Course overview -- Wintersemester 2011/2012 (Archive)

Klassische Permutationstests für Zweistichprobenprobleme (wie Pitman's Permutationstests zum Vergleich von Mittelwerten) besitzen im Gegensatz zu vielen anderen Resamplingverfahren den Vorteil, dass sie das Niveau auch für endliche Stichproben exakt einhalten, falls die Daten austauschbar sind. In heterogenen Modellen ist dies allerdings nicht der Fall und der Test hält häufig nicht einmal asymptotisch die vorgegebene Fehlerwahrscheinlichkeit 1.Art ein. Der Vortrag gibt hierfür eine theoretische Erklärung und zeigt, wie dieses Problem in verschiedenen Situationen gelöst werden kann.

The hazard has been the basis for non- and semiparametric estimation of survival with (right) censored and (left) truncated data, as reflected in the dominant use of the Kaplan-Meier estimator and the Cox proportional hazards model. We show that the Kaplan-Meier has an equivalent representation as an inverse probability weighted empirical cumulative distribution function (ipw-ecdf), which has an immediate extension to the competing risks setting. The weights in this ipw-ecdf form suggest an estimator of the subdistribution hazard in the situation of competing risks. The resulting nonparametric product-limit estimator is also equivalent to the standard nonparametric estimator for the cause specific cumulative incidence function. Furthermore, by using the proper filtration, a martingale property is derived for the weighted counting process that corresponds to the subdistribution. Using this martingale property, several results and proofs from standard survival analysis are easily extended to the competing risks setting. As an example, we briefly discuss asymptotics in the proportional subdistribution hazards model.

The NPMLE of a distribution function under random one-sided (left or right) truncation is the well-known Lynden-Bell estimator. However, in some applications, left and right truncation occur simultaneously, and Lynden-Bell is inconsistent. In Survival Analysis and Epidemiology, this double truncation is typically encountered when the recruited inter-event times correspond to terminating events (e.g. cancer or AIDS diagnosis) falling between two fixed dates. Another interesting example of random double truncation is the problem of two-sided detection limits for quasar luminosities in Astronomy. Under double truncation, the NPMLE has no explicit form, and its computation and the derivation of its statistical properties are far from obvious. In this talk I will review the Efron-Petrosian NPMLE and a semiparametric counterpart which is based on a parametric specification of the joint distribution of the (random) truncation limits. Relative advantages and disadvantages will be discussed, as well as the existing software for the computation of the estimators. Time permitting, I will also consider the problem of estimating the underlying density. Extensive simulations and applications to epidemiological and astronomical data will be provided. This is joint work with Carla Moreira (University of Vigo) and Rosa Crujeiras (University of Santiago de Compostela).

In this talk I will discuss the problem of Optimal Stopping (OS) of Markov Chains (MCs), the methods for its solution, the classical and the generalized Gittins indices and related problems: the Katehakis-Veinott Restart Problem and the Whittle family of Retirement Problems. The celebrated Gittins index, its generalizations and related techniques play an important role in applied probability models, resource allocation problems, optimal portfolio management problems as well as other problems of financial mathematics. It is well known that a connection exists between the Ratio (cycle) maximization problem, the Katehakis-Veinott (KV) Restart Problem and the Whittle family of Retirement Problems, and that their key characteristics, the classical Gittins index, the KV index, and the Whittle index are equal in a classical setting. These indices were generalized by the author (Statistics and Probability Letters, 2008) in such a way that it is possible to use the so called State Elimination algorithm, developed earlier to solve the OS of MCs problem to calculate this common index. One of the goals of this talk is to demonstrate also that the equality of these indices is a special case of a similar equality for three simple abstract optimization problems. A more general - continue, quit, restart problem will be also discussed.