Testing the Equality and Exchangeability of Bivariate Distributions

We consider three methods (oments, cut-points, and ranks) for testing the hypotheses of equality of two bivariate distribution functions (H _0a ) and exchangeability (H _0b ). To test H _0a , the asymptotic normality of the vector of mixed moments provides a statistic with an asymptotic chi-square distribution. With every observation, method of cut-points associates three 2 × 2 tables to record the proportions of the X, Y, and the combined samples that fall in the four regions around the observation. We measure the total squared deviations of the proportions in the combined sample from X and Y samples. The two methods are compared with the method of ranks based on the Puri and Sen (1971 Puri , M. L. , Sen , P. K. ( 1971 ). Nonparametric Methods in Multivariate Analysis . New York : John Wiley and Sons . [Google Scholar]) multivariate two-sample rank test for location.

To test H _0b we identify two bivariate distributions, one above and the other below the line of symmetry X = Y, to which a test of H _0a is applied. Under H _0b , matrix of mixed moments is symmetric and a quadratic form in differences of (r,s)-th and (s, r)-th mixed moments provides an asymptotic chi-square distribution. A permutation test is devised to apply the method of cut-points to the observations above and below the line of symmetry after they are folded. We also describe an adaption of the Puri-Sen rank test to assess H _0b . To estimate the power of the above methods under different types of alternatives and compare them to existing tests, we report on a Monte Carlo experiment that evaluates the finite-sample performance of these methods under the Plackett's family of bivariate distributions.     

Testing the difference between two independent regression models

ABSTRACT

In some situations, for example, in biology or psychology studies, we wish to determine whether the linear relationship between response variable and predictor variables differs in two populations. The analysis of the covariance (ANCOVA) or, equivalently, the partial F-test approaches are the commonly used methods. In this study, the asymptotic distribution for the difference between two independent regression coefficients was established. The proposed method was used to derive the asymptotic confidence set for the difference between coefficients and hypothesis testing for the equality of the two regression models. Then a simulation study was conducted to compare the proposed method with the partial F method. The performance of the new method was comparable with that of the partial F method.     

An objective look at obtaining the plotting positions for QQ-plots

ABSTRACT

Choosing the plotting positions for the QQ-plot has been a subject of much debate in the statistical and engineering literature. This paper looks at this problem objectively by considering three frameworks: distribution-theoretic; decision-theoretic; game-theoretic. In each framework, we derive the plotting positions and show that there are more than one legitimate solution depending on the practitioner’s objective. This work clarifies the choice of the plotting positions by allowing one to easily find the mathematical equivalent of their view and choose the corresponding solution. This work also discusses approximations to the plotting positions when no closed form is available.     

Geostatistical analysis of disease data: a case study of tuberculosis incidence in Iran

The main objective of this study is to introduce two advanced statistical methods and to consider geographical distribution of tuberculosis incidence in Iran. With the knowledge that environmental and climatic conditions in each region are affective for the incidence and spread of the disease, the study has been taken into consideration. The disease incidences in different counties are realizations of spatial data, therefore we apply the Poisson kriging and ordinary kriging for prediction of tuberculosis incidence rates map in Iran. To identify high risk areas using statistical map of disease, our results show that tuberculosis incidences are not uniformly distributed in whole of the country and estimated risk is high in the eastern parts. Assessing geographical distribution of a disease is essential for health officials to recognize high-risk areas, and improve case management and resource allocation.     

Majorization and the spectral radius of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by \(\lambda (G)\), is the largest eigenvalue of G. Let k and \(n_1,\ldots ,n_k\) be some positive integers. Let \(T(n_1,\ldots ,n_k)\) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1-1},\ldots ,P_{n_k-1}\) where every neighbor of v in T has degree one or two. Let \(P=(p_1,\ldots ,p_k)\) and \(Q=(q_1,\ldots ,q_k)\), where \(p_1\ge \cdots \ge p_k\ge 1\) and \(q_1\ge \cdots \ge q_k\ge 1\) are integer. We say P majorizes Q and let \(P\succeq _M Q\), if for every j, \(1\le j\le k\), \(\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i\), with equality if \(j=k\). In this paper we show that if P majorizes Q, that is \((p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)\), then \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).     

A comparison of parametric and semi-parametric survival models with artificial neural networks

Survival models are used to examine data in the event of an occurrence. These are discussed in various types including parametric, non-parametric and semi-parametric models. Parametric models require a clear distribution of survival time, and semi-parametric models assume proportional hazards. Among these models, the non-parametric model of artificial neural network has the fewest assumptions and can be often replaced by other models. Given the importance of distribution Weibull survival models in this study of simulation shape parameter of the Weibull distribution have been assumed as 1, 2 and 3, and also the average rate at levels of 0%–75% have been censored. The values predicted by the neural network forecasting model with parametric survival and Cox regression models were compared. This comparison considering levels of complexity due to the hazard model using the ROC curve and the corresponding tests have been carried out.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Three-level control charts with variable sample size,sampling interval,and control limits

Recently, three-level control chart has been developed for monitoring situations in which an appropriate quality measure uses three discrete levels to classify a product characteristic. In this paper, the variable parameters control charts for multinomial data are developed with a three-level classification scheme. We compare it with various adaptive charts to show that this modified scheme can improve the performance. In order to evaluate and compare the performance of this scheme, adjusted average time to signal is used as the performance measure. Results indicate that the proposed chart has improved performance and is relatively sensitive to small shifts.     

Application of Different Techniques of Factor Analysis to Develop Neighborhood Social Capital Rating Scale

Composite indicators are widely used to determine the ranking of countries, organizations or individuals in terms of overall performance on multiple criteria. Their calculation requires standardization of the individual statistical criteria and aggregation of the standardized indicators. These operations introduce a potential propagation effect of extreme values on the calculation of the composite indicator of all entities. In this paper, we propose robust composite indicators for which this propagation effect is limited. The approach uses winsorization based on a robust estimate of the distribution of the sub-indicators. It is designed such that the winsorization affects only the composite indicator rank but has no effect on the entities ranking in each sub-indicator. The simulation study documents the benefits of distribution-based winsorization in the presence of outliers. It leads to a ranking that is closer to the clean data ranking when compared to the ranking obtained using either no winsorization or the traditional winsorization based on empirical quantiles. In the empirical application, we illustrate the use of winsorization for ranking countries based on the United Nations Industrial Development Organization’s Competitive Industrial Performance index. We show that even though the sub-indicator ranking does not change, the robust winsorization approach has a material impact on the ranking of the composite indicator for countries with large discrepancies in the scores of the sub-indicators.