Maximum likelihood estimates with order restrictions on probabilities and odds ratios: A geometric programming approach.
The paper investigates the relation between maximum likelihood and minimum -divergence estimates of unknown parameters and studies the asymptotic behaviour of the likelihood ratio maximum. Observations are assumed to be done in the continuous time.
We consider a Köthe space of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give some applications....
In some cases, the estimators obtained in compound tests have better features than the traditional ones, obtained from individual tests, cf. Sobel and Elashoff (1975), Garner et al. (1989) and Loyer (1983). The bias, the efficiency and the robustness of these estimators are investigated in several papers, e.g. Chen and Swallow (1990), Hung and Swallow (1999) and Lancaster and Keller-McNulty (1998). Thus, the use of estimators based on compound tests not only allows a substantial saving of...
A problem of minimax prediction for the multinomial and multivariate hypergeometric distribution is considered. A class of minimax predictors is determined for estimating linear combinations of the unknown parameter and the random variable having the multinomial or the multivariate hypergeometric distribution.
A class of minimax predictors of random variables with multinomial or multivariate hypergeometric distribution is determined in the case when the sample size is assumed to be a random variable with an unknown distribution. It is also proved that the usual predictors, which are minimax when the sample size is fixed, are not minimax, but they remain admissible when the sample size is an ancillary statistic with unknown distribution.