On a class exhibiting trend change in average failure rate

Abstract

Weak convergence and moment convergence issues are investigated for the New Better than Average Failure Rate (NBAFR) family (introduced by Loh (1984 Loh, W. Y. 1984. A new generalization of the class of NBU distributions. IEEE Transactions on Reliability R-33 :97–113[Crossref], [Web of Science ®] , [Google Scholar])). We explore the validity of these results in the context of a more general ageing class that we introduce. We prove some new properties of this class and derive its interrelationships with other non-monotonic ageing families. Reliability and moment bounds are obtained and an interesting characterization of exponentiality is proved. Special cases of our results lead to new theorems for the NBAFR class. Finally weak convergence and related issues are established for this class.     

Test for comparing complete expectations of life of two groups

Abstract

Complete expectation of life of an individual gives an intuitive and interesting perspective on the ageing process and is an important concept in the insurance sector for determination of premium. We propose a new test for testing equality of complete expectations of life of two groups/populations. Power of the new test is calculated through simulations and compared with the power of the tests given by Berger, Boos, and Guess (1988 Berger, R. L., D. D. Boos, and F. M. Guess. 1988. Tests and confidence sets for comparing two mean residual life functions. Biometrics 44 (1):103–15.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Aly (1997 Aly, E. E. A. A. 1997. Nonparametric tests for comparing two mean residual life functions. Lifetime Data Analysis 3 (4):353–66.[Crossref], [PubMed] , [Google Scholar]). It is observed that the proposed test statistic is more powerful than the competing tests for the cases considered in this paper. A real life illustration is included.     

On high-dimensional tests for mutual independence based on Pearson’s correlation coefficient

Abstract

This paper investigates a class of statistics based on Pearson’s correlation coefficient for testing the mutual independence of a random vector in high dimensions. Two existing statistics, proposed by Schott (2005 Schott, J. R. 2005. Testing for complete independence in high dimensions. Biometrika 92 (4):951–6. doi:10.1093/biomet/92.4.951.[Crossref], [Web of Science ®] , [Google Scholar]) and Mao (2014 Mao, G. 2014. A new test of independence for high-dimensional data. Statistics & Probability Letters 93:14–8. doi:10.1016/j.spl.2014.05.024.[Crossref], [Web of Science ®] , [Google Scholar]) respectively, are special cases of the class. A generic testing theory for the class of statistics is developed, which clarifies under what conditions the class of statistics can be employed for the testing purpose. By virtue of the theory, three new tests are introduced, and related statistical properties are discussed. To examine our theoretical findings and check the performance of the new tests, simulation studies are applied. The simulation results justify the theoretical findings and show that the newly introduced tests perform well, as long as both the dimension and the sample size of the data are moderately large.     

Some improved tail bounds for the sum of variables with geometric distribution

Abstract

In this paper, compared with the results in Janson (2018 Janson, S. 2018. Tail bounds for sums of geometric and exponential variables. Statistics & Probability Letters 135 (C):1–6. doi:10.1016/j.spl.2017.11.017.[Crossref] , [Google Scholar]), we provide some improved explicit bounds for the tail probabilities of the sum of independent geometric variables with their expectations and variances. Particularly, in some cases, we demonstrate that our bounds are sharper than the ones in Janson (2018 Janson, S. 2018. Tail bounds for sums of geometric and exponential variables. Statistics & Probability Letters 135 (C):1–6. doi:10.1016/j.spl.2017.11.017.[Crossref] , [Google Scholar]).     

Optimal nonparametric estimator of the area under ROC curve based on clustered data

Abstract

In diagnostic trials, clustered data are obtained when several subunits of the same patient are observed. Intracluster correlations need to be taken into account when analyzing such clustered data. A nonparametric method has been proposed by Obuchowski (1997 Obuchowski, N. A. 1997. Nonparametric analysis of clustered ROC curve data. Biometrics 53 (2):567–78.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to estimate the Receiver Operating Characteristic curve area (AUC) for such clustered data. However, Obuchowski’s estimator is not efficient as it gives equal weight to all pairwise rankings within and between cluster. In this paper, we propose a more efficient nonparametric AUC estimator with two sets of optimal weights. Simulation results show that the loss of efficiency of Obuchowski’s estimator for a single AUC or the AUC difference can be substantial when there is a moderate intracluster test correlation and the cluster size is large. The efficiency gain of our weighted AUC estimator for a single AUC or the AUC difference is further illustrated using the data from a study of screening tests for neonatal hearing.     

Variance estimation in adaptive cluster sampling

Abstract

In this paper, a class of variance estimator is proposed of a finite population variance under an adaptive cluster sampling design in the presence of information on an auxiliary variable. We obtain expressions for the mean square error and bias for the developed estimators and their performance is evaluated on a Poisson clustered process and a real data set. The simulation study evaluates the efficiency of the suggested estimators for an adaptive cluster sampling (ACS) design and the Isaki (1983 Isaki, C. T. 1983. Variance estimation using auxiliary information. Journal of the American Statistical Association 78 (381):117–23. doi: 10.1080/01621459.1983.10477939.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator of the variance for SRSWOR over the sample variance for SRSWOR.     

Quantile estimation for a progressively censored exponential distribution

Abstract

In this paper, we consider the problem of estimating the quantile of a two-parameter exponential distribution with respect to an arbitrary strictly convex loss function under progressive type II censoring. Inadmissibility of the best affine equivariant (BAE) estimator is established through a conditional risk analysis. In particular we provide dominance results for quadratic, linex and absolute value loss functions. Further, a class of dominating estimators is derived using the IERD (integral expression of risk difference) approach of Kubokawa (1994 Kubokawa, T. 1994. A unified approach to improving equivariant estimators. The Annals of Statistics 22 (1):290–9. doi:10.1214/aos/1176325369.[Crossref], [Web of Science ®] , [Google Scholar]). In sequel the generalized Bayes estimator is shown to improve the BAE estimator.     

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.     

Log-epsilon-skew normal: A generalization of the log-normal distribution

Abstract

The log-normal distribution is widely used to model non-negative data in many areas of applied research. In this paper, we introduce and study a family of distributions with non-negative reals as support and termed the log-epsilon-skew normal (LESN) which includes the log-normal distributions as a special case. It is related to the epsilon-skew normal developed in Mudholkar and Hutson (2000 Mudholkar, G. S., and A. D. Hutson. 2000. The epsilon-skew-normal distribution for analyzing near-normal data. Journal of Statistical Planning and Inference 83 (2):291–309. doi:10.1016/S0378-3758(99)00096-8.[Crossref], [Web of Science ®] , [Google Scholar]) the way the log-normal is related to the normal distribution. We study its main properties, hazard function, moments, skewness and kurtosis coefficients, and discuss maximum likelihood estimation of model parameters. We summarize the results of a simulation study to examine the behavior of the maximum likelihood estimates, and we illustrate the maximum likelihood estimation of the LESN distribution parameters to two real world data sets.     

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.     

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.     

Discriminant coordinates analysis for multivariate functional data

Abstract

One of the basic statistical methods of dimensionality reduction is analysis of discriminant coordinates given by Fisher (1936 Fisher, R. A. 1936. The use of multiple measurements in taxonomic problem. Annals of Eugenics 7 (2):179–88. doi:10.1111/j.1469-1809.1936.tb02137.x.[Crossref] , [Google Scholar]) and Rao (1948). The space of discriminant coordinates is a space convenient for presenting multidimensional data originating from multiple groups and for the use of various classification methods (methods of discriminant analysis). In the present paper, we adapt the classical discriminant coordinates analysis to multivariate functional data. The theory has been applied to analysis of textural properties of apples of six varieties, measured over a period of 180?days, stored in two types of refrigeration chamber.     

Elementary renewal theorems for widely dependent random variables with applications to precise large deviations

Abstract

On the basis of Wang and Cheng (J. Math. Anal. Appl. 384 (2011) 597–606), this paper further investigates elementary renewal theorems for counting processes generated by random walks with widely orthant dependent increments. The obtained results improve the corresponding ones of the above-mentioned paper mainly in the sense of weakening the moment conditions on the positive parts of the increments. Meanwhile, a revised version of strong law of large numbers for random walks with widely orthant dependent increments is established, which improves Theorem 1.4 of Wang and Cheng (2011 Wang, Y., and D. Cheng. 2011. Basic renewal theorems for a random walk with widely dependent increments and their applications. Journal of Mathematical Analysis and Applications 384 (2):597–606. doi:10.1016/j.jmaa.2011.06.010.[Crossref], [Web of Science ®] , [Google Scholar]) by enlarging the regions of dominating coefficients. Finally, by using the above results, some precise large deviation results for a nonstandard renewal risk model are established, in which the innovations are widely orthant dependent random variables with common heavy tails, and the inter-arrival times are also widely orthant dependent.     

Some imputation methods to compensate with non-response for estimation of population mean in two-occasion successive sampling

Abstract

Non-response is an unavoidable phenomenon in almost all sample survey, and if it is not dealt carefully at the very starting of the analysis, results may lead to significant biases in the survey estimates. Keeping these facts in mind and inspired by the work of Singh (2009 Singh, S. 2009. A new method of imputation in survey sampling. Statistics 43 (5):499–511. doi:10.1080/02331880802605114.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), this article suggests some alternative efficient methods to deal with the missing data problems at the beginning of the analysis and proposes estimation procedures of the current population mean in two-occasion successive sampling. The properties of the resultant estimation procedures have been examined and supplemented with empirical studies. Results have been critically analyzed, and recommendations are made to the survey practitioners.     

Asymptotic Properties of Conditional Quantile Estimator Under Left-Truncated and α-Mixing Conditions

In this article, we establish strong uniform convergence and asymptotic normality of estimators of conditional quantile and conditional distribution function for a left truncated model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The results of Lemdani et al. (2009 Lemdani , M. , Ould-Saïd , E. , Poulin , N. ( 2009 ). Asymptotic properties of a conditional quantile estimator with randomly truncated data . J. Multivariate Anal. 100 : 546 – 559 .[Crossref], [Web of Science ®] , [Google Scholar]) are relaxed from the i.i.d. assumption to α-mixing setting. Finite sample behavior of the estimators is investigated via simulations as well.     

Combining answers to direct and indirect questions: An implementation of Kuk’s randomized response model

Abstract

In this article, we propose a two folded approach for estimation of the population proportion of a sensitive attribute. The rationale of proposed technique is to give respondents choice if they want to avail randomized device or not. Thus, our technique is inherently capable of entertaining the responses attained through direct questioning and by employing randomization device, as well. We believe that free to choose policy will be helpful to develop a curtsy between interviewer and respondent (which is highly desirable, especially when sensitive issue is under consideration) and thus enhances the survey reliability. The proposed technique is developed for simple random sampling, at first, and then extended to stratified random sampling. The superior performance of proposed technique in comparison to the Kuk (1990 Kuk, A. Y. 1990. Asking sensitive questions indirectly. Biometrika 77 (2):436–8. doi:10.1093/biomet/77.2.436.[Crossref], [Web of Science ®] , [Google Scholar]) method is demonstrated throughout this article. We provide algebraic, graphical and numerical evidences in support of our proposed technique. Furthermore, we also offer cost analysis considering most commonly used cost functions in the literature of survey research.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

The exponential distribution analog of the Grubbs–Weaver method

Abstract

Grubbs and Weaver (1947 Grubbs, F. E., and C. L. Weaver. 1947. The best unbiased estimate of population standard deviation based on group ranges. Journal of the American Statistical Association 42 (238):224–41. doi: 10.2307/2280652.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically for samples of size up to 100, and conjectured that their “rule of eights” is valid for all sample sizes. Here we examine the analogous problem where the underlying distribution is exponential and find that a “rule of fours” yields optimality and prove the result rigorously.     

Survival analysis for a new compounded bivariate failure time distribution in shock and competing risk models via an EM algorithm

Abstract

In this paper, two bivariate models based on the proposed methods of Marshall and Olkin are introduced. In the first model, the new bivariate distribution is presented based on the proposed method of Marshall and Olkin (1967 Marshall, A. W., and I. Olkin. 1967. A multivariate exponential distribution. Journal of the American Statistical Association 62 (317):30–44. doi: 10.2307/2282907.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) which has natural interpretations, and it can be applied in fatal shock models or in competing risks models. In the second model, the proposed method of Marshall and Olkin (1997 Marshall, A. W., and I. Olkin. 1997. A new method of adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika 84 (3):641–52. doi: 10.1093/biomet/84.3.641.[Crossref], [Web of Science ®] , [Google Scholar]) is generalized to bivariate case and a new bivariate distribution is introduced. We call these new distributions as the bivariate Gompertz (BGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the BGP model can be obtained as a special case of the BGPG model. Then, we present various properties of the new bivariate models. In this regard, the joint and conditional density functions, the joint cumulative distribution function can be obtained in compact forms. Also, the aging properties and the bivariate hazard gradient are discussed. This model has five unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters, and it is computationally quite tractable. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze three real data sets for illustrative purposes.