## N oxt0 t0qq

s(x,t) < , ,,, , , .s , . (33) wp(z(t),x) o(x,t), ts < t< to x > 0 where the multiplier already depends on the stress at time t and on the age x. Different environments can be defined not necessarily by the switching point or by considering changing in time stresses. Let the stress zo(to) be a baseline stress at a baseline (fixed) time instant to. Denote the corresponding mortality rate, as previously, by io(x,to). Then the stress z(t) and the mortality rate (x,t) characterize the...

## The failure time model

In failure time analysis the response variable Y is a failure time. Measures of explained variation and predictive accuracy may be defined as above but when censoring occurs, as is usual in survival analysis, the estimation procedure gets complicated The explained residual variation cannot be determined because the loss corresponding to a censored failure time is unavailable. Since the estimated explained variation has not been accepted as an estimator of the explained variation other...

## Qi QI Q2 Qs

Power of Qopt versus Q3 for Different K Fig. 1. Power of Qopt versus Q3 for Different K VKVN02 Andersen,P.K., Gill, R.D. Cox's regression model for counting processes a large sample study. The Annals of Statistics, 10, 1100 20 (1981) VKVN02 Cai, J., Prentice, R.L. Estimating Equations for Hazard Ratio Parameters Based on Correlated Failure Time Data. Biometrika, 82, 151-64 (1995) VKVN02 Cook, R.J., Lawless, J.F., Nadeau, C. Robust Tests for Treatment Comparisons Based on Recurrent...

## Generalized Bayesian inference in case of fuzzy information

The generalization of Bayes' theorem by application of the extension principle from fuzzy set theory is not reasonable, because it is not keeping the sequential updating procedure of Bayesian inference. Therefore another method was developed which takes care of imprecision of a-priori distributions and fuzziness of data as well. This is presented in the paper VH04 . It is important to note that a more general concept of probability distributions, so-called fuzzy probability distributions, is...

## Y Sei1 sdSes

Where Se(-) is the Kaplan-Meier estimator of the survival function 1 Fe using the censored residuals ei,, Si . 3. Apply ordinary least squares (OLS) to ( S(Y ), Zi) . Update Y S Z. 4. Stop if Y converges or oscillates. Otherwise, return to step 2. Incorporating PCR into the Buckley-James algorithm is straightforward, since the calculation of the principal components uses only the matrix of co-variates and is done before any regression models are estimated. A forward stepwise regression using...

## D d 0 dxfxt m9tx

We continue our proof for non-uniform case. Let fx(t) satisfy Lipschitz condition in some neighborhood of the point ti. We have for ti < t2 gt2 (x) gt1 (x) fx(s) v (u,x) duds fx (s) v (u,x) duds fx(s) v(u,x) duds fx(s) v(u,x) duds fx(s) v(u,x) duds + fx(s) v(u,x) duds+ + fx(s) v(u,x) duds fx(s) v(u,x) duds v(u,x) fx(s) ds du v(u,x) fx(s) dsdu. J the first member in this sum has an order (t2 ti)fx(ti) v (u,x)du + o(t2 ti). ti The sum of the second and fifth members can be represented as...

## Jjifi

The simple case f c in found in BN04 where f (t) c(t) ln(1 +1). In this case the calculations reduce to UJ V(Y * YD (f f ) f(t ))) , and thus the estimate is not consistent in this case. In the nonlinear case, whatever the hypothesis about the correlation of the noises is, the predictor of 0l is found by least square minimization argminoeRP Y - g(f, a))'S-1(Yi - g(t, a)) i 1 n where g(ti,a) is the vector in Rfi of the values (g(tij,a))j i fi. Under Hl, LM93 provide direct estimation of the...

## A[0x undu x [0x1

M2(x) a2(x) E0(tx - A1(x))2 u2n(du x 0,x)), (2) M3(x) E0(tx - A1(x))3 u3n(du x 0,x)), (3) M4(x) E0(tx - A1(x))4 u4n(du x 0,x)) + 3M (x). (4) It is not difficult to evaluate other moments. Inverse gamma process In the works har04a, har04d and others we gave arguments justifying I-process to be used in reliability problems. We showed examples, where I-process analytical properties are useful in optimization problems of prophylaxis and reservation. For practical aim one should consider more narrow...

## Levy Processes as Degradation Models

Many degradation models are based on the concept of accumulated damage. Noortwijk Noo96 points out that in systems subject to shocks, the order in which the damage (i.e. the shocks) occurs is often immaterial so that the random deterioration incurred in equal time intervals forms a set of exchangeable random variables BS92 . This also implies that the distribution of the degradation incurred is independent of the time scale, i.e. the process has stationary increments. Exchangeable and...

## Survival Model With Change Point in Both Hazard and Regression Parameters

Laboratoire de Statistique et Probabilit s, Universit Paul Sabatier, 118, route de Narbonne, 31062 Toulouse cedex 4, France dupuy math.ups-tlse.fr In this paper, we consider a parametric survival regression model with a change-point in both hazard and regression parameters. Change-point occurs at an unknown time point. Estimators of the change-point, hazard and regression parameters are proposed and shown to be consistent. Let T be a random failure time variable. The distribution of T is...

## Nonparametric Estimation for Failure Rate Functions of Discrete Time semiMarkov Processes

1 Universit de Technologie de Compi gne, Laboratoire de Math matiques Appliqu es de Compi gne, BP 20529, 60205 Compi gne, France barbu dma.utc.fr 2 Universit de Technologie de Compi gne, Laboratoire de Math matiques Appliqu es de Compi gne Nikolaos.Limnios utc.fr Summary. We consider a semi-Markov chain, with a finite state space. Taking a censored history, we obtain empirical estimators for the discrete semi-Markov kernel, renewal function and semi-Markov transition function. We propose...

## Comparisons of Test Statistics Arising from Marginal Analyses of Multivariate Survival Data

Center for Drug and Evaluation Research, HFD-705 7500 Standish Place, Metro Park North (MPN) II, Rockville, MD 20855 liq cder.fda.gov 2 Department of Biostatistics, Harvard School of Public Health 655 Huntington Avenue, Boston MA 02115 lagakos hsph.harvard.edu Summary. We investigate the properties of several statistical tests for comparing treatment groups with respect to multivariate survival data, based on the marginal analysis approach introduced by Wei, Lin and Weissfeld WLW89 . We...

## M k x M kAxM kB x

LRk n-1 2 f Wk(u) dMkA - dMkA (u), (2) and the asymptotic distribution of the vector (LR , LR2)' will be derived from he asymptotic propertie the asymptotic properties of the martingales vector (n 1 2M1A,nB1 2Mib , 2.1 Preliminary results for the martingales under H0 Rebolledo's theorem ensures the convergence of the vector (n-1 2M 1a, n-1 2Mib,n 1 2M2A,n-1 2M2b)' to a Gaussian process (miA,miB,m2A, m2B)', with null expectation and with variances vj(x) where The covariance between m A and m B...

## Simulation studies

We used simulation studies to explore the predictive power of the accelerated failure time model using partial least squares and the Buckley-James fitting algorithm. Mean squared prediction error was used to measure how well the covariate effect was predicted, and mean absolute prediction error was used to measure how well the response was predicted. Simulations were done using different numbers of explanatory variables (p 10, 25, 40, 50, and 100), with different correlations among the...

## Optimization of Screening Strategies and Sensitivity Analyses

The focus of this investigation is to compare the effects of different breast cancer screening policies and the costs directly related to these policies, based on the models introduced in the last sections. The health outcome of interest is the expected gain in quality-adjusted survival. We interpret this quality adjustment to be relative to a typical health history rather than that of a state of perfect health PBW99 . Quality adjustments are important because they allow, with certain...

## PChW log pW hjx2udu6

Where L is the partial log-likelihood (in the sense of section 2.3) and h is a positive smoothing parameter which controls the tradeoff between the fit of the data and the smoothness of the function. Maximization of (6) over the desired class of functions defines the maximum penalized likelihood estimator (MPLE) X . The solution is then approximated on a basis of splines. The main advantage of the penalized likelihood approach over the kernel smoothing method is that there is no edge problem...

## Zor Z0j rj

We also denote by zo(r) zoj rj 0 a potential realization of the missing covariate. We shall treat the variable R as an extra covariate taking values in the set R 0,1 9. Denoting the sample space of covariates Zo and by Zo and Zi, respectively, the unobserved model is defined on the probability space (Q' x Q, G' Ft t< T, Pr), where Q' RxZo xZ1, G' is the Borel a-field of Q' and ''Pr'' is defined in the condition (2.1) below. The observable model corresponds to the transformation of this space...

## Nonparametric Estimation and Testing in Survival Models

Henning Lauter1 and Hannelore Liero2 1 Institute of Mathematics, University of Potsdam laeuter rz.uni-potsdam.de 2 Institute of Mathematics, University of Potsdam liero rz.uni-potsdam.de The aim of this paper is to demonstrate that nonparametric smoothing methods for estimating functions can be an useful tool in the analysis of life time data. After stating some basic notations we will present a data example. Applying standard parametric methods to these data we will see that this approach...

## Tests of Fit based on Products of Spacings

1 L.S.T.A., Universit Paris VI, 7 avenue du Ch teau, F 92340 Bourg-la-Reine, France pd ccr.jussieu.fr 2 Sanofi-Synth labo Recherche, 371 rue du Professeur Joseph Blayac, 34184 Montpellier Cedex 04, France Gerard.Derzko sanofi-aventis.com Summary. Let Z Z1, ,Zn be an i.i.d. sample from the distribution F(z) P(Z < z) and density f(z) dF(z). Let Zi,n < < Zn n be the order statistics generated by Z1, ,Zn. Let Z0> n a inf z F(z) > 0 and Z +1 b sup z F(z) < 1 denote the end-points of the...

## Phfnf e P sup fj exp cme2

6.2 Hellinger and Kullback-Leibler distances. Let P and Q be two measures both dominated by a a finite measure p, H2(P,Q) be the Hellinger distance between P and Q, Consider the Kullback-Leibler distance K f,q) J ln f q) fdp J ln f q) dP. Here P is the probability distribution with density f with respect to the measure j. Let Xi, ,Xn be i.i.d random variables with the common distribution P GP and density f gF, Pn be the empirical distribution Pn A A , where SXj A J, f X A Suppose we have to...

## On statistics of inverse gamma process as a model of wear

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint-Petersburg, harlamov random.ipme.ru Summary. Some aspects of statistics of inverse gamma process as a model of wear are considered. Formulae for finite-dimensional distribution densities of the process are given. Partial derivatives with respect to parameters of one-dimensional densities of both the direct, and inverse processes with independent positive increments are derived. Methods for estimation of...

## T Epq epTZhk YhkIp

Estimation in a Markov chain regression model with missing covariates 103 Here for any pair of adjacent states, h (i,j) G Eo, we have Nh,k (Ip) l(Jm-l,k i, Jm,k j)l(Tm,k G Ip) . With n (9, ft) parameters fixed, the equation VaQn('n 'q) 0 can be solved for ahp. The solution is given by at step (q + 1) of the EM algorithm, we obtain a pseudo- estimate of the hazard rate ah. Set hk,q+l(T,n) Yh,k(u) h,q+1(u,n)du V hp,q (n)Yh,k (Ip) j0 p 1 The profile likelihood score equation for the regression...

## List of Contributors

Arbeev Center for Demographic Studies, Duke University, 2117 Campus Drive, Box 90408, Durham, NC 27708-0408,USA arbeev cds.duke.edu L. S. Arbeeva Ulyanovsk State University, Leo Tolstoi St. 42, 432700 Ulyanovsk, Russia arbeev mail.ru V.N. Ardashev Burdenko Main Military Clinical Hospital, Moscow, Russia Department of Mathematical Statistics, Vilnius University, Lithuania 99 Sant Publique, Universit Victor Universit de Technologie de Compi gne, Laboratoire de Math matiques Appliqu es de...

## Compound Poisson With Exponential Damage

The process Y(t),t > 0 is a homogeneous compound Poisson process if F(t) 1 - e-xt, t > 0. In this case the damage distribution is D(y t) J2P(n At)G(n)(y), (17) where p(n At) e xt- is the probability function of the Poisson distri- bution with mean At. We consider the special case where Yi,Y2, have a common exponential distribution, i.e., G(y) 1 - e- v, y > 0. In this case G(n) (y) is the cdf of the Erlang distribution, and we have G(n)(y) 1 - P(n - 1 py), n > 1, (18) where P( iy) is the...

## Some recent results on joint degradation and failure time modeling

Equipe Statistique Math matique et ses Applications U.F.R. Sciences et Modelisation, Universite Victor Segalen Bordeaux 2 146 rue Leo Saignat 33076 Bordeaux cedex FRANCE couallier sm.u-bordeaux2.fr Key words degradation process, hazard rate, hitting time, regression, correlated errors, generalized least squares estimation, Nelson-Aalen estimate, semiparametric estimation Analyzing survival data is historically and classically based on a sample of n real and non negative random variables (T1,...

## Three Types of Hazard Functions Curves Described

Sidorovich G.I., Shamansky S.V., Pop V.P., Rukavicin O.A. Burdenko Main Military Clinical Hospital, Moscow, Russia name email.address Summary. Not doubts that measures of short-term treatment effects (remission or response rates) are presenting great interest to provide more efficient treatments. However, for all diseases with unfavorable prognosis, to which pertains hemoblasto-sis, life expectancy is the most important feature. The irrevocable decision about the choice between two different...

## L3 Lk fak

Denote the solution to this working likelihood by 3 ( fa1, fa2, , faK). WLW show that when Xk(t Z) Xk(t)exp(faZ), 3 is consistent and asymptotically normal as n > oo that is, 3 3 and ( 3 - 3) N(0, S), where S VD1VVD1, V is a K-dimensional matrix obtained from the working likelihood (see Appendix 1), and Vd is the diagonal matrix with the same diagonal elements as V. WLW also provide a consistent sandwich estimate of S, which we denote by S. WLW propose a directional and omnibus test of Ho,...

## Sequential Analysis of Quality of Life Rasch Measurements

Veronique Sebille1 and Mounir Mesbah2 1 Laboratoire de Biostatistiques, Facult de Pharmacie, Universit de Nantes, 1 rue Gaston Veil, BP 53508, 44035 Nantes Cedex 1, France. veronique.sebille univ-nantes.fr 2 Laboratoire de Statistique Th orique et Appliqu e (LSTA), Universit Pierre et Marie Curie - Paris VI, Bo te 158, - Bureau 8A25 - Plateau A. 175 rue du Chevaleret, 75013 Paris, France mesbah ccr.jussieu.fr Summary. Early stopping of clinical trials either in case of beneficial or deleterious...

## Forward and Backward Recurrence Times and Length Biased Sampling Age Specific Models

Harvard School of Public Health and the Dana-Farber Cancer Institute Boston, MA 02115, U.S.A. name email.address Summary. Consider a chronic disease process which is beginning to be observed at a point in chronological time. The backward recurrence and forward recurrence times are defined for prevalent cases as the time with disease and the time to leave the disease state respectively, where the reference point is the point in time at which the disease process is being observed. In this setting...

## 1 j je

Fisher's information V(X) statistic under 0) Fisher's information V(X) will be V (X) - l (Q, a*) -1 with l (Q,0 -1 l (Q,a*) - l (Q,a*) laa (Q,0 -1 (Q,a*) 8.1 Stopping boundaries for the one-sided SPRT and TT The stopping boundaries, allowing to detect an effect size (ES) with working significance level a and power 1-3 (with 3 a), are Z -a + bV (lower boundary) and Z a + bV (upper boundary) for the one-sided SPRT, Z -a + 3cV (lower boundary) and Z a + cV (upper boundary) for the one-sided...

## Hazards Regression

The two-sample transformation model can be viewed as a special case of the linear transformation model H(T) pz + ae, or, after reparameterization, aH(T) pz + e. Taking H(t) logA(t) and Fe(t) 1 e-e results in the 'a-proportional hazards model' (Hsieh HSI96c ) When p and a are further expressed as p exp(pTz) and a exp(YTx) for two sets of p- and -vectors z and x, model (9) evolves into in terms of the cumulative hazard or (when z and x are time-fixed) X(t z, x) Xo(t) Ao(t)y'TX-1e Tx, (11) in...

## Estimation in a Markov chain regression model with missing covariates

Elashoff1 and Donald L. Morton2 1 Department of Biostatistics, University of California, Los Angeles, CA 90095-1772 2 John Wayne Cancer Institute, Santa Monica, CA 90404 Summary. Markov chain proportional hazard regression model provides a powerful tool for analysis of multiple event times. We discuss estimation in absorbing Markov chains with missing covariates . We consider a MAR model assuming that the missing data mechanism depends on the observed covariates,...