Simultaneous decomposition of the variance by sources and subgroups – new insights

There are two types of decompositions: of linear combinations of random variables into contributions of individual variables (sources) and associations between them, and of populations into contributions of their subpopulations. Simultaneous treatment of the two types is called for, which takes into account the correlations between sources within subpopulations and between subpopulation means. The expected values of the subcomponents are derived and their sensitivity to correlations among sources within groups and among source group means is conducted. An example is provided, in which the correlations contribute 20--25% to total variability. This additional information is hidden when decompositions are not simultaneous.     

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.     

Fourier–type tests involving martingale difference processes

We develop testing procedures which detect if the observed time series is a martingale difference sequence. Furthermore, tests are developed that detect change–points in the conditional expectation of the series given its past. The test statistics are formulated following the approach of Fourier–type conditional expectations first proposed by Bierens (1982 Bierens, H. J. (1982). Consistent model speci?cation tests. J. Econometr. 20:105–134.[Crossref], [Web of Science ®] , [Google Scholar]) and have the advantage of computational simplicity. The limit behavior of the test statistics is investigated under the null hypothesis as well as under alternatives. Since the asymptotic null distribution contains unknown parameters, a bootstrap procedure is proposed in order to actually perform the test. The performance of the bootstrap version of the test is compared in finite samples with other methods for the same problem. A real–data application is also included.     

Detecting structural change with heteroskedasticity

The hypothesis of structural stability that the regression coefficients do not change over time is central to all applications of linear regression models. It is rather surprising that existing theory as well as practice focus on testing for structural change under homoskedasticity – that is, regression coefficients may change, but the variances remain the same. Since structural change can, and often does, involve changes in variances, this is a puzzling gap in the literature. Our main focus in this paper is to utilize a newly developed test (MZ) by Maasoumi et al. (2010 Maasoumi, E., Zaman, A., Ahmed, M. (2010). Tests for structural change, aggregation, and homogeneity. Econ Model. 27(6):1382–1391.[Crossref], [Web of Science ®] , [Google Scholar]) that tests simultaneously for break in regression coefficients as well as in variance. Currently, the sup F test is most widely used for structural change. This has certain optimality properties shown by Andrews (1993 Andrews, D.W.K. (1993). Test for parameter instability and structural change with unknown change point. Econometrica. 61(4):821–856.[Crossref], [Web of Science ®] , [Google Scholar]). However, this test assumes homoskedasticity across the structural change. We introduce the sup MZ test which caters to unknown breakpoints, and also compare it to the sup F. Our Monte Carlo results show that sup MZ test incurs only a low cost in case of homoskedasticity while having hugely better performance in case of heteroskedasticity. The simulation results are further supported by providing a real-world application. In real-world datasets, we find that structural change often involves heteroskedasticity. In such cases, the sup F test can fail to detect structural breaks and give misleading results, while the sup MZ test works well. We conclude that the sup MZ test is superior to current methodology for detecting structural change.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

Statistical analysis of dependent competing risks model in accelerated life testing under progressively hybrid censoring using copula function

This article considers statistical analysis of dependent competing risks model from Weibull distribution in accelerated life testing, in which copula function is used to examine the dependence structure between competing failure modes. We derive the maximum likelihood estimates, the approximate, and Bootstrap confidence intervals of the parameters. The effects of different dependence structures on the estimates of parameters are investigated. The simulation is given to compare the performance of the estimates when the competing failure modes are dependent with those when the failure modes are independent. Finally, one dataset was used for illustrative purpose in conclusion.     

Bootstrap approach to test the homogeneity of order restricted mean vectors when the covariance matrices are unknown

Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are unknown, arbitrary positive definite and unequal are considered. This problem of testing has been studied to some extent, for example, by Kulatunga and Sasabuchi (1984 Kulatunga, D. D. S., Sasabuchi, S. (1984). A test of homogeneity of mean vectors against multivariate isotonic alternatives. Mem Fac Sci, Kyushu Univ Ser A Mathemat 38:151–161. [Google Scholar]) when the covariance matrices are known and also Sasabuchi et al. (2003 Sasabuchi, S., Tanaka, K., Tsukamodo, T. (2003). Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown. Annals of Statistics. 31(5):1517–1536.[Web of Science ®] , [Google Scholar]) and Sasabuchi (2007 Sasabuchi, S. (2007). More powerful tests for homogeneity of multivariate normal mean vectors under an order restriction. Sankhya 69(4):700–716. [Google Scholar]) when the covariance matrices are unknown but common. In this paper, a test statistic is proposed and because of the main advantage of the bootstrap test is that it avoids the derivation of the complex null distribution analytically, a bootstrap test statistic is derived and since the proposed test statistic is location invariance the bootstrap p-value defined logical and some steps are presented to estimate it. Our numerical studies via Monte Carlo simulation show that the proposed bootstrap test can correctly control the type I error rates. The power of the test for some of the p-dimensional normal distributions is computed by Monte Carlo simulation. Also, the null distribution of test statistic is estimated using kernel density. Finally, the bootstrap test is illustrated using a real data.     

Parametric bootstrap inferences for the growth curve models with intraclass correlation structure

This paper presents parametric bootstrap (PB) approaches for hypothesis testing and interval estimation of the fixed effects and the variance component in the growth curve models with intraclass correlation structure. The PB pivot variables are proposed based on the sufficient statistics of the parameters. Some simulation results are presented to compare the performance of the proposed approaches with the generalized inferences. Our studies show that the PB approaches perform satisfactorily for various cell sizes and parameter configurations, and tends to outperform the generalized inferences with respect to the coverage probabilities and powers. The PB approaches not only have almost exact coverage probabilities and Type I error rates, but also have the shorter expected lengths and the higher powers. Furthermore, the PB procedure can be simply carried out by a few simulation steps. Finally, the proposed approaches are illustrated by using a real data example.     

Theory and methods for partitioned Gini coefficients computed on post-stratified data

The Gini coefficient is used to measure inequality in populations. However, shifts in the population distribution may affect subgroups differently. Consequently, it can be informative to examine inequality separately for these segments. Consider an independently and identically distributed sample split based on ranking and compute the Gini coefficient for each partition. These coefficients, calculated from post-stratified data, are not functions of U-statistics. Therefore, previous theoretical and methodological results cannot be applied. In this article, the asymptotic joint distribution is derived for the partitioned coefficients and bootstrap methods for inference are developed. Finally, an application to per capita income across census tracts is examined.     

A Unified Approach to Estimating and Testing Income Distributions With Grouped Data

We propose a unified approach that is flexibly applicable to various types of grouped data for estimating and testing parametric income distributions. To simplify the use of our approach, we also provide a parametric bootstrap method and show its asymptotic validity. We also compare this approach with existing methods for grouped income data, and assess their finite-sample performance by a Monte Carlo simulation. For empirical demonstrations, we apply our approach to recovering China's income/consumption distributions from a sequence of income/consumption share tables and the U.S. income distributions from a combination of income shares and sample quantiles. Supplementary materials for this article are available online.