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.     

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.     

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.     

An adept-stratified Kuk's randomized response model using Neyman allocation

This paper suggests a new stratified randomized response model based on Kuk's [Biometrika (1990), 77, 2, pp.436–438] model that has Neyman allocation and considerable gain in precision. It has been identified that the stratified randomized response models due to Kim and Warde (2004 Kim, J., Warde, W. (2004). A stratified Warner randomized response model. J. Stat. Plan. Inference 120:155–165.[Crossref], [Web of Science ®] , [Google Scholar]), Kim and Elam's (2005), and Kim and Elam's (2007) are members of the proposed model. It is shown that the proposed model is more efficient than Kuk's (1990) model both theoretically and empirically. The results of this paper are also extended in the situation when trials are repeated.     

Performance of some ridge regression estimators for the multinomial logit model

This article considers several estimators for estimating the ridge parameter k for multinomial logit model based on the work of Khalaf and Shukur (2005 Khalaf, G., and G. Shukur. 2005. Choosing ridge parameters for regression problems. Commun. Statist. Theor. Meth., 34:1177–1182.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Alkhamisi et al. (2006 Alkhamisi, M., G. Khalaf, and G. Shukur. 2006. Some modifications for choosing ridge parameters. Commun. Statist. Theor. Meth. 35:2005–2020.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Muniz et al. (2012 Muniz, G., B. M. G. Kibria, K. Månsson, and G. Shukur. 2012. On developing ridge regression parameters: A graphical investigation. in SORT. 36: 115–138.[Web of Science ®] , [Google Scholar]). The mean square error (MSE) is considered as the performance criterion. A simulation study has been conducted to compare the performance of the estimators. Based on the simulation study we found that increasing the correlation between the independent variables and the number of regressors has negative effect on the MSE. However, when the sample size increases the MSE decreases even when the correlation between the independent variables is large. Based on the minimum MSE criterion some useful estimators for estimating the ridge parameter k are recommended for the practitioners.     

On dynamic weighted survival entropy of order α

Recently, Abbasnejad et al. (2010 Abbasnejad, M., Arghami, N.R., Morgenthaler, S., Mohtashami Borzadaran, G.R. (2010). On the dynamic survival entropy. Stat. Probab. Lett. 80:1962–1971.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a measure of uncertainty based on survival function, called the survival entropy of order α. A dynamic form of the survival entropy of order α is also proposed by them. In this paper, we derive the weighted form of these measures. The properties of the new measures are also discussed.     

Almost everywhere convergence for sequences of pairwise NQD random variables

In this article, we are going to study the almost everywhere convergence for sequences of pairwise negatively quadrant dependent random variables by using truncation technique and Kolmogorov-type generalized three-series theorem. Our results generalize and improve the corresponding results of Wu (2002 Wu, Q. Y. (2002). Convergence properties of pairwise NQD random sequence. Acta. Math. Sin. 45:617–624 (in Chinese). [Google Scholar]) and Li and Yang (2008 Li, R., Yang, W. G. (2008). Strong convergence of pairwise NQD random sequences. J. Math. Anal. Appl. 334:741–747.[Crossref], [Web of Science ®] , [Google Scholar]). We also give some examples showing that our extensions are not trivial.     

Complete convergence for coordinatewise asymptotically negatively associated random vectors in Hilbert spaces

In this article, we study the complete convergence for sequences of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. We also investigate that some related results for coordinatewise negatively associated random vectors in Huan, Quang, and Thuan (2014 Huan, N. V., N. V. Quang, and N. T. Thuan. 2014. Baum–Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Mathematica Hungarica 144(1):132–419.[Crossref], [Web of Science ®] , [Google Scholar]) still hold under this concept.     

Optimality of ratio type estimation methods for population mean in the presence of missing data

This article proposes various Searls-type ratio imputation methods (STRIM) on the lines of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). It is a well-known fact that the optimal ratio type estimator attains the MSE of regression estimator (or optimal difference estimator) but while using Searls-type transformation (STT) (Searls (1964 Searls, D. T. 1964. The utilization of a known coefficient of variation in the estimation procedure. J. Am. Stat. Assoc. 59:1225–1226.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])) this may not always happen. These STRIM are shown to perform better than the imputation procedures of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). The STRIM may even outperform the Searls type difference imputation methods (STDIM) proposed by us in our earlier work, Bhushan and Pandey (2016 Bhushan, S., and A. P. Pandey. 2016. Optimal imputation of the missing data for estimation of population mean. Journal of Statistics and Management System 19 (6):755–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This study is concluded with the numerical study along with the theoretical comparison.     

Characterization results based on dynamic Tsallis cumulative residual entropy

In this article, the concept of cumulative residual entropy (CRE) given by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: A new measure of information. IEEE Trans. Inf. Theory 50:1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]) is extended to Tsallis entropy function and dynamic version, both residual and past of it. We study some properties and characterization results for these generalized measures. In addition, we provide some characterization results of the first-order statistic based on the Tsallis survival entropy.     

A new estimator of finite population mean based on the dual use of the auxiliary information

When a sufficient correlation between the study variable and the auxiliary variable exists, the ranks of the auxiliary variable are also correlated with the study variable, and thus, these ranks can be used as an effective tool in increasing the precision of an estimator. In this paper, we propose a new improved estimator of the finite population mean that incorporates the supplementary information in forms of: (i) the auxiliary variable and (ii) ranks of the auxiliary variable. Mathematical expressions for the bias and the mean-squared error of the proposed estimator are derived under the first order of approximation. The theoretical and empirical studies reveal that the proposed estimator always performs better than the usual mean, ratio, product, exponential-ratio and -product, classical regression estimators, and Rao (1991 Rao, T.J. (1991). On certail methods of improving ration and regression estimators. Commun. Stat. Theory Methods 20(10):3325–3340.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (2009 Singh, R., Chauhan, P., Sawan, N., Smarandache, F. (2009). Improvement in estimating the population mean using exponential estimator in simple random sampling. Int. J. Stat. Econ. 3(A09):13–18. [Google Scholar]), Shabbir and Gupta (2010 Shabbir, J., Gupta, S. (2010). On estimating finite population mean in simple and stratified random sampling. Commun. Stat. Theory Methods 40(2):199–212.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Grover and Kaur (2011 Grover, L.K., Kaur, P. (2011). An improved estimator of the finite population mean in simple random sampling. Model Assisted Stat. Appl. 6(1):47–55. [Google Scholar], 2014) estimators.     

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.     

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.     

Some Fundamental Properties of a Multivariate von Mises Distribution

In application areas like bioinformatics, multivariate distributions on angles are encountered which show significant clustering. One approach to statistical modeling of such situations is to use mixtures of unimodal distributions. In the literature (Mardia et al., 2012 Mardia , K. V. , Kent , J. T. , Zhang , Z. , Taylor , C. , Hamelryck , T. ( 2012 ). Mixtures of concentrated multivariate sine distributions with applications to bioinformatics . J. Appl. Stat. 39 : 2475 – 2492 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), the multivariate von Mises distribution, also known as the multivariate sine distribution, has been suggested for components of such models, but work in the area has been hampered by the fact that no good criteria for the von Mises distribution to be unimodal were available. In this article we study the question about when a multivariate von Mises distribution is unimodal. We give sufficient criteria for this to be the case and show examples of distributions with multiple modes when these criteria are violated. In addition, we propose a method to generate samples from the von Mises distribution in the case of high concentration.     

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.     

On the estimation of a sensitive quantitative mean using blank cards

In this article, we introduce for the first time, the blank card methods for estimation of finite population mean of a sensitive variable. Two generic randomization devices are suggested, and for each device we identify the choices of special models. We introduce additive, multiplicative, and combination of both additive and multiplicative scrambling models that require use of a non sensitive variable. We derive the basic statistical properties of each model. It is interesting to note that various existing estimators can be viewed as the special cases of those presented here. The statistical efficiency of proposed techniques is compared with Greenberg et al. (1971 Greenberg, B.G., Kuebler Jr, R.R., Abernathy, J.R., Horvitz, D.G. (1971). Application of the randomized response technique in obtaining quantitative data. J. Am. Stat. Assoc. 66(334):243–250.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and modified Perri (2008 Perri, P.F. (2008). Modified randomized devices for Simmons’ model. Model Assisted Stat. Appl. 3(3):233–239. [Google Scholar]) model. The proposed devices can easily be adjusted to achieve the required efficiency level by making suitable choices of different design parameters.     

Monitoring means and covariances of multivariate non linear time series with heavy tails

In this paper, the focus is on sequential analysis of multivariate financial time series with heavy tails. The mean vector and the covariance matrix of multivariate non linear models are simultaneously monitored by modifying conventional control charts to identify structural changes in the data. The considered target process is a constant conditional correlation model (cf. Bollerslev, 1990 Bollerslev, T. (1990). Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Rev. Econ. Stat. 72:498–505.[Crossref], [Web of Science ®] , [Google Scholar]), an extended constant conditional correlation model (cf. He and Teräsvirta, 2004 He, C., Teräsvirta, T. (2004). An extended constant conditional correlation GARCH model and its fourth-moment structure. Economet. Theory 20:904–926.[Crossref], [Web of Science ®] , [Google Scholar]), a dynamic conditional correlation model (cf. Engle, 2002 Engle, R.F. (2002). Dynamic conditional correlation: A simple class of multivariate GARCH models. J. Bus. Econ. Stat. 20(3):339–350.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), or a generalized dynamic conditional correlation model (cf. Capiello et al., 2006 Capiello, L., Engle, R., Sheppard, K. (2006). Asymmetric correlations in the dynamics of global equity and bond returns. J. Financial Economet. 4(4):537–572.[Crossref] , [Google Scholar]). For statistical surveillance we use control charts based on residuals. Further, the procedures are constructed for t-distribution. The detection speed of these charts is compared via Monte Carlo simulation. In the empirical study, the procedure with the best performance is applied to log-returns of the stock market indices FTSE and CAC.     

A new lifetime model with different types of failure rate

A new class of lifetime distributions, which can exhibit with upside-down bathtub-shaped, bathtub-shaped, decreasing, and increasing failure rates, is introduced. The new distribution is constructed by compounding generalized Weibull and logarithmic distributions, leading to improvement on the lifetime distribution considered in Dimitrakopoulou et al. (2007 Dimitrakopoulou, T., K. Adamidis, and S. Loukas. 2007. A lifetime distribution with an upside-down bathtub-shaped hazard function. IEEE Transactions on Reliability 56:308–11.[Crossref], [Web of Science ®] , [Google Scholar]) by having no restriction on the shape parameter and extending the result studied by Tahmasbi and Rezaei (2008 Tahmasbi, R., and S. Rezaei. 2008. A two-parameter lifetime distribution with decreasing failure rate. Computational Statistics and Data Analysis 52:3889–901.[Crossref], [Web of Science ®] , [Google Scholar]) in the general form. The proposed model includes the exponential–logarithmic and Weibull–logarithmic distributions as special cases. Various statistical properties of the proposed class are discussed. Furthermore, estimation via the maximum likelihood method and the Fisher information matrix are discussed. Applications to real data demonstrate that the new class of distributions is more flexible than other recently proposed classes.     

A modified Hosmer–Lemeshow test for large data sets

The Hosmer–Lemeshow test is a widely used method for evaluating the goodness of fit of logistic regression models. But its power is much influenced by the sample size, like other chi-square tests. Paul, Pennell, and Lemeshow (2013 Paul, P., M. L. Pennell, and S. Lemeshow. 2013. Standardizing the power of the Hosmer–Lemeshow goodness of fit test in large data sets. Statistics in Medicine 32:67–80.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) considered using a large number of groups for large data sets to standardize the power. But simulations show that their method performs poorly for some models. In addition, it does not work when the sample size is larger than 25,000. In the present paper, we propose a modified Hosmer–Lemeshow test that is based on estimation and standardization of the distribution parameter of the Hosmer–Lemeshow statistic. We provide a mathematical derivation for obtaining the critical value and power of our test. Through simulations, we can see that our method satisfactorily standardizes the power of the Hosmer–Lemeshow test. It is especially recommendable for enough large data sets, as the power is rather stable. A bank marketing data set is also analyzed for comparison with existing methods.     

Testing for Random Effects in Growth Curve Models

Growth curve models (GCMs) are useful and Demidenko (2004 Demidenko, E. (2004). Mixed Models: Theory and Applications. New York: Wiley.[Crossref] , [Google Scholar]) considered the presence of random effects under the normal assumptions about random effects and random errors. It is also of interest to remove distribution assumptions to investigate the same problem. A difference-based test is constructed for GCMs, which can be regarded as an extension of Li and Zhu (2010 Li, Z.X., Zhu, L.X. (2010). On variance components in semiparametric mixed models for longitudinal data. Scand. J. Statist. 37:442–457.[Crossref], [Web of Science ®] , [Google Scholar])’s method and a complement to Demidenko (2004 Demidenko, E. (2004). Mixed Models: Theory and Applications. New York: Wiley.[Crossref] , [Google Scholar]) where his test is exact in small samples. Without any distribution assumptions, our test derived for GCMs is asymptotically a standard normal. The power properties are also investigated. Besides, simulations are carried out to examine its performance.