By Rainer Winkelmann; Stefan Boes

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Is the ML estimator a good estimator to use? Is it unbiased? Is it consistent? Does it use the information provided in the data eﬃciently? What is its distribution? One important aspect of ML estimation is that the method indeed has desirable properties and a known sampling distribution, regardless of the particulars of the probability model, as long as the model is correctly speciﬁed. For then, the maximum likelihood estimator is 1. consistent 2. asymptotically normal 3. eﬃcient Consistency means that, as the sample size increases, the ML estimator tends in probability toward the true parameter value.

Yi |xi is Bernoulli distributed with parameter πi = exp(xi β)/[1+exp(xi β)]. • yi |xi is Poisson distributed with parameter λi = exp(xi β) • yi |xi is normally distributed with parameters µi = xi β and σ 2 . In order to accommodate such models within the previous framework, we have to extend the assumption of random sampling to pairs of observations (yi , xi ), requiring that the i-th draw is independent from all other draws i = i. All we need to do then is to replace the marginal probability or density function f (yi ; θ) with the conditional probability or density function f (yi |xi ; θ) implied by the model.

1. Sampling from a Bernoulli Distribution (Part I) Assume that a random sample of size n has been drawn from a Bernoulli distribution with parameter π. Then the likelihood function and the log-likelihood function have the form n (1 − π)1−yi π yi L(π; y) = i=1 n (1 − yi ) log(1 − π) + yi log π log L(π; y) = i=1 In order to illustrate that L(π; y1 , . . 1 plots the likelihood function for two diﬀerent samples of size n = 5. The sample (0, 0, 0, 1, 1) has the likelihood function L1 (π) = (1−π)3 π 2 .