A Central Tolerance Region for the Multivariate Normal Distribution†

Abstract

In a multivariate normal distribution the probability density takes any particular value at all points on the surface of an ellipsoid whose centre is the mean of the distribution. Among such ellipsoids there is one,R, say, that contains 100α per cent of the individuals in the population. The paper deals with the problem of determining from a random sample a region which containsRwith prescribed probability β. In the univariate case,Rreduces to an interval with the mean as midpoint. Any linear function of variables whose joint distribution is normal has again a normal distribution. LetTadenote the interval corresponding toRin the case of the linear combination with coefficients vectora. A method of determining from the sample intervals which simultaneously includeTafor allawith probability β is indicated.