Stochastic Comparison on General Inactivity Time and General Residual Life of k-Out-of-n Systems

This paper conducts stochastic comparison on general residual life and general inactivity time of (n ? k + 1)-out-of-n systems and investigates the stochastic behavior of the general inactivity time of a system with units having decreasing reversed hazard rate. These results strengthen some conclusions in both Khaledi and Shaked (2006 Khaledi , B. , Shaked , M. ( 2006 ). Ordering conditional residual lifetimes of coherent systems . Journal of Statistical Planning and Inference 137 : 1173 – 1184 .[Crossref], [Web of Science ®] , [Google Scholar]) and Hu et al. (2007 Hu , T. , Jin , W. , Khaledi , M. ( 2007 ). Ordering conditional distributions of generalized order statistics . Probability in the Engineering and Informational Sciences 21 : 401 – 417 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Vector error correction heterogeneous autoregressive forecast model of realized volatility and implied volatility

A vector error correction model is proposed for forecasting realized volatility which takes advantage of the cointegration relation between realized volatility and implied volatility. The model is constructed by adding a cointegration error term to a vector-and-unit-root version of the heterogeneous autoregressive (HAR) model of Corsi (2009 Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7 (2):174–96. [Google Scholar]). The proposed model is easier to implement, extend, and interpret than fractional cointegration models. A Monte Carlo simulation and real data analysis reveal advantages of the proposed model over other existing models of Corsi (2009 Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7 (2):174–96. [Google Scholar]), Busch Christensen and Nielsen (2011 Busch, T., B. J. Christensen, and M. Nielsen. 2011. The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. Journal of Econometrics 160 (1):48–57. [Google Scholar]), Cho and Shin (2016 Cho, S. J. and D. W. Shin. 2016. An integrated heteroscedastic autoregressive model for forecasting long-memory volatilities. Journal of the Korean Statistical Society, 45:371–380. [Google Scholar]), and Bollerslev Patton, and Quaedvlieg (2016 Bollerslev, T., A. J. Patton, and R. Quaedvlieg. 2016. Exploiting the errors:A simple approach for improved volatility forecasting. Journal of Econometrics 192:1-18. [Google Scholar]).     

Multivariate Generalized Distributions of Order k

Abstract

The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1988. Multivariate distributions of order k. Statistics and Probability Letters, 7(3): 207–216.  [Google Scholar]). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1990. Multivariate distributions of order k, part II. Statistics and Probability Letters, 10(1): 29–35.  [Google Scholar]). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996 Sen, K. and Jain, R. 1996. “Cluster generalized negative binomial distribution”. In Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday Edited by: Borthakur, A. C. 227–241. New Delhi: New Age International Publishers.  [Google Scholar]). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997 Sen, K. and Jain, R. 1997. A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics—Theory and Methods, 26: 871–884. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.     

A study of uniformity pattern for extended designs

The objective of this paper is to investigate the issue of projection discrepancy for extended U-type or nearly U-type extended designs along the line of Fang and Qin (2005 Fang, K. T., and H. Qin. 2005. Uniformity pattern and related criteria for two-level factorials. Science China Series A 48:1–11.[Crossref] , [Google Scholar]) based on the centered L₂-discrepancy proposed in Hickernell (1998 Hickernell, F. J. 1998. A generalized discrepancy and quadrature error bound. Mathematics of Computation 67:299–322.[Crossref], [Web of Science ®] , [Google Scholar]). Extended designs are obtained through augmenting optimally few runs (or points) to an optimal U-type design. Lower bounds to projection discrepancy with reference to the centered L₂-discrepancy of extended designs have been obtained. Some illustrative examples are also provided.     

A general algorithm for non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data

In this paper, we study an algorithm to compute the non-parametric maximum likelihood estimator of stochastically ordered survival functions from case 2 interval-censored data. The algorithm, simply denoted by SQP (sequential quadratic programming), re-parameterizes the likelihood function to make the order constraints as a set of linear constraints, approximates the log-likelihood function as a quadratic function, and updates the estimate by solving a quadratic programming. We particularly consider two stochastic orderings, simple and uniform orderings, although the algorithm can also be applied to many other stochastic orderings. We illustrate the algorithm using the breast cancer data reported in Finkelstein and Wolfe (1985 Finkelstein, D. M., and R. A. Wolfe. 1985. A semiparametric model for regression analysis of interval-censored failure time data. Biometrics 41:933–45. [Google Scholar]).     

Testing economic “genetic pleiotropy” for Box-Cox linear model

Abstract

Genetic pleiotropy occurs when a single gene influences two or more seemingly unrelated phenotypic traits. It is significant to detect pleiotropy and understand its causes. However, most current statistical methods to discover pleiotropy mainly test the null hypothesis that none of the traits is associated with a variant, which departures from the null to test just one associated trait or k associated traits. Schaid et al. (2016 Schaid, D. J., X. Tong, B. Larrabee, R. B. Kennedy, G. A. Poland, and J. P. Sinnwell. 2016. Statistical methods for testing genetic pleiotropy. Genetics 204 (2):483–97. doi:10.1534/genetics.116.189308.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) first proposed a sequential testing framework to analyze pleiotropy based on a linear model and a multivariate normal distribution. In this paper, we analyze the Economic pleiotropy which occurs when an economic action or policy influences two or more economic phenomena. In this paper, we extend the linear model to Box-Cox transformation model and proposed a new decision method. It improves the efficiency of hypothesis test and controls the Type I error. We then apply the method using economic data to multivariate sectoral employments in response to governmental expenditures and provide a quantitative assessment and some insights of different impacts from economic policy.     

Some new results on aggregate claim amounts from two heterogeneous Marshall–Olkin extended exponential portfolios

In this work, we discuss some stochastic comparisons of two aggregate claim amounts. Applications of our results to the value-at-risk and tail-value-at-risk are also mentioned. It is also shown that the aggregate claim amounts of risks exhibiting a weak form of dependence known as positive cumulative dependence (negatively associated) is larger (smaller) in convex order than the corresponding aggregate claim amounts under the theoretical independence assumption. The obtained results show that the correlations between individual risks increase stop-loss premiums corresponding to aggregate claim amounts. The results established here complete and extend the results of Barmalzan, Payandeh Najafabadi, and Balakrishnan (2017 Barmalzan, G., A. T. Payandeh Najafabadi, and N. Balakrishnan. 2017. Ordering results for aggregate claim amounts from two heterogeneous Marshall–Olkin extended exponential portfolios and their applications in insurance analysis. Theory of Probability and its Applications(to appear). [Google Scholar], Theory of Probability and its Applications, to appear).     

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.     

Analytical results for IGFR and DRPFR distributions,with actuarial applications

An increasing generalized failure rate of a lifetime X defines an ageing concept, denoted by IGFR. Another notion, denoted by DRPFR, is defined by the decreasingness of the reversed proportional failure rate. In this article, we provide characterizations for both IGFR and DRPFR absolutely continuous lifetimes, based on monotonicity of quotients of probabilistic functionals and a result by Nanda and Shaked (2001 Nanda, A.K., Shaked, M. (2001). The hazard rate and the reversed hazard rate orders, with applications to order statistics. Ann. Inst. Stat. Math. 53:853–864.[Crossref], [Web of Science ®] , [Google Scholar]). We derive the necessary conditions for the IGFR notion, based on stochastic orderings of truncated distributions, and we prove that the product of DRPFR lifetimes is also DRPFR; that the IGFR property is preserved by composition with certain risk aversion utility functions; and that the order statistics and the records (and the subsequent order statistic (record)) are IGFR under suitable assumptions, with similar results for DRPFR lifetimes. Also, we provide sufficient conditions for the hazard rate ordering of products and random products of IGFR lifetimes, and similar results for the reversed hazard rate order and DRPFR lifetimes, with a complementary result for the mean residual life order of random products of two families of IGFR lifetimes, we derive the upper and lower bounds for the cumulative distribution function of the product of IGFR lifetimes, and we provide the lower bounds for the risk function of an IGFR lifetime based on the distribution moments, and these bounds are extended for the product of IGFR lifetimes. We discuss extensively the applications of the results in insurance portfolios.     

The Johnson System of Frequency Curves—Historical,Graphical, and Limiting Perspectives

Abstract

The idea of transforming one random variate to another with a more convenient density has been developed in the first half of the 20th century. In his thesis, Norman L. Johnson (1917–2004) developed a pioneering system of transformations of the standard normal distribution which gained substantial popularity in the second half of the 20th century and beyond. In Johnson’s 1949 Johnson, N. L. (1949), “Systems of Frequency Curves Generated by Methods of Translation,” Biometrika, 36, 149–176.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar] Biometrika paper entitled Systems of frequency curves generated by methods of translation, summarizing that thesis, one of his primary interests was the behavior of the shape of the probability density functions as their parameter values change. Herein, we attempt to further elucidate this behavior through a series of geometric expositions of that transformation process. In these expositions insight is obtained into the behavior of Johnson’s density functions, and their skewness and kurtosis, as they converge to their limiting distributions, a topic which received little attention.     

The structural Sharpe model under t-distributions

In this paper we consider Sharpe's single-index model or Sharpe's model, by assuming that the returns obtained follow a multivariate t elliptical distribution. Also, given that the returns of the market are not observable, the statistical analysis was made in the context of an errors-in-variables model. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators the local influence method [10 Cook, R. D. 1986. Assessment of local influence. J. Roy. Statist. Soc. B, 48: 133–169.  [Google Scholar]] was implemented. The results are illustrated by using a set of shares of companies belonging to the Chilean Stock Market. The main conclusion is that the t model with small degrees of freedom is able to incorporate possible outliers and influential returns in the data.     

Lower Bounds on the Symmetric L 2-Discrepancy and Their Application

The role of uniformity measured by the symmetric L ₂-discrepancy given in Hickernell (1998 Hickernell , F. J. (1998). A generalized discrepancy and quadrature error bound. Math. Computat. 67:299–322.[Crossref], [Web of Science ®] , [Google Scholar]) has been studied in fractional factorial designs. The issue of lower bounds on the symmetric L ₂-discrepancy is crucial in the construction of uniform designs. This article reports some new lower bounds on the symmetric L ₂-discrepancy for symmetric fractional factorials and for a set of asymmetric fractional factorials. It is valuable to use these lower bounds to measure uniformity of given designs.     

A cost-effective computational approach with non response on two occasions

Abstract

The present study is an attempt to equip a strategy with a cost-effective computational approach when non response is present under two occasion sampling. We have applied our computational cost strategy over Choudhary et al. (2004 Choudhary, R. K., H. V. L. Bathla, and U. C. Sud. 2004. On non-response in sampling on two occasions. Journal of the Indian Society of Agricultural Statistics 58(3):331–43. [Google Scholar])’s non response setup for fixed precision and evaluated cost. In addition, we have also computed variance for some fixed cost. We have discussed the aforementioned procedure for three cases as when there is non response present on both occasions, first occasion and second occasion. A numerical illustration is demonstrated for validation of improved cost methodology where we also work out with optimum unmatched or matched fraction while Choudhary et al. (2004 Choudhary, R. K., H. V. L. Bathla, and U. C. Sud. 2004. On non-response in sampling on two occasions. Journal of the Indian Society of Agricultural Statistics 58(3):331–43. [Google Scholar]) do not provide the direct optimal result.     

Dimension reduction boosting

L₂Boosting is an effective method for constructing model. In the case of high-dimensional setting, Bühlmann and Yu (2003 Bühlmann, P., Yu, B. (2003). Boosting with the L₂-loss: regression and classification. J. Amer. Stat. Assoc. 98:324–339.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proposed the componentwise L₂Boosting, but componentwise L₂Boosting can only fit a special limited model. In this paper, by combining a boosting and sufficient dimension reduction method, e.g., sliced inverse regression (SIR), we propose a new method for regression, called dimension reduction boosting (DRBoosting). Compared with L₂Boosting, the computation of DRBoosting is less intensive and its prediction is better, especially for high-dimensional data. Simulations confirm the advantage of the new method.     

Complete convergence for weighted sums of negatively dependent random variables under the sub-linear expectations

In this article, we study complete convergence theorems for weighted sums of negatively dependent random variables under the sub-linear expectations. Our results extend the corresponding results of Sung (2012 Sung, S. H. 2012. A note on the Complete convergence for weighted sums of negatively dependent random variables. Journal of Inequalities and Applications 2012:158, 10 pages. [Google Scholar]) relative to the classical probability.     

Bayesian estimation of sensitivity level and population proportion of a sensitive characteristic in a binary optional unrelated question RRT model

Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]) proposed an unrelated question binary optional randomized response technique (RRT) model for estimating the proportion of population that possess a sensitive characteristic and the sensitivity level of the question. In our work, decision theoretic approach has been followed to obtain Bayes estimates of the two parameters along with their corresponding minimal Bayes posterior expected losses (BPEL) using beta prior and squared error loss function (SELF). Relative losses are also examined to compare the performances of the Bayes estimates with those of the classical estimates obtained by Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]). The results obtained are illustrated with the help of real survey data using non informative prior.     

New Generalizations of Cauchy Distribution

The generalized skew-normal distribution introduced by Balakrishnan (2002 Balakrishnan , N. ( 2002 ). Discussion on ‘Skew multivariate models related to hidden truncation and/or selective reporting’ by B. C. Arnold and R. J. Beaver . Test 11 : 37 – 39 .[Web of Science ®] , [Google Scholar]) is used to obtain new generalizations of univariate Cauchy distribution with two parameters, denoted by GC _{m, n} (a, b) with m and n non-negative integer numbers and a, b ∈ R. For cases (m, n) = (1, 2), (m, n) = (2, 1), (m, n) = (0, 3) and (m, n) = (3, 0) explicit forms of the density functions are derived and compared to previous generalizations of Cauchy and skew-Cauchy distributions.     

Use of scrambled response for estimating mean of the sensitivity variable

Abstract

In this article, we propose new efficient and more generalized difference-cum-exponential type estimator and generalized-difference-cum-generalized exponential type estimators for estimating the mean of sensitivity variable using the auxiliary information. We also discuss theoretically that proposed generalized estimators are more efficient than Sousa et al. (2010 Sousa, R., J. Shabbir, P. C. Real, and S. Gupta. 2010. Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information. Journal of Statistical Theory and Practice 4 (3):495–507.[Taylor & Francis Online] , [Google Scholar]), Gupta et al. (2012 Gupta, S., J. Shabbir, R. Sousa, and P. C. Real. 2012. Estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics-Theory and Methods 41:1–12.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Koyuncu, Gupta, and Sousa (2014 Koyuncu, N., S. Gupta, and R. Sousa. 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 43 (7):1583–94. doi: 10.1080/03610918.2012.737492.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimators. Results from a real life application and simulation study are presented to demonstrate the performance of the proposed mean estimators in relation to some of the existing mean estimators.