Real valued $M$-estimators ${\widehat{\theta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(\theta \right))$ in a statistical model with observations $Y_i\sim F_{{\theta }_0}$ are replaced by ${ℝ}^p$-valued $M$-estimators ${\widehat{\beta }}_n:=min{\sum }_1^n\rho (Y_i-\tau \left(u\left(z_i^T\beta \right)\right))$ in a new model with observations $Y_i\sim F_{u\left(z_i^t{\beta }_0\right)}$, where $z_i\in {ℝ}^p$ are regressors, ${\beta }_0\in {ℝ}^p$ is a structural parameter and $u:ℝ\to ℝ$ a structural function of the new model. Sufficient conditions for the consistency of ${\widehat{\beta }}_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” ${\widehat{\theta }}_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If...

En este trabajo se acomete una generalización de la definición de Shannon-Lindley para la información esperada proporcionada por un experimento que presupone la existencia de estratificación en el espacio muestral. Ante la evidente dificultad de cálculo de la información esperada en la situación planteada -dificultad que se deriva de la existencia de un vector como parámetro de interés y de un resultado muestral que es un conjunto de muestras obtenidas de poblaciones distintas- en este artículo...

We consider a deconvolution problem of estimating a signal blurred with a random noise. The noise is assumed to be a stationary Gaussian process multiplied by a weight function function εh where h ∈ L2(R1) and ε is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [G.K. Golubev and R.Z. Khasminskii,...

The problems of minimax mutual prediction are considered for binomial and multinomial random variables and for sums of limited random variables with unknown distribution. For the loss function being a linear combination of quadratic losses minimax mutual predictors are determined where the parameters of predictors are obtained by numerical solution of some equations.

The problem of minimax mutual prediction is considered for multinomial random variables with the loss function being a linear combination of quadratic losses connected with prediction of particular variables. The basic parameter of the minimax mutual predictor is determined by numerical solution of some equation.

In this paper, the problem of inference with misclassified multinomial data is addressed. Over the last years there has been a significant upsurge of interest in the development of Bayesian methods to make inferences with misclassified data. The wide range of applications for several sampling schemes and the importance of including initial information make Bayesian analysis an essential tool to be used in this context. A review of the existing literature followed by a methodological discussion is...

In the Bayesian approach, the Bayes factor is the main tool for model selection and hypothesis testing. When prior information is weak, "default" or "automatic" priors, which are typicaIly improper, are commonly used but, unfortunately, the Bayes factor is defined up to a multiplicative constant. In this paper we revise some recent but already popular methodologies, intrinsic and lractional, to deal with improper priors in model selection and hypothesis testing. Special attention is paid to the...

Updating probabilities by information from only one hypothesis and thereby ignoring alternative hypotheses, is not only biased but leads to progressively imprecise conclusions. In psychology this phenomenon was studied in experiments with the “pseudodiagnosticity task”. In probability logic the phenomenon that additional premises increase the imprecision of a conclusion is known as “degradation”. The present contribution investigates degradation in the context of second order probability distributions....

Let X=(X₁,..., Xₙ) be a sample from a distribution with density f(x;θ), θ ∈ Θ ⊂ ℝ. In this article the Bayesian estimation of the parameter θ is considered. We examine whether the Bayes estimators of θ are pointwise ordered when the prior distributions are partially ordered. Various cases of loss function are studied. A lower bound for the survival function of the normal distribution is obtained.