Estimation in the Markov-Pólya scheme

The Markov-Pólya urn scheme is considered, in which the balls are sequentially and equiprobably drawn from an urn initially containing a given number $a_j$ of balls of the $j$th color, $j=1,\dots,N$, and after each draw the ball is returned into the urn together with $s$ new balls of the same color. It is assumed that at the beginning only the total number of balls in the urn is known and one must estimate its structure $\overline\theta=(\theta_1,\dots,\theta_N)$ by observing the frequencies in $n$ trials of the balls of corresponding colors. Various approaches including the Bayes and minimax ones for estimating $\overline\theta$ under a quadratic loss function are discussed. The connection of the obtained results with known ones for multinomial and multivariate hypergeometric distributions is also discussed.