Estimation of the ratio of a geometric process

We propose some estimators of the ratio parameter of a geometric process, i.e. of a stochastic process $\{X_{i},i=1,2,\ldots \}$ for which there exists a positive real number $a,$ called the ratio parameter, such that $\{Y_{i}=a^{i-1}X_{i},\,i=1,2,\ldots \}$ forms a renewal process. We assume that the cumulative distribution function $F$ of the random variables $Y_{i},$ $i=1,2, \ldots ,$ is completely unknown. We compare the accuracy of the proposed estimators of the ratio with known estimators given by Lam (1992) and by Chan et al. (2006), and also with the maximum likelihood estimators derived under the assumption that $F$ has a known form.