مرکز منطقه ای اطلاع رساني علوم و فناوري - Confidence intervals based on one or more observations

Abstract :

On the basis of $n\\geq 2$ observations, confidence limits of the form $\\overline{X} \\pm tS/ \\sqrt {n}$ are constructed for the location (e.g., the median) of any distribution of known form with unknown location and dispersion (scale), where $\\overline{X}$ and $S$ are the sample mean and "unbiased" standard deviation. Particular attention is given to the values of $t$ needed for the Cauchy and uniform distributions. The latter $t$ suffices for any (unknown) symmetric unimodal distribution if $t \\geq n - 1$ . A table compares these values of $t$ for $n=2,3,4$ , and $5$ with those for the normal case, which are derived here very simply and are identical with those found by "Student." We are also able to include the case of a single observation $(n=1)$ , where confidence intervals of various forms are made just wide enough for the least favorable dispersion. They, therefore, include the true location with at least but, in general, not exactly the desired probability; these intervals involve a predetermined value that plays a role reminiscent of but quite different from that of the prior distribution that would enter into a Bayesian analysis. In addition, upper confidence limits for the dispersion are constructed for $n \\geq 1$ .