An objective Bayesian estimation in a two-period crossover design

Two-period crossover design is one of the commonly used designs in clinical trials. But, the estimation of treatment effect is complicated by the possible presence of carryover effect. It is known that ignoring the carryover effect when it exists can lead to poor estimates of the treatment effect. The classical approach by Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) consists of two stages. First, a preliminary test is conducted on carryover effect. If the carryover effect is significant, analysis is based only on data from period one; otherwise, analysis is based on data from both periods. A Bayesian approach with improper priors was proposed by Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) which uses a mixture of two models: a model with carryover effect and another without. The indeterminacy of the Bayes factor due to the arbitrary constant in the improper prior was addressed by assigning a minimally discriminatory value to the constant. In this article, we present an objective Bayesian estimation approach to the two-period crossover design which is also based on a mixture model, but using the commonly recommended Zellner–Siow g-prior. We provide simulation studies and a real data example and compare the numerical results with Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s and Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s approaches.     

On the Optimality of Circular Block Designs Under a Mixed Interference Model

Filipiak and Markiewicz (2012 Filipiak, K., Markiewicz, A. (2012). On universal optimality of circular weakly neighbor balanced designs under an interference model. Comm. Stat. Theor Methods 41: 2356–2366.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) proved the universal optimality of circular weakly neighbor balanced designs (CWNBDs) under the interference model with fixed neighbor effects among the class of complete block designs. In two special cases where a CWNBD cannot exist, Filipiak et al. (2012 Filipiak, K., Markiewicz, A., Ró?ański, R. (2012). Maximal determinant over a certain class of matrices and its application to D-optimality of designs. Linear Algebra Appl. 436(4): 874–887.[Crossref], [Web of Science ®] , [Google Scholar]) characterized D-optimal designs. The aim of this paper is to show the universal optimality of CWNBDs and to characterize D-optimal designs under the interference model with random neighbor effects.     

A Modification of a Percentile Estimation Procedure Based on Generalized Polya Urns

Adaptive designs find an important application in the estimation of unknown percentiles for an underlying dose-response curve. A nonparametric adaptive design was suggested by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to simultaneously estimate multiple percentiles of an unknown dose-response curve via generalized Polya urns. In this article, we examine the properties of the design proposed by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) when delays in observing responses are encountered. Using simulations, we evaluate a modification of the design under varying group sizes. Our results demonstrate unbiased estimation with minimal loss in efficiency when compared to the original compound urn design.     

Some extensions of circular balanced and circular strongly balanced repeated measurements designs

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this paper the works of Iqbal and Tahir (2009 Iqbal, I., and M. H. Tahir. 2009. Circular strongly balanced repeated measurements designs. Communications in Statistics—Theory and Methods 38:3686–96.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Iqbal, Tahir, and Ghazali (2010 Iqbal, I., M. H. Tahir, and S. S. A. Ghazali. 2010. Circular first- and second-order balanced repeated measurements designs. Communications in Statistics—Theory and Methods 39:228–40.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) are generalized for the construction of circular-balanced and circular strongly balanced repeated measurements designs through the method of cyclic shifts for three periods.     

Linear transformation models for survival analysis with tumor growth information in cancer screening study

The complication in analyzing tumor data is that the tumors detected in a screening program tend to be slowly progressive tumors, which is the so-called left-truncated sampling that is inherent in screening studies. Under the assumption that all subjects have the same tumor growth function, Ghosh (2008 Ghosh, D. (2008). Proportional hazards regression for cancer studies. Biometrics 64:141–148.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) developed estimation procedures for the Cox proportional hazards model. Shen (2011a Shen, P.-S. (2011a). Proportional hazards regression for cancer screening data. J. Stat. Comput. Simul. 18:367–377.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) demonstrated that Ghosh (2008 Ghosh, D. (2008). Proportional hazards regression for cancer studies. Biometrics 64:141–148.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])'s approach can be extended to the case when each subject has a specific growth function. In this article, under linear transformation model, we present a general framework to the analysis of data from cancer screening studies. We developed estimation procedures under linear transformation model, which includes Cox's model as a special case. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators.     

Multivariate Bayesian U-type asymmetric designs for non parametric response surface prediction under correlated errors

The objective of this paper is to study U-type designs for Bayesian non parametric response surface prediction under correlated errors. The asymptotic Bayes criterion is developed in terms of the asymptotic approach of Mitchell et al. (1994 Mitchell, T., Sacks, J., Ylvisaker, D. (1994). Asymptotic Bayes criteria for nonparametric response surface design. Ann. Stat. 22:634–651.[Crossref], [Web of Science ®] , [Google Scholar]) for a more general covariance kernel proposed by Chatterjee and Qin (2011 Chatterjee, K., Qin, H. (2011). Generalized discrete discrepancy and its applications in experimental designs. J. Stat. Plann. Inference 141:951–960.[Crossref], [Web of Science ®] , [Google Scholar]). A relationship between the asymptotic Bayes criterion and other criteria, such as orthogonality and aberration, is then developed. A lower bound for the criterion is also obtained, and numerical results show that this lower bound is tight. The established results generalize those of Yue et al. (2011 Yue, R.X., Qin, H., Chatterjee, K. (2011). Optimal U-type design for Bayesian nonparametric multiresponse prediction. J. Stat. Plann. Inference 141:2472–2479.[Crossref], [Web of Science ®] , [Google Scholar]) from symmetrical case to asymmetrical U-type designs.     

An efficient use of moment's ratios of scrambling variables in a randomized response technique

This paper aimed at providing an efficient new unbiased estimator for estimating the proportion of a potentially sensitive attribute in survey sampling. The suggested randomization device makes use of the means, variances of scrambling variables, and the two scalars lie between “zero” and “one.” Thus, the same amount of information has been used at the estimation stage. The variance formula of the suggested estimator has been obtained. We have compared the proposed unbiased estimator with that of Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. Relevant conditions are obtained in which the proposed estimator is more efficient than Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. The optimum estimator (OE) in the proposed class of estimators has been identified which finally depends on moments ratios of the scrambling variables. The variance of the optimum estimator has been obtained and compared with that of the Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. It is interesting to mention that the “optimum estimator” of the class of estimators due to Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) depends on the parameter π under investigation which limits the use of Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) OE in practice while the proposed OE in this paper is free from such a constraint. The proposed OE depends only on the moments ratios of scrambling variables. This is an advantage over the Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Numerical illustrations are given in the support of the present study when the scrambling variables follow normal distribution. Theoretical and empirical results are very sound and quite illuminating in the favor of the present study.     

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982 Bassett, G., Koenker, R. (1982). An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–415.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; Koenker and Bassett, 1978 Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica 46(1):33–50.[Crossref], [Web of Science ®] , [Google Scholar]). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.     

Using the Inverse Maximum Ratio-Λ as a Technique to Quantify Surface Uniformity

Techniques used in variability assessment are subsequently used to draw conclusions regarding the “spread”/uniformity of data curves. Due to the limitations of these techniques, they are not adequate for circumstances where data manifest with multiple peaks. Examples of these manifestations (in three-dimensional space) include under-foot pressure distributions recorded for different types of footwear (Becerro-de-Bengoa-Vallejo et al., 2014 Biau, D.J. (2011). In brief: Standard deviation and standard error. Clinical Orthopaedics and Related Research 469(9):2661–2664.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Cibulka et al., 1994 Cibulka, M.T., Sinacore, D.R., Mueller, M.J. (1994). Shin splints and forefoot contact running: A case report. Journal of Orthopaedic &; Sports Physical Therapy 20(2):98–102.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Davies et al., 2003 Davies, M.B., Betts, R.P., Scott, I.R. (2003). Optical plantar pressure analysis following internal fixation for displaced intra-articular os calcis fractures. Foot &; Ankle International 24(11):851–856.[PubMed], [Web of Science ®] , [Google Scholar]), surface textures and interfaces designed to impact friction, and and and molecular surface structures such as viral epitopes (Torras and Garcia-Valls, 2004 Torras, C., Garcia-Valls, R. (2004). Quantification of membrane morphology by interpretation of scanning electron microscopy images. Journal of Membrane Science 233(1–2):119–127.[Crossref], [Web of Science ®] , [Google Scholar]; Pacejka, 1997; Fustaffson, 1997). This article proposes a technique for generating a single variable – Λ that will quantify the uniformity of such surfaces. We define and validate this technique using several mathematical and graphical models.     

Sample size estimation for comparing rates of change in K-group repeated count outcomes

Sample size estimation for comparing the rates of change in two-arm repeated measurements has been investigated by many investigators. In contrast, the literature has paid relatively less attention to sample size estimation for studies with multi-arm repeated measurements where the design and data analysis can be more complex than two-arm trials. For continuous outcomes, Jung and Ahn (2004 Jung, S., Ahn, C. (2004). K-sample test and sample size calculation for comparing slopes in data with repeated measurements. Biometrical J. 46(5):554–564.[Crossref], [Web of Science ®] , [Google Scholar]) and Zhang and Ahn (2013 Zhang, S., Ahn, C. (2013). Sample size calculation for comparing time-averaged responses in k-group repeated measurement studies. Comput. Stat. Data Anal. 58:283–291.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) have presented sample size formulas to compare the rates of change and time-averaged responses in multi-arm trials, using the generalized estimating equation (GEE) approach. To our knowledge, there has been no corresponding development for multi-arm trials with count outcomes. We present a sample size formula for comparing the rates of change in multi-arm repeated count outcomes using the GEE approach that accommodates various correlation structures, missing data patterns, and unbalanced designs. We conduct simulation studies to assess the performance of the proposed sample size formula under a wide range of designing configurations. Simulation results suggest that empirical type I error and power are maintained close to their nominal levels. The proposed method is illustrated using an epileptic clinical trial example.     

A Realized Stochastic Volatility Model With Box–Cox Transformation

This article presents a new class of realized stochastic volatility model based on realized volatilities and returns jointly. We generalize the traditionally used logarithm transformation of realized volatility to the Box–Cox transformation, a more flexible parametric family of transformations. A two-step maximum likelihood estimation procedure is introduced to estimate this model on the basis of Koopman and Scharth (2013 Koopman, S.J., Scharth, M. (2013), The Analysis of Stochastic Volatility in the Presence of Daily Realised Measures, Journal of Financial Econometrics, 11, 76–115.[Crossref], [Web of Science ®] , [Google Scholar]). Simulation results show that the two-step estimator performs well, and the misspecified log transformation may lead to inaccurate parameter estimation and certain excessive skewness and kurtosis. Finally, an empirical investigation on realized volatility measures and daily returns is carried out for several stock indices.     

Fubini theorem for non additive measures in the framework of g-expectation

Since the seminal paper of Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), Chateauneuf and Lefort (2008 Chateauneuf, A., and J. P. Lefort. 2008. Some Fubini theorems on product σ-algebras for non-additive measures. International Journal of Approximate Reasoning 48:686–96.[Crossref], [Web of Science ®] , [Google Scholar]) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.     

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.     

Matrix mean squared error comparisons of some biased estimators with two biasing parameters

To deal with multicollinearity problem, the biased estimators with two biasing parameters have recently attracted much research interest. The aim of this article is to compare one of the last proposals given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) with Liu-type estimator (Liu 2003 Liu, K. 2003. Using Liu-type estimator to combat collinearity. Communications in Statistics: Theory and Methods 32 (5):1009–20.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and k ? d class estimator (Sakallioglu and Kaciranlar 2008 Sakallioglu, S., and S. Kaciranlar. 2008. A new biased estimator based on ridge estimation. Statistical Papers 49:669–89.[Crossref], [Web of Science ®] , [Google Scholar]) under the matrix mean squared error criterion. As well as giving these comparisons theoretically, we support the results with the extended simulation studies and real data example, which show the advantages of the proposal given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) over the other proposals with increasing multicollinearity level.     

On the comparison of Warner's and unrelated question randomized response plans for estimating sensitive finite population proportions

We consider the problem of estimation of a finite population proportion (P) related to a sensitive attribute under Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) randomized response plan and the unrelated question plan due to Horvitz et al. (1967 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and prove that for a given probability sampling design, given any linear unbiased estimator (LUE) of P based on Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) plan with any given value of the plan parameter, there exists an LUE of P based on the unrelated question plan with a uniformly smaller variance for suitable choices of the plan parameters. Assuming that only the attribute is sensitive but its complement is innocuous, the same is also shown to be true when the plan parameters for the two plans are so chosen so that both offer the same specified level of privacy.     

On extreme order statistics from heterogeneous beta distributions with applications

This paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007 Kochar, S.C., Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab. Eng. Inf. Sci. 21:597–609.[Crossref], [Web of Science ®] , [Google Scholar]), Mao and Hu (2010 Mao, T., Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probab. Eng. Inf. Sci. 24:245–262.[Crossref], [Web of Science ®] , [Google Scholar]), Balakrishnan et al. (2014 Balakrishnan, N., Barmalzan, G., Haidari, A. (2014). On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables. J. Multivariate Anal. 127:147–150.[Crossref], [Web of Science ®] , [Google Scholar]), and Torrado (2015 Torrado, N. (2015). On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J. Math. Anal. Appl. 426:824–838.[Crossref], [Web of Science ®] , [Google Scholar]). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.     

Buffered Autoregressive Models With Conditional Heteroscedasticity: An Application to Exchange Rates

This article introduces a new model called the buffered autoregressive model with generalized autoregressive conditional heteroscedasticity (BAR-GARCH). The proposed model, as an extension of the BAR model in Li et al. (2015 Li, G.D., Guan, B., Li, W.K., and Yu, P. L.H. (2015), “Hysteretic Autoregressive Time Series Models,” Biometrika, 102, 717–723.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), can capture the buffering phenomena of time series in both the conditional mean and variance. Thus, it provides us a new way to study the nonlinearity of time series. Compared with the existing AR-GARCH and threshold AR-GARCH models, an application to several exchange rates highlights the importance of the BAR-GARCH model.     

Analysis of Order Statistics from Distributions with Regularly Varying Tails

In this article, necessary conditions for comparing order statistics from distributions with regularly varying tails are discussed in terms of various stochastic orders. A necessary and sufficient condition for stochastically comparing tail behaviors of order statistics is derived. The main results generalize and recover some results in Kleiber (2002 Kleiber, C. (2002). Variability ordering of heavy-tailed distributions with applications to order statistics. Statist. Probab. Lett. 58:381388.[Crossref], [Web of Science ®] , [Google Scholar], 2004 Kleiber, C. (2004). Lorenz ordering of order statistics from log-logistic and related distributions. J. Statist. Plann. Infer. 120:2004.[Crossref], [Web of Science ®] , [Google Scholar]). Extensions to coherent systems are mentioned as well.     

Shared inverse Gaussian frailty models based on additive hazards

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this article, we assume that frailty acts additively to hazard rate. We introduce the shared inverse Gaussian frailty models with three different baseline distributions, namely the generalized log-logistic, the generalized Weibull, and exponential power distribution. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival dataset 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.     

Bahadur Representation of Linear Kernel Quantile Estimator for Stationary Processes

In this article, we discuss the method of linear kernel quantile estimator proposed by Parzen (1979 Parzen, E. (1979). Nonparametric statistical data modeling. J. Amer. Statist. Assoc. 74:105121.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We establish a Bahadur representation in sense of almost surely convergence with the rate log^{? α}n under the case of S-mixing random variable sequence which was proposed by Berkes (2009 Berkes, I., Hörmann, S., (2009). Asymptotic results for the itpirical process of stationary sequences. Stoch. Process. Their Applic. 119:12981324.[Crossref], [Web of Science ®] , [Google Scholar]). We also obtain the strong consistence of this estimator and its convergence rate.