A MONTE CARLO COMPARISON OF VARIOUS ASYMPTOTIC APPROXIMATIONS TO THE DISTRIBUTION OF INSTRUMENTAL VARIABLES ESTIMATORS

ABSTRACT

We examine empirical relevance of three alternative asymptotic approximations to the distribution of instrumental variables estimators by Monte Carlo experiments. We find that conventional asymptotics provides a reasonable approximation to the actual distribution of instrumental variables estimators when the sample size is reasonably large. For most sample sizes, we find Bekker[11] Bekker, P. A. 1994. Alternative Approximations to the Distributions of Instrumental Variable Estimators. Econometrica, 62: 657–681. [Crossref], [Web of Science ®] , [Google Scholar] asymptotics provides reasonably good approximation even when the first stage R ² is very small. We conclude that reporting Bekker[11] Bekker, P. A. 1994. Alternative Approximations to the Distributions of Instrumental Variable Estimators. Econometrica, 62: 657–681. [Crossref], [Web of Science ®] , [Google Scholar] confidence interval would suffice for most microeconometric (cross-sectional) applications, and the comparative advantage of Staiger and Stock[5] Staiger, D. and Stock, J. H. 1997. Instrumental Variables Regression with Weak Instruments. Econometrica, 65: 556–586. [Crossref], [Web of Science ®] , [Google Scholar] asymptotic approximation is in applications with sample sizes typical in macroeconometric (time series) applications.     

Quantile-based Chernoff distance for truncated random variables

Several probability distributions such as power-Pareto distribution (see Gilchrist 2000 Gilchrist, W. 2000. Statistical modelling with quantile functions. Boca Raton, FL: Chapman and Hall/CRC.[Crossref] , [Google Scholar] and Hankin and Lee 2006 Hankin, R. K. S., and A. Lee. 2006. A new family of non-negative distributions. Australian and New Zealand Journal of Statistics 48:67–78.[Crossref], [Web of Science ®] , [Google Scholar]), various forms of lambda distributions (see Ramberg and Schmeiser 1974 Ramberg, J. S., and B. W. Schmeiser. 1974. An appropriate method for generating asymmetric random variables. Communications of the ACM 17:78–82.[Crossref], [Web of Science ®] , [Google Scholar] and Freimer et al. 1988 Freimer, M., S. Mudholkar, G. Kollia, and C. T. Lin. 1988. A study of the generalized lambda family. Communications in Statistics - Theory and Methods 17:3547–67.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Govindarajulu distribution (see Nair, Sankaran, and Vineshkumar 2012 Nair, U. N., P. G. Sankaran, and B. Vineshkumar. 2012. The Govindarajulu distribution: some properties and applications. Communications in Statistics—Theory and Methods 41:4391–406.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), etc., do not have manageable distribution functions, though they have tractable quantile functions. Hence, analytical study of the properties of Chernoff distance of two random variables associated with these distributions via traditional distribution function-based tool becomes difficult. To make this simple, in this paper, we introduce quantile-based Chernoff distance for (left or right) truncated random variables and study its various properties. Some useful bounds as well as characterization results are obtained.     

Modeling bivariate survival data using shared inverse Gaussian frailty model

ABSTRACT

The shared frailty models are often used to model heterogeneity in survival analysis. The most common shared frailty model is a model in which hazard function is a product of a random factor (frailty) and the baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and the distribution of frailty. In this paper, we consider inverse Gaussian distribution as frailty distribution and three different baseline distributions, namely the generalized Rayleigh, the weighted exponential, and the extended Weibull distributions. With these three baseline distributions, we propose three different inverse Gaussian shared frailty models. We also compare these models with the models where the above-mentioned distributions are considered without frailty. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these three baseline distributions with a shared inverse Gaussian frailty so far. We also apply these three models by using a real-life bivariate survival data set of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data and a better model is suggested for the data using the Bayesian model selection criteria.     

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010 Lin, G.D., Huang, J.S. (2010). A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals. J. Multivariate Anal. 101: 22272233.[Crossref], [Web of Science ®] , [Google Scholar]) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013 Bairamov, I., Bayramoglu, K. (2013). From Huang-Kotz distribution to Baker’s distribution. J. Multivariate Anal. 113: 106115.[Crossref], [Web of Science ®] , [Google Scholar]) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.     

Adaptive Procedures for the Wilcoxon–Mann–Whitney Test: Seven Decades of Advances

The Wilcoxon–Mann–Whitney test has dominated non parametric analyses in behavioral sciences for the past seven decades. Its widespread use masks the fact that there exist simple “adaptive” procedures which use data-dependent statistical decision rules to select an optimal non parametric test. This paper discusses key adaptive approaches for testing differences in locations in two-sample environments. Our Monte Carlo analysis shows that adaptive procedures often perform substantially better than t-tests, even with moderately sized samples (80 observations). We illustrate adaptive approaches using data from Gneezy and Smorodinsky (2006 Gneezy, U., Smorodinsky, R. (2006). All-pay auctions: an experimental study. J. Economic Behav. Organizat. 61(2): 255275.[Crossref], [Web of Science ®] , [Google Scholar]), and offer a Stata package to researchers interested in taking advantage of these techniques.     

Generalized linear regression models incorporating original outcome distributions

ABSTRACT

In this article, we propose an approach for incorporating continuous and discrete original outcome distributions into the usual exponential family regression models. The new approach is an extension of the works of Suissa (1991 Suissa, S. (1991). Binary methods for continuous outcomes: A parametric alternative. J. Clin. Epidemiol. 44:241–248.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Suissa and Blais (1995 Suissa, S., Blais, L. (1995). Binary regression with continuous outcomes. Stat. Med. 14:247–255.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which present methods to estimate the risk of an event defined in a sample subspace of an original continuous outcome variable. Simulation studies are presented in order to illustrate the performance of the developed methodology. Real data sets are analyzed by using the proposed models.     

On General Continuous Triangular and Two-sided Power Distributions

In this article, we directly introduce the continuous version of the general discrete triangular distributions (Kokonendji and Zocchi, 2010 Kokonendji, C.C., Zocchi, S.S. (2010). Extensions of discrete triangular distribution and boundary bias in kernel estimation for discrete functions. Statist. Probab. Lett. 80:1655–1662.[Crossref], [Web of Science ®] , [Google Scholar]). It is bounded and, in general, unimodal with pike. It contains thus a very useful class of two-sided power distributions (van Dorp and Kotz, 2002a Van Dorp, J.R., Kotz, S. (2002a). A novel extension of the triangular distribution and its parameter estimation. Statistician 51:1–17. [Google Scholar],b Van Dorp, J.R., Kotz, S. (2002b). The standard two-sided power distribution and its properties; with applications in financial engineering. Amer. Statistician 56:90–99.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2003 Van Dorp, J.R., Kotz, S. (2003). Generalization of two-sided power distributions and their convolution. Commun. Statist. Theor. Meth. 32:1703–1723.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Moments, particular cases, limit distributions, and relations between parameters are straightforwardly derived.     

ESTIMATION OF PARAMETERS IN ARFIMA PROCESSES: A SIMULATION STUDY

It is known that, in the presence of short memory components, the estimation of the fractional parameter d in an Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q), process has some difficulties (see [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar]). In this paper, we continue the efforts made by Smith et al. [1] Smith, J., Taylor, N. and Yadav, S. 1997. Comparing the bias and misspecification in ARFIMA models. Journal of Time Series Analysis, 18(5): 507–527. [Crossref] , [Google Scholar] and Beveridge and Oickle [2] Beveridge, S. and Oickle, C. 1993. Estimating fractionally integrated time series models. Economics Letters, 43: 137–142.  [Google Scholar] by conducting a simulation study to evaluate the convergence properties of the iterative estimation procedure suggested by Hosking [3] Hosking, J. 1981. Fractional differencing. Biometrika, 68(1): 165–176. [Crossref], [Web of Science ®] , [Google Scholar]. In this context we consider some semiparametric approaches and a parametric method proposed by Fox-Taqqu[4] Fox, R. and Taqqu, M. S. 1986. Large-sample properties of parameter estimates for strongly dependent stationary gaussian time series. The Annals of Statistics, 14(2): 517–532. [Crossref], [Web of Science ®] , [Google Scholar]. We also investigate the method proposed by Robinson [5] Robinson, P. M. 1995a. Log-periodogram regression of time series with long range dependence. The Annals of Statistics, 23(3): 1048–1072. [Crossref], [Web of Science ®] , [Google Scholar] and a modification using the smoothed periodogram function.     

Estimation for exponential distribution based on competing risk middle censored data

Abstract

For a general censoring scheme called “middle censoring” scheme which was proposed by Jammalamadaka and Mangalam (2003 Jammalamadaka, S.R., Mangalam, V. (2003). Nonparametric estimation for middle censored data. J. Nonparametric Statist. 15:253–265.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in nonparametric set up. In this article, point and interval estimation problems are considered for the exponential distribution when the failure data is middle censored with two independent competing failure risks. Different methods are introduced to estimate the unknown model parameters such as maximum likelihood estimation, midpoint approximation, equivalent quantities estimation. The Bayesian estimation is also considered with gamma priors. Two numerical examples are analyzed to show the performance of the proposed methods.     

Credibility premium for rate-making systems

Credibility formula has been developed in many fields of actuarial sciences. Based upon Payandeh (2010 Payandeh, A.T. (2010). A new approach to the credibility formula. Insur.: Math. Econ. 46(2):334–338.[Crossref], [Web of Science ®] , [Google Scholar]), this article extends concept of credibility formula to relatively premium of a given rate-making system. More precisely, it calculates Payandeh’s (2010 Payandeh, A.T. (2010). A new approach to the credibility formula. Insur.: Math. Econ. 46(2):334–338.[Crossref], [Web of Science ®] , [Google Scholar]) credibility factor for zero-inflated Poisson gamma distributions with respect to several loss functions. A comparison study has been given.     

The McDonald Gumbel model

ABSTRACT

We propose a new model called the McDonald Gumbel distribution, the major advantage of which is its ability to fit asymmetric real data that can not be properly adjusted by existing distributions. This model contains as special models the Gumbel, exponentiated Gumbel (Persson and Rydén, 2010 Persson, K., Rydén, J. (2010). Exponentiated Gumbel distribution for estimation of return levels of significant wave height. J. Environ. Statist. 1:1–12. [Google Scholar]), beta Gumbel (Nadarajah and Kotz, 2004 Nadarajah, S., Kotz, S. (2004). The beta Gumbel distribution. Mathemat. Prob. Eng.323–332.[Crossref], [Web of Science ®] , [Google Scholar]), Kumaraswamy Gumbel distributions, among others. We obtain the ordinary moments, quantile and generating functions and mean deviations. The method of maximum likelihood is used to fit the proposed distribution. The applicability of the new model is illustrated by means of two real data sets.     

Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

Inference in log-alpha-power and log-skew-normal multivariate models

ABSTRACT

Random vectors with positive components are common in many applied fields, for example, in meteorology, when daily precipitation is measured through a region Marchenko and Genton (2010 Marchenko, Y., Genton, M. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21:318–340.[Crossref], [Web of Science ®] , [Google Scholar]). Frequently, the log-normal multivariate distribution is used for modeling this type of data. This modeling approach is not appropriate for data with high asymmetry or kurtosis. Consequently, more flexible multivariate distributions than the log-normal multivariate are required. As an alternative to this distribution, we propose the log-alpha-power multivariate and log-skew-normal multivariate models. The first model is an extension for positive data of the fractional order statistics model Durrans (1992 Durrans, S. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]). The second one is an extension of the log-skew-normal model studied by Mateu-Figueras and Pawlowsky-Glahn (2007 Mateu-Figueras, G., Pawlowsky-Glahn, V. (2007). The skew-normal distribution on the simplex. Commun. Stat.-Theory Methods 36:1787–1802.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We study parameter estimation for these models by means of pseudo-likelihood and maximum likelihood methods. We illustrate the proposal analyzing a real dataset.     

POOLED SLICING METHODS VERSUS SLICING METHODS

We consider the semiparametric regression model introduced by [1] Duan, N. and Li, K. C. 1991. Slicing regression: a link-free regression method. The Annals of Statistics, 19: 505–530. [Crossref], [Web of Science ®] , [Google Scholar]. The dependent variable y is linked to the index x′ β through an unknown link function. [1] Duan, N. and Li, K. C. 1991. Slicing regression: a link-free regression method. The Annals of Statistics, 19: 505–530. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Li, K. C. 1991. Sliced inverse regression for dimension reduction, with discussions. Journal of the American Statistical Association, 86: 316–342. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] present Slicing methods (the Sliced Inverse Regression methods SIR-I, SIR-II and SIR_α) in order to estimate the direction of the unknown slope parameter β. These methods are computationally simple and fast but depend on the choice of an arbitrary slicing fixed by the user. When the sample size is small, the number and the position of slices have an influence on the estimated direction. In this paper, we suggest to use the corresponding Pooled Slicing methods: PSIR-I (proposed by [3] Aragon, Y. and Saracco, J. 1997. Sliced Inverse Regression (SIR): an appraisal of small sample alternatives to slicing. Computational Statistics, 12: 109–130. [Web of Science ®] , [Google Scholar]), PSIR-II and PSIR_α. These methods combine the results from a number of slicings. We compare the sample behaviour of Slicing and Pooled Slicing methods on simulations. We also propose a practical choice of α in SIR_α and PSIR_α methods.     

EVALUATION OF POWER OF TESTS INVOLVING COVARIATES UNDER NONCONSTANT VARIANCE

In many experiments where pre-treatment and post-treatment measurements are taken, investigators wish to determine if there is a difference between two treatment groups. For this type of data, the post-treatment variable is used as the primary comparison variable and the pre-treatment variable is used as a covariate. Although most of the discussion in this paper is written with the pre-treatment variable as the covariate the results are applicable to other choices of a covariate. Tests based on residuals have been proposed as alternatives to the usual covariance methods. Our objective is to investigate how the powers of these tests are affected when the conditional variance of the post-treatment variable depends on the magnitude of the pre-treatment variable. In particular, we investigate two cases. [1] Crager, Michael R. 1987. Analysis of Covariance in Parallel-Group Clinical Trials With Pretreatment Baselines. Biometrics, 43: 895–901. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable gradually increases as the magnitude of the pre-treatment variable increases. (In many biological models this is the case.) [2] Knoke, James D. 1991. Nonparametric Analysis of Covariance for Comparing Change in Randomized Studies with Baseline Values Subject to Error. Biometrics, 47: 523–533. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar] The conditional variance of the post-treatment variable is dependent upon natural or imposed subgroups contained within the pre-treatment variable. Power comparisons are made using Monte Carlo techniques.     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.     

More on the two-parameter estimation in the restricted regression

ABSTRACT

In this paper, we introduce a new restricted two-parameter (RTP) estimator for the vector of parameters in a linear model when additional linear restrictions on the parameter vector are assumed to hold. We show that our new biased estimator is superior in the matrix mean square error criterion to the restricted ridge estimator proposed by Groß (2003 Groß, J. (2003). Restricted ridge estimation. Stat. Probab. Lett. 65:57–64.[Crossref], [Web of Science ®] , [Google Scholar]), restricted Liu estimator introduced by Kaçiranlar et al. (1999 Kaçiranlar, S., Sakall?oglus, S., Akdeniz, F., Styan, G.P.H., Werner, H.J. (1999). A new biased estimator in linear regression and a detailed analysis of the widely-analysed dataset on Portland cement. Sankhya Ser. B., Ind. J. Stat. 61:443–459. [Google Scholar]), and RTP estimator introduced by Özkale and Kaçiranlar (2007 Özkale, M., Kaçiranlar, S. (2007). The restricted and unrestricted two-parameter estimators. Commun. Stat. Theory Methods 36:2707–2725.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A numerical example and a Monte Carlo simulation have been analyzed to illustrate some of the theoretical results.     

Bootstrap test of hypothesis for the multi-state models in survival analysis

Abstract

In the present paper we develop bootstrap tests of hypothesis, based on simulation, for the transition probability matrix arising in the context of a multi-state model. The bootstrap test statistic is based on the paper of Tattar and Vaman (2008 Tattar, P. N., Vaman, H. J. (2008). Testing transition probability matrix of a multi-state model with censored data. Lifetime Data Anal. 14(2):216–230.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which develops a statistic for the testing problems concerning the transition probability matrix of the non homogeneous Markov process.     

An adaptive multivariate CUSUM control chart for signaling a range of location shifts

ABSTRACT

In this work, we proposed an adaptive multivariate cumulative sum (CUSUM) statistical process control chart for signaling a range of location shifts. This method was based on the multivariate CUSUM control chart proposed by Pignatiello and Runger (1990 Pignatiello, J.J., Runger, G.C. (1990). Comparisons of multivariate CUSUM charts. J. Qual. Technol. 22(3):173–186.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), but we adopted the adaptive approach similar to that discussed by Dai et al. (2011 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]), which was based on a different CUSUM method introduced by Crosier (1988 Crosier, R.B. (1988). Multivariate generalizations of cumulative sum quality-control schemes. Technometrics 30(3):291–303.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The reference value in this proposed procedure was changed adaptively in each run, with the current mean shift estimated by exponentially weighted moving average (EWMA) statistic. By specifying the minimal magnitude of the mean shift, our proposed control chart achieved a good overall performance for detecting a range of shifts rather than a single value. We compared our adaptive multivariate CUSUM method with that of Dai et al. (2001 Dai, Y., Luo, Y., Li, Z., Wang, Z. (2011). A new adaptive CUSUM control chart for detecting the multivariate process mean. Qual. Reliab. Eng. Int. 27(7):877–884.[Crossref], [Web of Science ®] , [Google Scholar]) and the non adaptive versions of these two methods, by evaluating both the steady state and zero state average run length (ARL) values. The detection efficiency of our method showed improvements over the comparative methods when the location shift is unknown but falls within an expected range.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.