
3.4.4 The Joint Distribution of a Sample Mean and an Extremum, With Applications to Quality Control
Mark G. Vangel Statistical Engineering Division, ITL In some industrial applications one compares a sample mean and minimum, or a mean and maximum, to reference values, and determines if the lot from which the sample was taken is acceptable, or if further investigation of this lot is indicated. For example, emphasis has long been placed on checking sample means and minima of lots of various packaged goods. Also, the sample means and minima are used in the testing of batches of raw material by many manufacturers of composite materials. And means and maxima of power loss of sampled motors have been recently proposed for use in testing whether manufactured motors comply with labeled motor efficiencies. Because the exact joint distribution of an extremum and the mean of a sample is usually complicated, establishing these reference values using statistical considerations typically involves crude approximations or simulation, even under the assumption of normality. Saddlepoint approximations, however, can be used to develop fairly simple and very accurate approximations to the joint cdf of the mean and an extremum.
Let
denote the normal density, and let
be the normal hazard function. For a normal model, the joint cdf of the mean and the minimum is well approximated by where
and where t_{*} is the (unique) solution to the equation In the figure, this approximation is compared with the contours of the exact distribution for a sample size of 2. A corresponding approximation has also been obtained for an exponential model. Figure 28: Contours of the joint distribution of the mean and the minimum (broken lines), along with the saddlepoint approximation (solid lines), for a sample size of 2 from a normal population. The approximation is already very accurate, and this accuracy improves rapidly with increasing sample size.
Date created: 7/20/2001 