A flexible semiparametric transformation model for

Authors: Thomas H Scheike

Publish Date: 2006/09/20

Volume: 12, Issue: 4, Pages: 461-480

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Abstract

I suggest an extension of the semiparametric transformation model that specifies a timevarying regression structure for the transformation and thus allows timevarying structure in the data Special cases include a stratified version of the usual semiparametric transformation model The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given The method is illustrated by a worked example and a small simulation study A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggestedI would like to thank Martin Jacobsen for pointing my direction to the product integration formulae I also appreciate discussions with Torben Martinussen Part of the this work was done while the author visited the Center for Advanced Study in Oslo and was partly supported by an NIH grant I would also like to thank two referees and the associate editor One for a detailed reading and making several suggestions for an improved presentation and one for providing me with several recent references that deals with the asymptotic analysis Their comments helped improve the manuscriptI here sketch the main arguments of the proof that establishes the asymptotic variances extending that of Bagdonavicius and Nikulin 1999 2001 and Chen et al 2002 to more than one dimension A detailed consistency proof for the standard transformation model can be found in Dabrowska 2005 and it appears that these arguments can be extended to our setting but this is no trivial exercise The key to the multivariate extension is the use of productlimit integration formulae and I here focus on establishing the key formulae that gives expressions for the standard deviations of the estimated quantitiesAssume that subjects are iid with covariates that are uniformly bounded Define varvec DbetaA = hboxdiagexp2 Z iTbetadotlambda 0HtX iexpZ iTbeta varvec Sbeta A = n1varvec XTtvarvec DtbetaA varvec Xt varvec S jbeta A = n1varvec XTtvarvec DX ijvarvec DtbetaAvarvec Xt varvec SZbeta A = n1varvec ZTtvarvec DtbetaA varvec Xt and varvec S jZbeta A = n1varvec ZTt varvec DX ijvarvec DtbetaAvarvec XtOne of the needed assumptions is that all S matrices converge uniformly to deterministic matrices in both time and the parameter space for β Let Θ denote an open ball around the true parameter value β0 then it is assumed that the limit of all large S’s exist and are denoted by small s’s Such that for example lim p varvec S jZbeta A = s jZ uniformly in Thetatimes 0tau Define the limit of E jt = varvec S 01tbetavarvec S jtbeta as e jubeta I also assume that the covariates are uniformly bounded

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