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# mle - Outline Maximum Likelihood Estimation November 8 2011...

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Outline Maximum Likelihood Estimation November 8, 2011 Maximum Likelihood Estimation

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Outline Outline 1 Maximum Likelihood Estimation - introduction 2 Properties of MLEs 3 Delta Method Maximum Likelihood Estimation
Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method Overview Up to now, we’ve been dealing with ad hoc point estimators Largely, we’ve dealt the sample mean, sample variance and sample proportion To some extent these are “natural” quantities that don’t require much justification In other cases, it may not be clear exactly how to estimate some parameter Maximum Likelihood Estimation

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Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method Overview-Example For instance, assume we have 2 groups of people and they either experience an outcome or don’t For group 1: X 1 . . . X n 1 iid Bernoulli ( p 1 ) For group 2: Y 1 . . . X n 2 iid Bernoulli ( p 2 ) If we care about the risk difference (RD), we learned how to estimate this ˆ RD = ˆ p 1 - ˆ p 2 and that this is unbiased We also computed the variance What if we care about RR= p 1 / p 2 or OR = p 1 / ( 1 - p 1 ) p 2 / ( 1 - p 2 ) ? Is there a natural estimate? Is it unbiased? And how about the variance? Maximum Likelihood Estimation
Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method The solution to this (and other) problems was developed by RA Fisher Fisher was a renowned jerk Early pioneer or the eugenics movement Ardent campaigner for sterilization Hated a fellow statistician (Karl Pearson) so vehemently he continued to attack him years after his death Famously, took money from tobacco companies and opposed all evidence connecting smoking to lung cancer He was, however, a noted geneticist and statistician Maximum Likelihood Estimation

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Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method Maximum Likelihood Estimation (MLE) is the most common way of deriving point estimators Developed by RA Fisher in the 1920s It is a general and flexible framework within which to operate It is at the core of frequentist statistics As such, it is likely to be the basis for the rest of your biostats and econometrics courses Maximum Likelihood Estimation
Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method Example MLE, more generally Variance estimates for MLEs Outline 1 Maximum Likelihood Estimation - introduction 2 Properties of MLEs 3 Delta Method Maximum Likelihood Estimation

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Maximum Likelihood Estimation - introduction Properties of MLEs Delta Method Example MLE, more generally Variance estimates for MLEs An example Imagine that we’re planning a study where we’ll enroll 3 men with the aim of estimating the proportion of the population who are obese (We clearly didn’t learn anything about sample size from the last lecture) What’s the pdf for the data before we collect it?
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