Statistical Inference in Bernoulli Trials with Dependence

Abstract

A model for Bernoulli trials with Markov dependence is developed which possesses the usual frequency parameter $p = P\lbrack X_i = 1\rbrack$ and an additional dependence parameter $\lambda = P\lbrack X_i = 1 \mid X_{i-1} = 1\rbrack$. Sufficient statistics for the model with $p$ and $\lambda$ unknown are found and an exact closed form expression for their small sample joint distribution is given. Large sample distribution theory is also given and small sample variances compared with large sample approximations. Easily computed estimators of $p$ and $\lambda$ are recommended and shown to be asymptotically efficient. With $p$ unknown the u.m.p. unbiased test of independence is noted to be the run test. An application to a rainfall example is given.