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.     

Stress–strength models with more than two states under exponential distribution

In the stress–strength models, analysis is based on the reliability of the system where the system is either in operational state or in failure state. Ery?lmaz (2011 Ery?lmaz, S. (2011). A new perspective to stress–strength models. Ann. Inst. Stat. Math. 63(1):101–115.[Crossref], [Web of Science ®] , [Google Scholar]) introduced the stress–strength reliability in a different framework assigning more than two states to the system depending on the difference between strength and stress values. Unlike Ery?lmaz (2011 Ery?lmaz, S. (2011). A new perspective to stress–strength models. Ann. Inst. Stat. Math. 63(1):101–115.[Crossref], [Web of Science ®] , [Google Scholar]), the present article deals with the ratio of the strength and stress values when the stress and strength follow independent exponential distributions. This article presents in detail the estimation aspect of the multistate stress–strength reliability function.     

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.     

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.     

Bayesian Estimation for Linear Functions of Poisson Rates

The probability matching prior for linear functions of Poisson parameters is derived. A comparison is made between the confidence intervals obtained by Stamey and Hamilton (2006 Stamey, J., Hamilton, C. (2006). A note on confidence intervals for a linear function of Poisson rates. Commun. Statist. Simul. &; Computat. 35(4):849–856.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and the intervals derived by us when using the Jeffreys’ and probability matching priors. The intervals obtained from the Jeffreys’ prior are in some cases fiducial intervals (Krishnamoorthy and Lee, 2010 Krishnamoorthy, K., Lee, M. (2010). Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases. J. Statist. Plann. Infere. 140(5):1182–1192.[Crossref], [Web of Science ®] , [Google Scholar]). A weighted Monte Carlo method is used for the probability matching prior. The power and size of the test, using Bayesian methods, is compared to tests used by Krishnamoorthy and Thomson (2004 Krishnamoorthy, K., Thomson, J. (2004). A more powerful test for comparing two Poisson means. J. Statist. Plann. Infere. 119(1):23–35.[Crossref], [Web of Science ®] , [Google Scholar]). The Jeffreys’, probability matching and two other priors are used.     

Multiple dependent state repetitive sampling plans based on one-sided process capability indices

In this research, multiple dependent state and repetitive group sampling are used to design a variable sampling plan based on one-sided process capability indices, which consider the quality of the current lot as well as the quality of the preceding lots. The sample size and critical values of the proposed plan are determined by minimizing the average sample number while satisfying the producer's risk and consumer's risk at corresponding quality levels. In addition, comparisons are made with the existing sampling plans [Pearn and Wu (2006a Pearn, W. L., and C. W. Wu. 2006a. Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega: International Journal of Management Science 34 (1):90–101.[Crossref], [Web of Science ®] , [Google Scholar]), Yen et al. (2015 Yen, C. H., C. H. Chang, and M. Aslam. 2015. Repetitive variable acceptance sampling plan for one-sided specification. Journal of Statistical Computation and Simulation 85 (6):1102–16.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])] in terms of average sample number and operating characteristic curve. Finally, an example is provided to illustrate the proposed plan.     

Universally optimal designs under interference models with and without block effects

The concept of neighbor designs was introduced and defined by Rees (1967 Rees, D.H. (1967). Some designs of use in serology. Biometrics 23:779–791.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) along with giving some methods of their construction. Henceforth, many methods of construction of neighbor designs as well as of their generalizations are available in the literature. However, there are only few results on their optimality. Therefore, the purpose of this article is to give an overview of study on this problem. Recent results on optimality of specified neighbor balanced designs under various interference models with block effects are presented and then these results are compared with respective models where block effects are not significant.     

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.     

An Approximation Model of the Collective Risk Model with INAR(1) Claim Process

Cossette et al. (2010 Cossette, H., Marceau, E., Maume-Deschamps, V. (2010). Discerte-time risk models based on time series for count random variables. ASTIN Bull. 40:123–150.[Crossref], [Web of Science ®] , [Google Scholar], 2011 Cossette, H., Marceau, E., Toureille, F. (2011). risk models based on time series for count random variables. Insur. Math. Econ. 48:19–28.[Crossref], [Web of Science ®] , [Google Scholar]) gave a novel collective risk model where the total numbers of claims satisfy the first-order integer-valued autoregressive process. For a risk model, it is interesting to investigate the upper bound of ruin probability. However, the loss increments of the above model are dependent; it is difficult to derive the upper bound of ruin probability. In this article, we propose an approximation model with stationary independent increments. The upper bound of ruin probability and the adjustment coefficient are derived. The approximation model is illustrated via four simulated examples. Results show that the gap of the approximation model and dependent model can be ignored by adjusting values of parameters.     

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.     

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.     

A Bayesian control chart for a common coefficient of variation

By using the medical data analyzed by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), a Bayesian procedure is applied to obtain control limits for the coefficient of variation. Reference and probability matching priors are derived for a common coefficient of variation across the range of sample values. By simulating the posterior predictive density function of a future coefficient of variation, it is shown that the control limits are effectively identical to those obtained by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for the specific dataset they used. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining posterior distributions, predictive intervals, and run-lengths in the case of the coefficient of variation. A simulation study shows that the 95% Bayesian confidence intervals for the coefficient of variation have the correct frequentist coverage.     

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.     

Confidence bands for the logistic and probit regression models over intervals

This article presents methods for the construction of two-sided and one-sided simultaneous hyperbolic bands for the logistic and probit regression models when the predictor variable is restricted to a given interval. The bands are constructed based on the asymptotic properties of the maximum likelihood estimators. Past articles have considered building two-sided asymptotic confidence bands for the logistic model, such as Piegorsch and Casella (1988 Piegorsch, W.W., Casella, G. (1988). Confidence bands for logistic regression with restricted predictor variables. Biometrics 44:739–750.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). However, the confidence bands given by Piegorsch and Casella are conservative under a single interval restriction, and it is shown in this article that their bands can be sharpened using the methods proposed here. Furthermore, no method has yet appeared in the literature for constructing one-sided confidence bands for the logistic model, and no work has been done for building confidence bands for the probit model, over a limited range of the predictor variable. This article provides methods for computing critical points in these areas.     

An extension of almost sure central limit theorem for the maximum of stationary Gaussian random fields

Let X₁, X₂, … be a sequence of stationary standardized Gaussian random fields. The almost sure limit theorem for the maxima of stationary Gaussian random fields is established. Our results extend and improve the results in Csáki and Gonchigdanzan (2002 Csáki, E., Gonchigdanzan, K. (2002). Almost sure limit theorems for the maximum of stationary Gaussian sequences. Stat. Probab. Lett. 58:195–203.[Crossref], [Web of Science ®] , [Google Scholar]) and Choi (2010 Choi, H. (2010). Almost sure limit theorem for stationary Gaussian random fields. J. Korean Stat. Soc. 39:449–454.[Crossref], [Web of Science ®] , [Google Scholar]).     

Complete and complete moment convergence for i.i.d. random variables under exponential moment conditions

In this paper, we establish a complete convergence result and a complete moment convergence result for i.i.d. random variables under moment condition which is slightly weaker than the existence of the moment generating function. The main results extend and improve the related known results of Lanzinger (1998 Lanzinger, H. (1998). A Baum-Katz theorem for random variables under exponential moment conditions. Stat. Probab. Lett. 39(2):89–95.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut and Stadtmüller (2011 Gut, A., Stadtmüller, U. (2011). An intermediate Baum-Katz theorem. Stat. Probab. Lett. 81(10):1486–1492.[Crossref], [Web of Science ®] , [Google Scholar]).     

Nonparametric Inference for an Inverse-Probability-Weighted Estimator with Doubly Truncated Data

Efron and Petrosian (1999 Efron, B., Petrosian, V. (1999). Nonparametric methods for doubly truncated data. Journal of the American Statistical Association 94:824–834.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) formulated the problem of double truncation and proposed nonparametric methods on testing and estimation. An alternative estimation method was proposed by Shen (2010a Shen, P.S. (2010a). Nonparametric analysis of doubly truncated data. Annals of the Institute of Statistical Mathematics 62:835–853.[Crossref], [Web of Science ®] , [Google Scholar]), utilizing the inverse-probability-weighting technique. One aim of this paper was to assess the computational complexity of the existing estimation methods. Through a simulation study, we found that these two estimation methods have the same level of computational efficiency. The other aim was to study the noniterative IPW estimator under the condition that truncation variables are independent. The IPW estimator and the interval estimation was proved satisfactory in the simulation study.     

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 new efficient unrelated randomized response model

The present paper suggests an interesting and useful ramification of the unrelated randomized response model due to Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) [A new unrelated question randomized response model. Statistics 46 (1), 99–109] that can be used for any sampling scheme. We have shown theoretically and numerically that the proposed model is more efficient than Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) model.