Simultaneous Confidence Intervals for the Parameters of Pareto Distribution under Progressive Censoring

In this article, we consider the progressive Type II right censored sample from Pareto distribution. We introduce a new approach for constructing the simultaneous confidence interval of the unknown parameters of this distribution under progressive censoring. A Monte Carlo study is also presented for illustration. It is shown that this confidence region has a smaller area than that introduced by Ku? and Kaya (2007 Ku? , C. , Kaya , M. F. ( 2007 ). Estimation for the parameters of the Pareto distribution under progressive censoring . Commun. Statist. Theor. Meth. 36 : 1359 – 1365 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

A Note on Simultaneous Confidence Intervals for Ratio of Variances

An explicit formula for confidence intervals for ratios of variances of several populations is presented. The intervals are based on jackknife statistics and the critical point of the studentized range distribution. The asymptotic probability of coverage is not less than the nominal value provided that the distributions of the sampled populations belong to a location-scale family of probabilities with finite fourth moment.     

Optimal Simultaneous Confidence Bands in Multiple Linear Regression with Predictor Variables Constrained in an Ellipsoidal Region

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model function, just as a confidence interval gives the plausible range of an unknown parameter. For a multiple linear regression model, confidence bands of different shapes, such as the hyperbolic band and the constant width band, can be constructed and the predictor variable region over which a confidence band is constructed can take various forms. One interesting but unsolved problem is to find the optimal (shape) confidence band over an ellipsoidal region χ_E under the Minimum Volume Confidence Set (MVCS) criterion of Liu and Hayter (2007 Liu, W., Hayter, A.J. (2007). Minimum area confidence set optimality for confidence bands in simple linear regression. J. Amer. Statist. Assoc. 102:181–190.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Liu et al. (2009 Liu, W., Bretz, F., Hayter, A.J., Wynn, H.P. (2009). Assessing non-superiority, non-inferiority or equivalence when comparing two regression models over a restricted covariate region. Biometrics 65:1279–1287.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). This problem is challenging as it involves optimization over an unknown function that determines the shape of the confidence band over χ_E. As a step towards solving this difficult problem, in this paper, we introduce a family of confidence bands over χ_E, called the inner-hyperbolic bands, which includes the hyperbolic and constant-width bands as special cases. We then search for the optimal confidence band within this family under the MVCS criterion. The conclusion from this study is that the hyperbolic band is not optimal even within this family of inner-hyperbolic bands and so cannot be the overall optimal band. On the other hand, the constant width band can be optimal within the family of inner-hyperbolic bands when the region χ_E is small and so might be the overall optimal band.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

Amemiya‘s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

Testing Equality of Mean Vectors in Two Sample Problem with Missing Data

Finding the influence of traffic accident on the road is helpful to analyze the characteristics of traffic flow, and take reasonable and effective control measures. Here, the detrended fluctuation analysis method is applied to investigate the complexity of time series in mixed traffic flow with a blockage induced by an accident. As a parameter to depict the long-term evolutionary behavior of the time series in traffic flow, the scaling exponent is analyzed. According to the scaling exponent, it is shown that the traffic flow time series can display long-range correlation characteristics, short-range correlation characteristics, and non-power-law relation in the long-range correlation characteristics, which is strongly dependent on the entering probability of vehicle, the ratio of slow vehicle and the blockage duration time.     

Correlation-Type Goodness of Fit Test for Extreme Value Distribution Based on Simultaneous Closeness

In reliability studies, one typically would assume a lifetime distribution for the units under study and then carry out the required analysis. One popular choice for the lifetime distribution is the family of two-parameter Weibull distributions (with scale and shape parameters) which, through a logarithmic transformation, can be transformed to the family of two-parameter extreme value distributions (with location and scale parameters). In carrying out a parametric analysis of this type, it is highly desirable to be able to test the validity of such a model assumption. A basic tool that is useful for this purpose is a quantile–quantile (QQ) plot, but in its use, the issue of the choice of plotting position arises. Here, by adopting the optimal plotting points based on Pitman closeness criterion proposed recently by Balakrishnan et al. (2010b Balakrishnan , N. , Davies , K. F. , Keating , J. P. , Mason , R. L. ( 2010b ). Computation of optimal plotting points based on Pitman Closeness with an application to goodness of fit for location-scale families. Submitted to Computational Statistics & Data Analysis.  [Google Scholar]), and referred to as simultaneous closeness probability (SCP) plotting points, we propose a correlation-type goodness of fit test for the extreme value distribution. We compute the SCP plotting points for various sample sizes and use them to determine the mean, standard deviation and critical values for the proposed correlation-type test statistic. Using these critical values, we carry out a power study, similar to the one carried out by Kinnison (1989 Kinnison , R. ( 1989 ). Correlation coefficient goodness of fit test for extreme value distribution . The American Statistician 43 : 98 – 100 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), through which we demonstrate that the use of SCP plotting points results in better power than with the use of mean ranks as plotting points and nearly the same power as with the use of median ranks. We then demonstrate the use of the SCP plotting points and the associated correlation-type test for Weibull analysis with an illustrative example. Finally, for the sake of comparison, we also adapt two statistics proposed by Gan and Koehler (1990 Gan , F. F. , Koehler , K. J. ( 1990 ). Goodness of fit based on P-P probability plots . Technometrics 32 : 289 – 303 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), in the context of probability–probability (PP) plots, based on SCP plotting points and compare their performance to those based on mean ranks. The empirical study also reveals that the tests from the QQ plot have better power than those from the PP plot.     

Further Results on Order Statistics Generated by Two Simulation Methods

Uniform order statistics generated by two simulation methods are compared by means of Pitman’s measure of closeness. This measure, as a probability, is shown to be asymptotically 1/2. Some results are also established for fixed points of the cumulative distribution function (CDF) for a uniform order statistic. These fixed points are important for calculations involving the joint distribution of these order statistics.     

A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.