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Showing posts from May, 2016

### The James-Stein Estimator

This write up is about an estimator. A statistical estimator, is used when you already have a model in mind, data at hand and want to estimate some parameters needed for the model. For example, you want to predict how many runs a batter would make in a game given recent history $$x_1,x_2,x_3,\ldots$$ . If we assume (we are making a choice of a model now) that the scores come from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$ then the probability density function for a given value $$x$$ is

The likelihood that a series of points $$x_1,x_2,x_3,\ldots$$ come from such a distribution can be expressed as

Next is basic calculus. Take the logarithm on both sides, set the partial derivative w.r.t. $$\mu$$ to zero yields (excluding the algebra)

To verify, you also need to check the second derivative's sign to see if its negative to ensure that it is indeed a maxima you have found.

So a simple estimator would be to use the average runs scored from the past $$n$$ days.…