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.

Some problems of point and interval prediction in a trend-renewal process (TRP) are considered. TRP's, whose realizations depend on a renewal distribution as well as on a trend function, comprise the non-homogeneous Poisson and renewal processes and serve as useful reliability models for repairable systems. For these processes, some possible ideas and methods for constructing the predicted next failure time and the prediction interval for the next failure time are presented. A method of constructing the predictors is also presented in the case when the renewal distribution of a TRP is unknown (and consequently, the likelihood function of this process is unknown). Using the prediction methods proposed, simulations are conducted to compare the predicted times and prediction intervals for a TRP with completely unknown renewal distribution with the corresponding results for the TRP with a Weibull renewal distribution and power law type trend function. The prediction methods are also applied to some real data.

The problem of finding optimal stopping times and the corresponding sequential estimators for the unknown parameter vector p = (p 1 , . . . , p m ) of m independent multinomial distributions M(1, p i ), i = 1, . . . , m, is considered in the special case when the data arrive at random times. In the problem of finding optimal sequential estimation procedures, the intermediate approach between the Bayes and the minimax principle is applied in which it is assumed that a vague prior information on the distribution of the unknown parameter is available. It is supposed that the set of all prior distributions of the parameter p is restricted to a set consisting of the priors for which certain conditions on the moments are imposed. Using the weighted squared error loss functions defined by Eqs. (4) and (13) and the observation cost determined by a convex function of the moment of stopping, several classes of -minimax sequential procedures for estimating p are established, respectively, to different conditions imposed on the set of prior distributions.