A homogeneous Poisson process (N(t),t ≥ 0) with the intensity function m(t)=θ is observed on the interval [0,T]. The problem consists in estimating θ with balancing the LINEX loss due to an error of estimation and the cost of sampling which depends linearly on T. The optimal T is given when the prior distribution of θ is not uniquely specified.

The Bayesian sequential estimation problem for an exponential family of processes is considered. Using a weighted square error loss and observing cost involving a linear function of the process, the Bayes sequential procedures are derived.

The method of least wquares is usually used in a linear regression model $𝐘=𝐗\beta +ϵ$ for estimating unknown parameters $\beta$. The case when $ϵ$ is an autoregressive process of the first order and the matrix $𝐗$ corresponds to a linear trend is studied and the Bayes approach is used for estimating the parameters $\beta$. Unbiased Bayes estimators are derived for the case of a small number of observations. These estimators are compared with the locally best unbiased ones and with the usual least squares estimators.

2000 Mathematics Subject Classification: 62E16,62F15, 62H12, 62M20.This paper is concerned with the problem of deriving Bayesian prediction bounds for the future observations (two-sample prediction) from the inverse Weibull distribution based on generalized order statistics (GOS). Study the two side interval Bayesian prediction, point prediction under symmetric and asymmetric loss functions and the maximum likelihood (ML) prediction using "plug-in" procedure for future observations from the inverse...

The first-order autoregressive model with uniform innovations is considered. In this paper, we propose a family of BAYES estimators based on a class of prior distributions. We obtain estimators of the parameter which perform better than the maximum likelihood estimator.

We consider semi-Markov control models with Borel state and action spaces, possibly unbounded costs, and holding times with a generalized exponential distribution with unknown mean θ. Assuming that such a distribution does not depend on the state-action pairs, we introduce a Bayesian estimation procedure for θ, which combined with a variant of the vanishing discount factor approach yields average cost optimal policies.

The paper concerns estimation of the rose of directions of a stationary fibre process in $R^3$ from the intersection counts of the process with test planes. A new approach is suggested based on Bayesian statistical techniques. The method is derived from the special case of a Poisson line process however the estimator is shown to be consistent generally. Markov chain Monte Carlo (MCMC) algorithms are used for the approximation of the posterior distribution. Uniform ergodicity of the algorithms used is...

Hydrology and water resources management are inherently affected by uncertainty in many of their involved processes, including inflows, rainfall, water demand, evaporation, etc. Statistics plays, therefore, an essential role in their study. We review here some recent advances within Bayesian statistics and decision analysis which will have a profound impact in these fields.

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X{̃}_t=\theta ̃X{̃}_{t-1}+\epsilon {̃}_t$. In this paper we study the convergence in distribution of the linear operator $I(\theta {̃}_T,\theta ̃)=(\theta {̃}_T-\theta ̃)\theta {̃}^{T-2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator.

We introduce the function $Z(x;\xi ,\nu ):={\int }_{-\infty }^x\varphi (t-\xi )·\Phi \left(\nu t\right)\text{d}t$, where $\varphi$ and $\Phi$ are the pdf and cdf of $N(0,1)$, respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables...

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1)...

It is important that the estimates of the parameters of an autoregressive moving-average (ARMA) model should satisfy the conditions of stationarity and invertibility. It can be shown that the unconditional maximum-likelihood estimates are bound to fill these conditions regardless of the size of the sample from which they are derived; and, in some quarters, it has been argued that they should be used in preference to any other estimates when the size of he sample is small. However, the maximum-likelihood...

Boolean cluster point processes with various cluster distributions are examined by means of their spherical contact distribution function. Special attention is paid to clusters with strong independence properties (Poisson clusters) and regular clusters.

The first-order autoregression model with heteroskedastic innovations is considered and it is shown that the classical bootstrap procedure based on estimated residuals fails for the least-squares estimator of the autoregression coefficient. A different procedure called wild bootstrap, respectively its modification is considered and its consistency in the strong sense is established under very mild moment conditions.