Stat 544 Spring 2006
Mini-Project #2
The US government regulates the amounts of 6 pollutants that may be legally emitted by
a type of internal combustion engine. A manufacturer of such engines periodically pays
for a very expensive set of tests to determine emission levels for an engine sampled from
its production line. Below are test results for n = 6 engines. These are (coded) logs of
measured emission levels x1 , . . . , x6 for these selected engines. Notice that complete test
results are available on only 4 of the engines.
Engine
x1
x2
x3
x4
x5
x6
1
.95
.50
.02
2
.88
.50
.92
3
−1.41 −1.78 −.94 −1.26 −1.41 −1.72
4
.36 −.14 1.41
.50
.39
.52
5
.93
.32 −.32
1.02
.92
.80
6
.11 −.22 −.14 −1.26
.11 −.53
On the same (coded log) scales as are used in the table above, regulatory limits on the
various kinds of pollutants are
Variable
x1 x2 x3 x4 x5 x6
Regulatory Limit (U ) 3.9 9.1 6.6 2.4 2.7 1.5
It is a plausible assumption that logarithms of measured pollutant levels for these engines
are multivariate normal. Of primary concern is the fraction of engines produced by this
company meeting emission standards. For the single pollutant i, this is the parametric
function
pi (μi , σ i ) = Pμi ,σi [Xi < Ui ]
¶
µ
Ui − μi
= Φ
σi
For
I [Xi < Ui ] =
½
1 if Xi < Ui
0 if Xi ≥ Ui
the fraction of engines produced by this company simultaneously meeting all 6 emission
standards is
" 6
#
Y
p (μ, Σ) = Pμ,Σ
I [Xi < Ui ] = 1
i=1
(This is the value of a multivariate normal cdf that does not in general have a convenient
closed form.)
Begin by considering the emission levels of the pollutants one at a time. Make and interpret
an appropriate (univariate) Bayes analysis for each pi (μi , σ i ). (Note that WinBUGS has a
standard normal cdf.)
1
Then consider the joint behavior of the various types of engine emissions. Make what seem
to be minimally informative prior assumptions on the 6-dimensional mean vector μ and the
covariance matrix Σ (you will probably want to investigate how strongly your choices of prior
parameters affect the posteriors). Then consider p (μ, Σ). You can’t compute this directly
inside WinBUGS but for fixed μ, Σ you can approximate it by simulating
a large number
Q6
of multivariate normal vectors and finding the fraction that have i=1 I [Xi < Ui ] = 1.
Make and interpret an appropriate Bayes analysis for p (μ, Σ) based on such approximate
evaluation of p (μ, Σ).
Suppose that this manufacturer is planning a design change in these engines. A single
prototype will be made and tested in an attempt to learn something about the fraction of
engines of the new design that would meet all 6 emission limits. What would you suggest
for a prior for a Bayes analysis of those test results?
Limit what you type up to turn in to a cover page plus at most 6 typewritten pages (including
whatever figures you want to include). Use at least 11 point fonts and 1 inch left and right
margins. Also include an Appendix with "commented" WinBUGS and/or R code that you
have used. (This Appendix does not count in the above "6 typewritten pages" limit.)
2