Eric Beutner

Order Restricted Inference for Exponential Step-Stress Models

In the context of multiple step-stress models, which is a special type of accelerated life-testing model, interest lies on the expected lifetimes of the experimental units under different stress levels. Although the expected lifetime is shortened as the stress level increases, this information has not been incorporated so far into the associated inferential procedures. For this reason, we develop here the order restricted maximum likelihood estimation (MLE) for multiple step-stress models with exponentially distributed lifetimes under Type-I, and Type-II censored sampling situations. Moreover, the existence of the unrestricted MLE for a certain stress level is conditional on observing failures at that particular stress level. Under the order restriction, MLE exist even for stress levels without observed failures, provided that these stress levels are internal. We also discuss hypothesis testing problems under order restrictions.

Deriving the asymptotic distribution of U- and V-statistics of dependent data using weighted empirical processes

It is commonly acknowledged that V-functionals with an unbounded kernel are not Hadamard differentiable and that therefore the asymptotic distribution of U- and V-statistics with an unbounded kernel cannot be derived by the Functional Delta Method (FDM). However, in this article we show that V-functionals are quasi-Hadamard differentiable and that therefore a modified version of the FDM (introduced recently in (J. Multivariate Anal. 101 (2010) 2452-2463)) can be applied to this problem. The modified FDM requires weak convergence of a weighted version of the underlying empirical process. The latter is not problematic since there exist several results on weighted empirical processes in the literature; see, for example, (J. Econometrics 130 (2006) 307-335, Ann. Probab. 24 (1996) 2098-2127, Empirical Processes with Applications to Statistics (1986) Wiley, Statist. Sinica 18 (2008) 313-333). The modified FDM approach has the advantage that it is very flexible w.r.t. both the underlying data and the estimator of the unknown distribution function. Both will be demonstrated by various examples. In particular, we will show that our FDM approach covers mainly all the results known in literature for the asymptotia distribution of U- and V-statistics based on dependent data - and our assumptions are by tendency even weaker. Moreover, using our FDM approach we extend these results to dependence concepts that are not covered by the existing literature.

Generalized order statistics: an exponential family in model parameters

Generalized order statistics, and thus sequential order statistics with conditional proportional hazard rates, are shown to form a regular exponential family in the model parameters. This structure is utilized to derive maximum likelihood estimators for these parameters or functions of them along with several properties of the estimators. The Fisher information matrix is stated, and asymptotic efficiency is shown.

When to Wait for More Evidence? Real Options Analysis in Proton Therapy

PURPOSE Trends suggest that cancer spending growth will accelerate. One method for controlling costs is to examine whether the benefits of new technologies are worth the extra costs. However, especially new and emerging technologies are often more costly, while limited clinical evidence of superiority is available. In that situation it is often unclear whether to adopt the new technology now, with the risk of investing in a suboptimal therapy, or to wait for more evidence, with the risk of withholding patients their optimal treatment. This trade-off is especially difficult when it is costly to reverse the decision to adopt a technology, as is the case for proton therapy. Real options analysis, a technique originating from financial economics, assists in making this trade-off. METHODS We examined whether to adopt proton therapy, as compared to stereotactic body radiotherapy, in the treatment of inoperable stage I non-small cell lung cancer. Three options are available: adopt without further research; adopt and undertake a trial; or delay adoption and undertake a trial. The decision depends on the expected net gain of each option, calculated by subtracting its total costs from its expected benefits. RESULTS In The Netherlands, adopt and trial was found to be the preferred option, with an optimal sample size of 200 patients. Increase of treatment costs abroad and costs of reversal altered the preferred option. CONCLUSION We have shown that real options analysis provides a transparent method of weighing the costs and benefits of adopting and/or further researching new and expensive technologies.

Modeling Parameters of a Load-Sharing System Through Link Functions in Sequential Order Statistics Models and Associated Inference

In a multi-sample experiment, we model the parameters of an equal load-sharing system by means of link functions in sequential order statistics models, and then discuss the estimation of these parameters based on a given link function. Different link functions are examined along with the corresponding maximum likelihood estimators, and their properties are studied both analytically and through Monte Carlo simulations.

Continuous mapping approach to the asymptotics of U- and V-statistics

We derive a new representation for U - and V -statistics. Using this representation, the asymptotic distribution of U - and V -statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate U - and V -statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general U - and V -statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of U - or V -statistics occurs at the rate a 3 n and a 4 n , respectively, when a n is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.

ON MOMENT CONDITIONS FOR QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF MULTIVARIATE ARCH MODELS

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest multivariate GARCH model, namely, the multivariate ARCH(1) model of the Baba, Engle, Kraft, and Kroner form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the log-likelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this nonstandard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sufficient for the stationarity of the score when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Walds test, and Raos score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.

Functional delta-method for the bootstrap of quasi-Hadamard differentiable functionals

The functional delta-method provides a convenient tool for deriving the asymptotic distribution of a plug-in estimator of a statistical functional from the asymptotic distribution of the respective empirical process. Moreover, it provides a tool to derive bootstrap consistency for plug-in estimators from bootstrap consistency of empirical processes. It has recently been shown that the range of applications of the functional delta-method for the asymptotic distribution can be considerably enlarged by employing the notion of quasi-Hadamard differentiability. Here we show in a general setting that this enlargement carries over to the bootstrap. That is, for quasi-Hadamard differentiable functionals bootstrap consistency of the plug-in estimator follows from bootstrap consistency of the respective empirical process. This enlargement often requires convergence in distribution of the bootstrapped empirical process w.r.t.\ a nonuniform sup-norm. The latter is not problematic as will be illustrated by means of examples.

The failure of the profile likelihood method for a large class of semi-parametric models

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector θ, say, exceeds the profile likelihood function evaluated at the parameter vector