SED navigation bar go to SED home page go to SED publications page go to NIST home page SED Home Page SED Contacts SED Projects SED Products and Publications Search SED Pages

contents     previous     next

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 $\phi(t)=\Phi^{\prime}(t)$ denote the normal density, and let

\begin{displaymath}h(t) = \frac{\phi(t)}{1-\Phi(t)}

be the normal hazard function. For a normal model, the joint cdf of the mean and the minimum is well approximated by

\begin{displaymath}F_{X_{(1)},\bar{X}}(t_1,t_2) = \frac{
A(t) dt}
{\int_{-\infty}^{\infty}A(t) dt},

A(t) = $\displaystyle h^{-(n-1)}(t)
\exp\left[\frac{(n-1)^2}{2n}(h(t)-t)^2 +
(n-1) t(h(t)-t)\right]$  
    $\displaystyle \cdot \sqrt{1-h^2(t)+t h(t)},$  

and where t* is the (unique) solution to the equation

\begin{displaymath}\frac{n-1}{n}(h(t_*)-t_*) = t_2-t_1.

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.

contents     previous     next

Date created: 7/20/2001
Last updated: 7/20/2001
Please email comments on this WWW page to