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What is best?

I have been learning a lot browsing through John D Cook’s blogs.  I found  Are men better than women at chess particularly illuminating, as well as the associated hyperlinks, esp. the 1961 paper he refers to (available in full text).

I reproduce the paper’s equations:

Consider, a random sample of size n  drawn from a standard normal distribution  (x_j \in N(0,1)) and let \phi (x) be the probability density function and and \Phi (x) the corresponding cumulative distribution function. In order to determine the order statistics, consider the sample ordered:

x_{(1)} \leq x_{(2)}\leq \ldots \leq x_{(n)}

The probability distribution of the k-order statistic can be derived (as shown in the paper) as follows:

  • the number of ways of choosing k statistic: \frac{n!}{(k-1)!(n-k)!} (there are (k-1)! ways of arranging the numbers below the k-statistic and (n-k)! ways of arranging the numbers above the k-statistic
  • the density function follows from: k-1 observations are less than y and n-k observations are greater than y and density of y
  • this leads to the probability density distribution:

f(y)= \frac{n!}{(k-1)!(n-k)!}\Phi (y)^{k-1} (1-\Phi(y))^{n-k}\phi (y)

  • the corresponding cumulative distribution function:

F(y)= \frac{n!}{(k-1)!(n-k)!} \int_{-\infty}^y \Phi (x)^{k-1} (1-\Phi (x))^{n-k} \phi (x) \, dx

The n-statistic is the maximum of the sample. As observed in the paper, and is evident from above this yields:

F(y)=\int_{0}^{\Phi (y)}n\Phi ^{n-1}\,d\Phi =\Phi (y)^n

Using the notation for the paper, let \alpha be F(y), e.g median \alpha=0.5.

y = \Phi^{-1} (\alpha^{1/n})

I explored these considerations by simulations and analytically. The 100 samples of sizes 100, 1000 and 10000 yielded the following:

The analytic relationship with the simulation data overlaid is shown in the following graphics (the second logarithmic horizontal axis):

In relation to the observations of John D Cook, the gap between median  of maximums of samples of size n and size 0.1 n are shown in the next graphic:

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Categories: Mathematica, Mathematics
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