Zdzisław Porosiński

On robust Bayesian estimation under some asymmetric and bounded loss function

The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.

On best choice problems having similar solutions

The purpose of the paper is to point out that best choice problems with different information structure may have similar solutions. A full-information best choice problem with a random number of objects having uniform distribution is considered. An optimal stopping rule, determined by decreasing sequence of levels, is found. Asymptotic behaviour of both an optimal stopping rule and a winning probability is examined in detail. Both the sequence of optimal levels determining optimal strategies and asymptotic winning probabilities are the same in the considered problem as well as in a best choice problem with partial information considered by Petruccelli (Ann. Statist. 8 (1980) 1171-1174).

Characterization of the monotone case for a best choice problem with a random number of objects

A class of distributions of N is characterized for a full-information best choice problem with a random number of objects N for which the so-called monotone case occurs. This class is shown to be a subclass of IFR distributions.

Trybula's papers related to optimal control

W pewnym okresie Trybula zwrocil uwage na zagadnienia adaptacyjnego sterowania (patrz [20, 21]). Wydaje sie, ze zainspirowala Go monografia Aoki [1]. Do tematu wrocil po dośc dlugim czasie. Zauwazyl, ze w literaturze zaklada sie, iz zaklocenia w systemach stochastycznych mają charakter gaussowski, podczas gdy w praktyce sygnaly, a wiec i zaklocenia, są dyskretne. Przypominamy tutaj typowy model analizowany w tej serii prac jako, ze wspolczesne zastosowania modeli liniowych w ekonomii i technice wymuszają sygnaly zarowno typu ciąglego, jak i dyskretne. Ograniczymy sie do szczegolowego przedstawienia konstrukcji sterowan bayesowskich przy kwadratowej funkcji kosztu i zakloceniach z wykladniczej klasy, spelniających dodatkowe warunki nalozone na momenty. W konkluzji podajemy odsylacze do prac, w ktorych wyznaczono sterowania minimaksowe. Slowa kluczowe: system liniowy, zaklocenia addytywne, wykladnicza klasa rozkladow, sterowanie bayesowskie, sterowanie minimaksowe.

Modified Strategies in a Competitive Best Choice Problem with Random Priority

A zero-sum game version of the full-information best choice problem is considered. Two players observe sequentially a stream of iid random variables (objects) from a known continuous distribution appearing according to some renewal process with the object of choosing the largest one. The horizon of observation is a positive random variable independent of objects. The observation of the random variables is imperfect and the players are informed only whether the object is greater than or less than some levels specified by both of them. Each player can choose at most one object. If both want to accept the same object, a random assignment mechanism is used. If some Player accepts an object, the other Player can change his level and continues the game alone. A similar game with discrete time and random number of objects is considered as a dual problem. The normal form of the game is derived. For the Poisson stream and the exponential horizon the value of the game and the form of the equilibrium strategy are obtained. In discrete-time case a game with geometric number of objects is completely solved.

The Full-Information Best Choice Problem with Two Choices

The following full-information best choice problem was studied by Gilbert and Mosteller (1966). A known number, say N, of iid rv’ s Y1, Y2,..., YN from a known continuous df F is observed sequentially. The objective is to maximize the probability of choosing the largest. After Yn is observed it must be chosen (and the observation is terminated) or rejected (and the observation is continued). Neither recall nor uncertainty of selection is allowed and one choice must be made.