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.


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


Zbl 0256.62029
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