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.     

U-Statistic Based Modified Information Criterion for Change Point Problems

The U-statistic based modified information criterion (MIC) is proposed and applied to detect the change point in a sequence of independent random variables. In this article, we show that the method is consistent in selecting the correct model, and the resulting test statistic has a simple limiting distribution. We investigate the method based on both symmetric and anti-symmetric kernel functions. The simulation results indicate that the new method has better power in detecting the changes compared to other methods, such as the likelihood based MIC (Chen et al., 2006 Chen , J. , Gupta , A. K. , Pan , J. ( 2006 ). Information criterion and change point problem for regular models . Sankhyā 68 : 252 – 282 . [Google Scholar]) and the Bayesian information criterion of Schwarz (BIC, Schwarz, 1978 Schwarz , G. ( 1978 ). Estimating the dimension of a model . Ann. Statist. 6 : 461 – 464 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Bayesian Inference for the Precision Matrix for Scale Mixtures of Normal Distributions

Baysian inference is considered for the precision matrix of the multivariate regression model with distribution of the random responses belonging to the multivariate scale mixtures of normal distributions. The posterior distribution and some identities involving expectations taken with respect to this posterior distribution are derived when the prior distribution of the parameters is from the conjugate family. The results are specialized to the case where the random responses have a matrix-t distribution and thus generalizing the results of Zellner (1976 Zellner , A. ( 1976 ). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms . J. Amer. Statist. Assoc. 71 : 400 – 405 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Muirhead (1986 Muirhead , R. J. ( 1986 ). A note on some Wishart expectations . Metrika 33 : 247 – 251 .[Crossref] , [Google Scholar]).     

Tests of Multivariate Independence for Ordinal Data

Population and sample versions of Kendall and Spearman measures of association suitable for multivariate ordinal data are defined. The latter generalize the indices of dependence of Ruymgaart and van Zuijlen (1978 Ruymgaart , F. H. , van Zuijlen , M. C. A. ( 1978 ). Asymptotic normality of multivariate linear rank statistics in the non-i.i.d. case . Ann. Statist. 6 : 588 – 602 .[Crossref], [Web of Science ®] , [Google Scholar]), Joe (1990 Joe , H. ( 1990 ). Multivariate concordance . J. Multivariate Anal. 35 : 12 – 30 .[Crossref], [Web of Science ®] , [Google Scholar]), and Schmid and Schmidt (2007 Schmid , F. , Schmidt , R. ( 2007 ). Multivariate extensions of Spearman's rho and related statistics . Statist. Probab. Lett. 77 : 407 – 416 .[Crossref], [Web of Science ®] , [Google Scholar]) by allowing atoms in the underlying distribution. The representation of the proposed empirical measures as U-statistics enables to establish their asymptotic normality under general distributions. A special attention is given to tests of independence for multivariate ordinal data, where the power of the new methodologies are investigated under fixed and contiguous alternatives.     

Nonnested Testing for Competing Autoregressive Dynamic Models Estimated by Instrumental Variables

In this work we investigate nonnested tests for two competing univariate dynamic linear models with autoregressive disturbances, where the motivation for instrumental variable estimation is mainly due to the recognized presence of current endogenous variables in the regression function, either in one or both models. As the previous transformation of both models yields regression functions which are nonlinear in the parameters, the attractive Gauss-Newton regression (GNR) approach, firstly advocated by Davidson and Mackinnon (1981 Davidson , R. , Mackinnon , J. G. ( 1981 ). Several tests for model specification in the presence of alternative hypotheses . Econometrica 49 : 781 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]), will be used to obtain the results.     

An Index of Local Sensitivity to Nonignorability for a Pseudolikelihood Method

For regression analysis of data with non response, sensitivity analysis is usually recommended. An index of local sensitivity to non ignorability (ISNI) (Troxel et al., 2004 Troxel , A. B. , Ma , G. , Heitjan , D. F. (2004). An index of local sensitivity to nonignorability. Statistica Sinica 14:1221–1237.[Web of Science ®] , [Google Scholar]) was derived to detect the sensitivity of maximum likelihood estimates to small departures from ignorability. However, ISNI requires specification of a parametric model for the missing-data mechanism. In this article, a local sensitivity index for a pseudolikelihood (PL) method that does not require specification of the mechanism is proposed. For bivariate data (x, y), when the non response mechanism is an arbitrary function of x + λy, this new index is defined as the first derivative of the PL estimate with respect to λ at λ = 0. The closed form was derived for normal regression data when the density function of the predictor x approximated by a kernel estimator in the PL method. The utility of this new local sensitivity index was illustrated through application on one dataset.     

Bounds on the efficiency of unbalanced ranked-set sampling

Abstract

Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) derived a sharp upper bound on the efficiency of the balanced ranked-set sampling (RSS) sample mean relative to the simple random sampling (SRS) sample mean under perfect rankings. The bound depends on the set size and is achieved for uniform distributions. Here we generalize the Takahasi and Wakimoto (1968 Takahasi, K., and K. Wakimoto. 1968. On unbiased estimates of the population mean based on the sample stratified by means of ordering. Annals of the Institute of Statistical Mathematics 20:1–31.[Crossref], [Web of Science ®] , [Google Scholar]) result by finding a sharp upper bound in the case of unbalanced RSS. The bound depends on the particular unbalanced design, and the distributions where the bound is achieved can be highly nonuniform. The bound under perfect rankings can be exceeded under imperfect rankings.     

A Class of Improved Estimators for the Scale Parameter of a Mixture Model of Exponential Distribution with Unknown Location

Under Stein's loss, a class of improved estimators for the scale parameter of a mixture of exponential distribution with unknown location is constructed. The method is analogous to Maruyama's (1998 Maruyama , Y. ( 1998 ). Minimax estimators of a normal variance . Metrika 48 : 209 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]) construction for the variance of a normal distribution and also an extension of the result produced in Petropoulos and Kourouklis (2002 Petropoulos , C. , Kourouklis , S. ( 2002 ). A class of improved estimators for the scale parameter of an exponential distribution with unknown location . Commun. Statist. Theor. Meth. 31 : 325 – 335 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Also, robustness properties are considered.     

Calibration of Proportional Hazards and Accelerated Failure Time Models

Given a prognostic model based on one population, one may ask: Can this model be used to accurately predict disease in a different population? When the underlying rate of disease differs in the new population, the model must be calibrated. van Houwelingen (2000 van Houwelingen , H. ( 2000 ). Validation, calibration, revision and combination of prognostic survival models . Statistics in Medicine 19 : 3401 – 3415 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) considered this calibration problem focusing on proportional hazards models. We extend the validation by calibration to the log-logistic accelerated failure time model. We use calibration of proportional hazards models and log-logistic accelerated failure time models to examine whether a survival model based on the Framingham Heart Study can be applied to diverse studies around the world.     

The Finite Time Ruin Probability of a New Risk Model Based on Entrance Process

In this article, we consider a new insurance risk model based on the entrance process proposed in Li et al. (2005 Li , Z. , Zhu , J. , Chen , F. ( 2005 ). Study of a risk model based on the entrance process . Statist. Probab. Lett. 72 : 1 – 10 .[Crossref], [Web of Science ®] , [Google Scholar]), and investigate the finite time ruin probabilities of this model. It is showed that an exponential upper bound for the finite time ruin probability exists, when the distributions of the claim size are light tailed. Furthermore, when the distributions of the claim size are heavy tailed, an asymptotic formula for the finite time ruin probability 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.     

Generalized method of moments for an extended gamma process

In reliability theory, a widely used process to model the phenomena of the cumulative deterioration of a system over time is the standard gamma process (SGP). Based on several restrictions, such as a constant variance-to-mean ratio, this process is not always a suitable choice to describe the deterioration. A way to overcome these restrictions is to use an extended version of the gamma process introduced by Cinlar (1980), which is characterized by shape and scale functions. In this article, the aim is to propose statistical methods to estimate the unknown parameters of parametric forms of the shape and scale functions. We here develop two generalized methods of moments (Hansen 1982 Hansen, L. P. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50 (4):1029–54.[Crossref], [Web of Science ®] , [Google Scholar]), based either on the moments or on the Laplace transform of an extended gamma process. Asymptotic properties are provided and a Wald-type test is derived, which allows to test SGPs against extended ones with a specific parametric shape function. Also, the performance of the proposed estimation methods is illustrated on simulated and real data.     

Bayesian Analysis of Instrumental Variable Models: Acceptance-Rejection within Direct Monte Carlo

We discuss Bayesian inferential procedures within the family of instrumental variables regression models and focus on two issues: existence conditions for posterior moments of the parameters of interest under a flat prior and the potential of Direct Monte Carlo (DMC) approaches for efficient evaluation of such possibly highly non-elliptical posteriors. We show that, for the general case of m endogenous variables under a flat prior, posterior moments of order r exist for the coefficients reflecting the endogenous regressors’ effect on the dependent variable, if the number of instruments is greater than m +r, even though there is an issue of local non-identification that causes non-elliptical shapes of the posterior. This stresses the need for efficient Monte Carlo integration methods. We introduce an extension of DMC that incorporates an acceptance-rejection sampling step within DMC. This Acceptance-Rejection within Direct Monte Carlo (ARDMC) method has the attractive property that the generated random drawings are independent, which greatly helps the fast convergence of simulation results, and which facilitates the evaluation of the numerical accuracy. The speed of ARDMC can be easily further improved by making use of parallelized computation using multiple core machines or computer clusters. We note that ARDMC is an analogue to the well-known “Metropolis-Hastings within Gibbs” sampling in the sense that one ‘more difficult’ step is used within an ‘easier’ simulation method. We compare the ARDMC approach with the Gibbs sampler using simulated data and two empirical data sets, involving the settler mortality instrument of Acemoglu et al. (2001 Acemoglu , D. , Johnson , S. , Robinson , J. A. ( 2001 ). The colonial origins of comparative development: An empirical investigation . American Economic Review 91 ( 5 ): 1369 – 1401 .[Crossref], [Web of Science ®] , [Google Scholar]) and father's education's instrument used by Hoogerheide et al. (2012a Hoogerheide , L. F. , Block , J. H. , Thurik , A. R. (2012a). Family background variables as instruments for education in income regressions: A Bayesian analysis. Economics of Education Review 31(5):515–523.[Crossref], [Web of Science ®] , [Google Scholar]). Even without making use of parallelized computation, an efficiency gain is observed both under strong and weak instruments, where the gain can be enormous in the latter case.     

Improved Estimation of Finite Population Variance Using Auxiliary Information

This article addresses the problem of estimating of finite population variance using auxiliary information in simple random sampling. A ratio-cum-difference type class of estimators for population variance has been suggested with its properties under large sample approximation. It has been shown that the suggested class of estimators is more efficient than usual unbiased, difference, Das and Tripathi (1978 Das , A. K. , Tripathi , T. P. ( 1978 ). Use of auxiliary information in estimating the finite population variance . Sankhya C 40 : 139 – 148 . [Google Scholar]), Isaki (1983 Isaki , C. T. ( 1983 ). Variance estimation using auxiliary information . J. Amer. Statist. Assoc. 78 : 117 – 123 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Singh et al. (1988 Singh , H. P. , Upadhyaya , L. N. , Namjoshi , U. D. ( 1988 ). Estimation of finite population variance . Curr. Sci. 57 : 1331 – 1334 .[Web of Science ®] , [Google Scholar]), Kadilar and Cingi (2006 Kadilar , C. , Cingi , H. ( 2006 ). Ratio estimators for the population variance in simple and stratified random sampling . Appl. Math. Comp. 173 ( 2 ): 1047 – 1059 .[Crossref], [Web of Science ®] , [Google Scholar]), and other estimators/classes of estimators. In addition, we support this theoretical result with the aid of a empirical study.     

Reliability characteristics of Farlie–Gumbel–Morgenstern family of bivariate distributions

Abstract

In this paper, we study the Farlie–Gumbel–Morgenstern family of bivariate distributions from a reliability point of view. The properties of this family of distributions and the association between the two variables are investigated by studying the local dependence function and the association measure defined by Clayton (1978 Clayton, D.G. (1978). A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151.[Crossref], [Web of Science ®] , [Google Scholar]). We also study the effect of the association parameter on the hazard components, the failure rate of the series system, and the regression mean residual life of a parallel system. Stochastic comparisons with respect to the association parameter are also studied. Some examples are provided to illustrate the results.     

Goodness-of-Fit Testing for Exponential Polynomial Growth Curves

A goodness-of-fit test is proposed for the family of exponential polynomial growth curve models (EPGCM; Heinen, 1999 Heinen , M. ( 1999 ). Analytical growth equations and their Genstat 5 equivalents . Netherlands J. Agricult. Sci. 47 : 67 – 89 .[Web of Science ®] , [Google Scholar]), which has wide applications in different areas of science. The exponential growth curve model (EGCM), the most prominent member of the EPGCM family, is a simple and biologically meaningful growth model. Other members of the EPGCM family also cover many realistic growth processes. Thus, a goodness-of-fit test for the EPGCM class has substantial practical value. The goodness-of-fit test developed here is based on the properties of finite differences. The performance of the theory developed is illustrated through simulation and analysis of real data.     

Stein-like 2SLS estimator

ABSTRACT

Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a Stein-like estimator for simultaneous equations and showed that his Stein shrinkage estimator has bounded finite sample risk, unlike the three-stage least square estimator. We revisit his proposal by investigating Stein-like shrinkage in the context of two-stage least square (2SLS) estimation of a structural parameter. Our estimator follows Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) in taking a weighted average of the 2SLS and ordinary least square estimators, with the weight depending inversely on the Hausman (1978 Hausman, J. A. (1978). Specification tests in econometrics. Econometrica 46:1251–1271.[Crossref], [Web of Science ®] , [Google Scholar]) statistic for exogeneity. Using a local-to-exogenous asymptotic theory, we derive the asymptotic distribution of the Stein estimator and calculate its asymptotic risk. We find that if the number of endogenous variables exceeds 2, then the shrinkage estimator has strictly smaller risk than the 2SLS estimator, extending the classic result of James and Stein (1961 James W, ., Stein, C. M. (1961). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1:361–380. [Google Scholar]). In a simple simulation experiment, we show that the shrinkage estimator has substantially reduced finite sample median squared error relative to the standard 2SLS estimator.     

A Euclidean Likelihood Estimator for Bivariate Tail Dependence

The spectral measure plays a key role in the statistical modeling of multivariate extremes. Estimation of the spectral measure is a complex issue, given the need to obey a certain moment condition. We propose a Euclidean likelihood-based estimator for the spectral measure which is simple and explicitly defined, with its expression being free of Lagrange multipliers. Our estimator is shown to have the same limit distribution as the maximum empirical likelihood estimator of Einmahl and Segers (2009 Einmahl , J. H. J. , Segers , J. ( 2009 ). Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution . Ann. Statist. 37 ( 5B ): 2953 – 2989 .[Crossref], [Web of Science ®] , [Google Scholar]). Numerical experiments suggest an overall good performance and identical behavior to the maximum empirical likelihood estimator. We illustrate the method in an extreme temperature data analysis.     

On Mixture Periodic Vector Autoregressive Models

This article deals with the study of some properties of a mixture periodically correlated n-variate vector autoregressive (MPVAR) time series model, which extends the mixture time invariant parameter n-vector autoregressive (MVAR) model that has been recently studied by Fong et al. (2007 Fong, P.W., Li, W.K., Yau, C.W., Wong, C.S. (2007). On a mixture vector autoregressive model. The Canadian Journal of Statistics 35:135–150.[Crossref], [Web of Science ®] , [Google Scholar]). Our main contributions here are, on the one side, the obtaining of the second moment periodically stationary condition for a n-variate MPVAR_S(n; K; 2, …, 2) model; furthermore, the closed-form of the second moment is obtained and, on the other side, the estimation, via the Expectation-Maximization (EM) algorithm, of the coefficient matrices and the error variance matrix.     

Linear Combinations,Products and Ratios of Simplicial or Spherical Variates

The density level sets of the two types of measures under consideration are l _{2, p} -circles with p = 1 and p = 2, respectively. The intersection-percentage function (ipf) of such a measure reflects the percentages which the level set corresponding to the p-radius r shares for each r > 0 with a set to be measured. The geometric measure representation formulae in Richter (2009 Richter , W.-D. (2009). Continuous l _{n, p} -symmetric distributions. Lithuanian Mathemat. J. 49:93–108.[Crossref], [Web of Science ®] , [Google Scholar]) is based upon these ipf's and will be used here for evaluating exact cdf's and pdf's for the linear combination, the product, and the ratio of the components of two-dimensional simplicial or spherically distributed random vectors.