Suppose you drew a random sample from a population. How large could we expect the sample's minimum and maximum to be? Obviously, the sample's minimum and maximum would change every time we drew the sample, but if we drew samples repeatedly, we might expect to see predictable patterns.
In fact, this is the case, and the minimum and maximum are just the two extreme members of the order statistics. The median is also a well-known order statistic, when the sample has odd size. The distributions of all order statistics are known.
If we wanted to determine whether a sample is likely to have been drawn from a normal distribution, for example, we might compare the numbers that were actually observed to the expected values of the order statistics. Too large a departure, and we might suspect that the observations are unlikely to have been drawn from a normal distribution.
This question is often judged by examining a Q-Q (quantile-quantile) plot, or less frequently a P-P (probability-probability) plot. These in turn depend on judgments as to where the order statistics should be plotted. This issue is the unsettled question of plotting positions. There are several plausible formulas for plotting positions that are commonly used. Most have the form Phiinv[(k-a)/(n+1-2a)], where 0<=a<1, k is the index of interest, n is the sample size, and Phiinv is the probit function, the quantile function for the standard normal distribution (and inverse to the cdf of the standard normal). There are strong arguments in favor of a=0 and a=1/2.
A popular estimate (Blom, 1958) of the order statistics of a sample from a normal population makes use of a=3/8. But it turns out this is a somewhat sloppy approximation of the accurate estimate (Elfving, 1947) in which a=pi/8. The approximation depends on the belief that 3 is sufficiently close to pi!
I discussed this question in a post on StatsExchange. Read the original question and the answers here:
http://stats.stackexchange.com/questions/9001/approximate-order-statistics-for-normal-random-variables/152834#152834
In fact, this is the case, and the minimum and maximum are just the two extreme members of the order statistics. The median is also a well-known order statistic, when the sample has odd size. The distributions of all order statistics are known.
If we wanted to determine whether a sample is likely to have been drawn from a normal distribution, for example, we might compare the numbers that were actually observed to the expected values of the order statistics. Too large a departure, and we might suspect that the observations are unlikely to have been drawn from a normal distribution.
This question is often judged by examining a Q-Q (quantile-quantile) plot, or less frequently a P-P (probability-probability) plot. These in turn depend on judgments as to where the order statistics should be plotted. This issue is the unsettled question of plotting positions. There are several plausible formulas for plotting positions that are commonly used. Most have the form Phiinv[(k-a)/(n+1-2a)], where 0<=a<1, k is the index of interest, n is the sample size, and Phiinv is the probit function, the quantile function for the standard normal distribution (and inverse to the cdf of the standard normal). There are strong arguments in favor of a=0 and a=1/2.
A popular estimate (Blom, 1958) of the order statistics of a sample from a normal population makes use of a=3/8. But it turns out this is a somewhat sloppy approximation of the accurate estimate (Elfving, 1947) in which a=pi/8. The approximation depends on the belief that 3 is sufficiently close to pi!
I discussed this question in a post on StatsExchange. Read the original question and the answers here:
http://stats.stackexchange.com/questions/9001/approximate-order-statistics-for-normal-random-variables/152834#152834
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