index (Marcotorchino, 1985) and Gray-Williams’ index (Gray and Williams, 1975 Gray , L. N. , Williams , J. S. ( 1975 ). Goodman and Kruskals Tau B: Multiple and partial analogy. Amer. Statist. Assoc. Proc. Soc. Statist. Sec. pp. 444 – 448 . [Google Scholar]).

In this article, the Multiple non Symmetric Correspondence Analysis (MNSCA), along with the decomposition of the TAU by Gray-Williams in main effects and interaction (D'Ambra et al., 2011 D'Ambra , L. , D'Ambra , A. , Sarnacchiaro , P. ( 2011 ). Visualising main effects and interaction term in multiple non symmetric correspondence analysis. Submitted.  [Google Scholar]), is used for the evaluation of the innovative performance of the manufacturing enterprises in Campania.

Finally, to identify a category which is statistically significant, the confidence ellipses have been proposed for the Multiple Non Symmetric Correspondence Analysis starting from the ellipses suggested by Beh (2010 Beh , E. J. ( 2010 ). Elliptical confidence regions for simple correspondence analysis . J. Statisti. Plann. Infer. [Web of Science ®] , [Google Scholar]) for the symmetrical analysis.     

Estimation of finite population mean using two auxiliary variables in stratified two-phase sampling

We propose a new ratio type estimator for estimating the finite population mean using two auxiliary variables in stratified two-phase sampling. Expressions for bias and mean squared error of the proposed estimator are derived up to the first order of approximation. The proposed estimator is more efficient than the usual stratified sample mean estimator, traditional stratified ratio estimator and some other stratified estimators including Bahl and Tuteja (1991 Bahl, S., Tuteja, R. K. (1991). Ratio and product type exponential estimators. Information and Optimization Sciences 12:159–163. [Google Scholar]), Chami et al. (2012 Chami, P. S., Singh, B., Thomas, D. (2012). A two-prameter ratio-product-ratio estimator using auxiliary information. ISRN Probability and Statistics 2012:1–15, doi: 10.5402/2012/103860.[Crossref] , [Google Scholar]), Chand (1975 Chand, L. (1975) Some Ratio Type Estimator Based on two or more Auxiliary Variables, Ph.D. dissertation, Iowa State University, Ames, Iowa (unpublished). [Google Scholar]), Choudhury and Singh (2012 Choudhury, S., Singh, B. K. (2012). A class of chain ratio-product type estimators with two auxiliary variables under double sampling scheme. Journal of the Korean Statistical Society 41:247–256. [Google Scholar]), Hamad et al. (2013 Hamad, N., Hanif, M., Haider, N. (2013). A regression type estimator with two auxiliary variables for two-phase sampling. Open Journal of Statistics, 3:74–78. [Google Scholar]), Vishwakarma and Gangele (2014 Vishwakarma, G. K., Gangele, R. K. (2014). A class of chain ratio-type exponential estimators in double sampling using two auxiliary variates. Applied Mathematics and Computation 227:171–175. [Google Scholar]), 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. Applied Mathematics and Computation 226:541–547. [Google Scholar]), and Chanu and Singh (2014 Chanu, W. K., Singh, B. K. (2014). Improved class of ratio-cum-product estimators of finite population mean in two phase sampling. Global Journal of Science Frontier Research: F Mathematics and Decision Sciences 14(2):1–14. [Google Scholar]).     

Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices

In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004 Basso , D. , Evangelaras , H. , Koukouvinos , C. , Salmaso , L. ( 2004 ). Nonparametric testing for main effects on inequivalent designs. Proc. 7th Int. Workshop Model-Oriented Design Anal. Heeze, Netherlands, June 14–18 . [Google Scholar]) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified.     

A two-stage algorithm for computation of the exact distribution of sums of spacings

Based on the recursions in Huffer (1988 Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacings. Journal of Applied Probability 25:346–354. [Google Scholar]) and Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]), we present a two-stage algorithm and two specialized methods for evaluating the probabilities involving linear combination of spacings of special forms. The two-stage algorithm combines the advantages of marking algorithm in Huffer and Lin (1997 Huffer, F. W., Lin, C. T. (1997). Computing the exact distribution of the extremes of sums of consecutive spacings. Computational Statistics and Data Analysis 26:117–132. [Google Scholar]) and general algorithm in Huffer and Lin (2001 Huffer, F. W., Lin, C. T. (2001). Computing the joint distribution of general linear combinations of spacings or exponential variates. Statistica Sinica 11:1141–1157. [Google Scholar]). The proposed methods can analytically derive the exact expressions for some specific problems, and efficiently handle problems such as the distribution of the circular scan statistic and multiple coverage probabilities.     

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.     

A New Generalized Exponential Geometric Distribution

In this article, another version of the generalized exponential geometric distribution different to that of Silva et al. (2010 Silva , R. B. , Barreto-Souza , W. , Cordeiro , G. M. ( 2010 ). A new distribution with decreasing, increasing and upside-down bathtub failure rate. Computat. Statist. Data Anal. 54: 935–944 . [Google Scholar]) is proposed. This new three-parameter lifetime distribution with decreasing, increasing, and bathtub failure rate function is created by compounding the generalized exponential distribution of Gupta and Kundu (1999 Gupta , R. D. , Kundu , D. ( 1999 ). Generalized exponential distributions . Austral. NZ J. Statist. 41 ( 2 ): 173 – 188 .[Crossref], [Web of Science ®] , [Google Scholar]) with a geometric distribution. Some basic distributional properties, moment-generating function, rth moment, and Rényi entropy of the new distribution are studied. The model parameters are estimated by the maximum likelihood method and the asymptotic distribution of estimators is discussed. Finally, an application of the new distribution is illustrated using the two real data sets.     

Searching effective rotation patterns for population median using exponential type estimators in two-occasion rotation sampling

ABSTRACT

The present article is an attempt to explore the rotation patterns using exponential ratio type estimators for the estimation of finite population median at current occasion in two occasion rotation sampling. Properties of the proposed estimators including the optimum replacement strategies have been elaborated. The proposed estimators have been compared with sample median estimator when there is no matching from previous occasion as well with the ratio type estimator proposed by Singh et al. (2007 Singh, H.P., Tailor, R., Singh, S., Kim, Jong-Min. (2007). Quintile estimation in successive sampling. J. Kor. Stat. Soc. 36(4):543–556. [Google Scholar]) for second quantile. The behaviors of the proposed estimators are justified by empirical interpretations and validated by means of simulation study with the help of some natural populations.     

Comparison of Frailty Models for Acute Leukemia Data under Gompertz Baseline Distribution

In this article, we consider two different shared frailty regression models under the assumption of Gompertz as baseline distribution. Mostly assumption of gamma distribution is considered for frailty distribution. To compare the results with gamma frailty model, we consider the inverse Gaussian shared frailty model also. We compare these two models to a real life bivariate survival data set of acute leukemia remission times (Freireich et al., 1963 Freireich, E.J., Gehan, E., Frei, E., Schroeder, L.R., Wolman, I.J., Anbari, R., Burgert, E.O., Mills, S.D., Pinkel, D., Selawry, O.S., Moon, J.H., Gendel, B.R., Spurr, C.L., Storrs, R., Haurani, F., Hoogstraten, B., Lee, S. (1963). The effect of 6-mercaptopurine on the duration of steroid-induced remissions in acute leukemia: a model for evaluation of other potentially useful therapy. Blood 21:699–716.[Web of Science ®] , [Google Scholar]). Analysis is performed using Markov Chain Monte Carlo methods. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the acute leukemia data.     

Almost sure convergence of vertex degree densities in the vertex splitting model

We study the limiting degree distribution of the vertex splitting model introduced in Ref.^[3 David, F.; Dukes, M.; Jonsson, T.; Stefansson, S.Ö. Random tree growth by vertex splitting. J. Statist. Mech. Theory Exp. 2009, 04. doi:10.1088/1742-5468/2009/04/P04009. [Google Scholar]^]. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this, we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.     

On evaluating the performance of different forecasters

In this article, we have evaluated the performance of different forecasters and tested association between their performances for different pairs of variables. We have used three data sets of track records of professional U.S. economic forecasters participating in the Blue Chip consensus forecasting service (the data sets contain the root mean square errors (RMSE) of different forecasters for different years). To evaluate the performance of forecasters we have covered three well-known tests, namely the usual F test (cf. Fisher (1923 Fisher, R. A., Mackenzie, M. A. (1923). Studied in crop variation II. The manurial response of different potato. Journal of Agricultural Science 13:311–320. [Google Scholar])), Kruskal Wallis test (cf. Kruskal and Wallis (1952 Kruskall, W. H., Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of American Statistical Association 47:583–621. [Google Scholar])), and Extension of Median test (cf. Daniel (1990 Daniel, W. W. (1990). Applied Nonparametric Statistics. Duxbury Classic Series. (2nd Ed.), Boston. [Google Scholar])). To test the association between the forecaster's performances for different pairs of variables, we have considered Gini mean correlation coefficient r_g1 (cf. Yitzhaki, S., and Olkin, I. (1991 Yitzhaki, S., Olkin, I. (1991). Concentration indices and concentration curves, in K. Mosler and M. Scarsini (eds.), Stochastic Orders and Decisions under Risk, Institute of Mathematical Statistics: Lecture-Notes Monograph Series, 19, 1991, 380–392. [Google Scholar]) and Yitzhaki (2003 Yitzaki, S. (2003). Gini mean difference: A superior measure of variability for non normal distribution. Metron-International Journal of Statistics, LXI:285–316. [Google Scholar])), Modified rank correlation coefficient (cf. Zimmerman (1994 Zimmerman, D. W. (1994). A Note on modified rank correlation. Journal of educational and Behavioral Statistics 19:357–362. [Google Scholar])) and three modifications of Spearman rank correlation coefficient. We have observed that different forecasters do not necessarily offer same average performance. Moreover, an evidence of association between two criteria does not always lead us reaching at the same decision. The outcomes of the study may help the practitioners in selecting the best forecaster(s) for policymaking purposes.     

Some new results on allocation of coverage limits and deductibles to mutually independent risks

This article further investigates the allocation of coverage limits and deductibles to multiple independent risks from the viewpoint of policyholders with increasing utility functions. In a more general setup, we develop the usual stochastic orders on the retained loss, which either generalize or supplement the corresponding results due to Lu and Meng (2011 Lu, Z., Meng, L. (2011). Stochastic comparisons for allocations of policy limits and deductibles with applications. Insur. Math. Econ. 48:338–343. [Google Scholar]) and Hu and Wang (2014 Hu, S., Wang, R. (2014). Stochastic comparisons and optimal allocation for policy limits and deductibles. Commun. Stat. Theory Methods 43:151–164. [Google Scholar]). Also, the most unfavorable and favorable allocations of coverage limits and deductibles are developed for multiple risks with dominated reversed hazard rates and hazard rates, respectively.     

Selection of Binary Variables and Classification by Boosting

We adopt boosting for classification and selection of high-dimensional binary variables for which classical methods based on normality and non singular sample dispersion are inapplicable. Boosting seems particularly well suited for binary variables. We present three methods of which two combine boosting with the relatively classical variable selection methods developed in Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]). Our primary interest is variable selection in classification with small misclassification error being used as validation of proposed method for variable selection. Two of the new methods perform uniformly better than Wilbur et al. (2002 Wilbur , J. D. , Ghosh , J. K. , Nakatsu , C. H. , Brouder , S. M. , Doerge , R. W. ( 2002 ). Variable selection in high-dimensional multivariate binary data with application to the analysis of microbial community DNA fingerprints . Biometrics 58 : 378 – 386 . [Google Scholar]) in one set of simulated and three real life examples.     

A Tail Quantile Approximation for the Student t Distribution

In this article, we derive a new formula for extreme Student t quantiles. We use the fact that the Student t distribution arises as the limit of a variance-mixture of normals. For the normal distribution there is already a tail quantile formula derived by Reiss (1989 Reiss , R.-D. ( 1989 ). Approximate Distributions of Order Statistics: With Applications to Non-parametric Statistics . New York : Springer .[Crossref] , [Google Scholar]). We generalize his procedure and transfer it to our scenario.

Eventually, we compare the quantile estimates of our formula to those from Gafer and Kafadar (1984 Gafer , D. P. , Kafadar , K. ( 1984 ). A retrievable recipe for inverse t. Amer. Statistician 38(4):308–311 . [Google Scholar]), who also derived a Student t quantile formula. Using R to generate a benchmark we find that our method is more accurate for very high quantiles.     

Steady-state Gibbs sampler estimation for lung cancer data

This paper is based on the application of a Bayesian model to a clinical trial study to determine a more effective treatment to lower mortality rates and consequently to increase survival times among patients with lung cancer. In this study, Qian et al. [13 J. Qian, D.K. Stangl, and S. George, A Weibull model for survival data: Using prediction to decide when to stop a clinical trial, in Bayesian Biostatistics, D. Berry and D. Stangl, eds., Marcel Dekker, New York, 1996, pp. 187–205. [Google Scholar]] strived to determine if a Weibull survival model can be used to decide whether to stop a clinical trial. The traditional Gibbs sampler was used to estimate the model parameters. This paper proposes to use the independent steady-state Gibbs sampling (ISSGS) approach, introduced by Dunbar et al. [3 M. Dunbar, H.M. Samawi, R. Vogel, and L. Yu, A more efficient Gibbs sampler estimation using steady state simulation: Application to public health studies, J. Stat. Simul. Comput. 10.1080/00949655.2013.770857.[Taylor &; Francis Online] , [Google Scholar]], to improve the original Gibbs sampler in multidimensional problems. It is demonstrated that ISSGS provides accuracy with unbiased estimation and improves the performance and convergence of the Gibbs sampler in this application.     

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.     

Selecting the best exponential population under Type-II progressive censoring scheme via empirical Bayes approach

The problem of selecting a population according to “selection and ranking” is an important statistical problem. The ideas in selecting the best populations with some demands having optimal criterion have been suggested originally by Bechhofer (1954 Bechhofer, R. E. (1954). A single-sample multiple-decision procedure for ranking means of normal populations with known variances. The Annals of Mathematical Statistics 25:16–39. [Google Scholar]) and Gupta (1956 Gupta, S. S. (1956). On a decision rule for a problem in ranking means. Mimeograph Series No. 150. Chapel Hill, North Carolina: University of North Carolina. [Google Scholar], 1965 Gupta, S. S. (1965). On some multiple decision (selection and ranking) rules. Technometrics 7:225–245. [Google Scholar]). In the area of ranking and selection, the large part of literature is connected with a single criterion. However, this may not satisfy the experimenter’s demand. We follow methodology of Huang and Lai (1999 Huang, W. T., Lai, Y. T. (1999). Empirical Bayes procedures for selecting the best population with multiple criteria. Annals of the Institute of Statistical Mathematics 51:281–299. [Google Scholar]) and the main focus of this article is to select a best population under Type-II progressively censored data for the case of right tail exponential distributions with a bounded and unbounded supports for μ_i. We formulate the problem and develop a Bayesian setup with two kinds of bounded and unbounded prior for μ_i. We introduce an empirical Bayes procedure and study the large sample behavior of the proposed rule. It is shown that the proposed empirical Bayes selection rule is asymptotically optimal.