Narrow Results

We consider the state complexity of several combined operations. Those results show that the state complexity of a combined operation is in general very different from the composition of the state complexities of the participating individual operations. We also consider general estimation methods for the state complexity of combined operations. In particular, estimation through nondeterministic state complexity is studied. It is shown that the method is very promising for a large class of combined operations.

The problem of statistical modeling of the geometric count data with a specific probability model of lifetimes is of interest and importance in reliability. In this paper, we construct a geometric process (GP), with parameter $a$, for modeling the geometric count data when the distribution of first occurrence time is a scaled Muth with parameters $\lambda$ and $\beta$. We investigate the estimators of the process parameters $a$, $\lambda$ and $\beta$ from a point of approximations of classical and modified approach by using the different estimation methodologies such as the maximum likelihood, moments, least-squares and maximum spacing. We perform a simulation study to compare the estimation performance of the estimators obtained. Finally, we provide an illustrative analysis conducted on a real-world dataset to show the efficiency of the GP model constructed in this paper against the alpha-series and renewal processes and exemplify the data modeling stages. Consequently, a forecasting to such data using the GP with the scaled Muth is investigated.