Estimation of parameters for trend-renewal processes

Methods of estimating unknown parameters of a trend function for trend-renewal processes are investigated in the case when the renewal distribution function is unknown. If the renewal distribution is unknown, then the likelihood function of the trend-renewal process is unknown and consequently the maximum likelihood method cannot be used. In such a situation we propose three other methods of estimating the trend parameters. The methods proposed can also be used to predict future occurrence times. The performance of the estimators based on these methods is illustrated numerically for some trend-renewal processes for which the statistical inference is analytically intractable.     

Pmc-optimal nonparametric quantile estimator

According to Pitman's Measure of Closeness, if T₁and T₂are two estimators of a real parameter $[d], then T₁is better than T₂if Po[d]{\T₁-o[d] < \T₂-0[d]\} > 1/2 for all 0[d]. It may however happen that while T₁is better than T₂and T₂is better than T₃, T₃is better than T₁. Given q ? (0,1) and a sample X₁, X₂, ..., X_nfrom an unknown F ? F, an estimator T* = T*(X₁,X₂...X_n)of the q-th quantile of the distribution F is constructed such that P_F{\F(T*)-q\ <[d] \F(T)-q\} >[d] 1/2 for all F?F and for all T€T, where F is a nonparametric family of distributions and T is a class of estimators. It is shown that T* =X_j:n'for a suitably chosen jth order statistic.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

Parameter estimation for switched counting processes

The paper investigates parameter estimation problems in special Markov modulated counting processes. The events occuring at any state of an underlying Markov chain can be equipped with marks performing additional information on the events. Specifying the model to the case of two-state Markov chain modulation, the so-called switched counting process, some statistical problems are studied:maximum likelihood estimators, Rao-Blackwell optimal estimators, test of equality of the counting intensities of the two states and minimax estimation procedures. Tne consideration could be applied in various practical problems, in particular, in queueing and in reliability models, for example in failure-repair processes with alternatively operating repair systems.     

Optimal sequential estimation procedures of a function of a probability of success under LINEX loss

In this paper, we investigate the problem of estimating a function g(p), where p is the probability of success in a sequential sample of independent identically Bernoulli distributed random variables. As a loss associated with estimation we introduce a generalized LINEX loss function. We construct a sequential procedure possessing some asymptotically optimal properties in the case when p tends to zero. In this approach to the problem, the conditions are given, under which the stopping time is asymptotically efficient and normal, and the corresponding sequential estimator is asymptotically normal. The procedure constructed guarantees that its sequential risk is asymptotically equal to a prescribed constant.     

Estimation procedures with delayed observations

The statistical model is considered in which the collection of data from several independent populations is available only at random times determined by order statistics of lifetimes of a given number of objects. Each of the populations is distributed according to a general multiparameter exponential family. The problem is to estimate the mean value vector parameter of the multiparameter exponential family of distributions of the forthcoming observations. Under the loss function involving a weighted squared error loss, the cost proportional to the events appeared and a cost of observing the process, a class of optimal sequential procedures is established. The procedures are derived in two situations: when the lifetime distribution is completely known and in the case when it is unknown but assumed to belong to an exponential subfamily with an unknown failure rate parameter.     

A Distribution-Free Median-Unbiased Quantile Estimator

Given λ∈(0-,l), let x_λ(F) denote the unique λ-quantile of the distribution F. A distribution-free median-unbiased estimator of x_λ(F) is explicitly constructed     

Two-phase nonlinear regression with smooth transition

A method of estimating parameters of twophase nonlinear regression with smooth transition between phases is described. It consists of two stages, both utilizing the least square fit. In the first one, each phase is fitted separately and, simultaneously, a transition point is determined. In the second stage, the two phases are joined smoothly by a proper transition function, which depends on the transition point chosen by the grid search. The practical aspects of the method proposed are demonstrated on the data concerning the soil bulk density in dependence on the soil water content.