Exact Statistics - The Approach

The Approach

All classical statistical procedures are constructed using statistics which depend only on observable random vectors, whereas Generalized Estimators, Tests, and Confidence Intervals used in exact statistics take advantage of the observable random vectors and the observed values both, as in the Bayesian approach but without having to treat constant parameters as random variables. For example, in sampling from a normal population with mean and variance, suppose and are the sample mean and the sample variance. Then, it is well known that

and that

Now suppose the parameter of interest is the coefficient of variation, . Then, we can easily perform exact tests and exact confidence intervals for based on the generalized statistic

$R = \frac {\overline{x} S} {s \sigma} - \frac{\overline{X}- \mu} {\sigma} = \frac {\overline{x}} {s} \frac {\sqrt{U}} {\sqrt{n}} ~-~ \frac {Z} {\sqrt{n}}$ ,

where is the observed value of and is the observed value of . Exact inferences on based on probabilities and expected values of are possible because its distribution and the observed value are both free of nuisance parameters.

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