## A note on the estimation of parameters in a Bernoulli model with dependence.(English)Zbl 0349.62053

Summary: A generalization of a Bernoulli process which incorporates a dependence structure was given by J. Klotz [Ann. Stat. 1, 373–379 (1973; Zbl 0256.62029)], in which he considered $$X_1, X_2, \cdots, X_n$$ as a stationary two-state Markov chain with state space $$\{0, 1\}$$. The parameters of the process are $$p = P(X_i = 1)$$ and $$\lambda$$, which measures the degree of persistence in the chain. Klotz was unable to solve the equations arising from the full likelihood for the M.L.E.’s of $$p$$ and $$\lambda$$, so proposed and investigated an ad hoc procedure. Here explicit solutions are obtained for M.L.E.’s based on a modified likelihood function, where the modification consists of neglecting the first term of the full likelihood. In addition it is observed that Klotz’s equations can in fact be solved explicitly.

### MSC:

 62M05 Markov processes: estimation; hidden Markov models 62F10 Point estimation

Zbl 0256.62029
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