## Generalized Maximum Likelihood Estimator

Cosslett (1983) proposed maximizing the log likelihood function (9.2.7) of a binary QR model with respct to P and F, subject to the condition that F is a distribution function. The log likelihood function, denoted here as у/, is

¥(fi, Л = І {у, log F(x’fl) + (1 – y,) log [1 – F(x’M). (9.6.33)

f-i

The consistency proof of Kiefer and Wolfowite (1956) applies to this kind of model. Cosslett showed how to compute MLE fi and F and derived conditions for the consistency of MLE, translating the general conditions of Kiefer and Wolfowitz into this particular model. The conditions Cosslett found, which are not reproduced here, are quite reasonable and likely to hold in most practical applications.

Clearly some kind of normalization is needed on fi and /’before we maximize (9.6.33)...

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