
3.3.5 Background Corrected Confidence Intervals For Particle Contamination Levels
Hungkung Liu Statistical Engineering Division, CAML
Kensei Ehara National Research Laboratory of Metrology, Japan In particle contamination monitoring using such instruments as laser particle counters, condensation nucleus counters, and liquid particle counters, typically an additive background noise contributes to the particle count. It frequently occurs that particlefree gas or liquid is available with which one can estimate the frequence of the instrument noise in a separate experiment.
Assume that the background noise inflated sample particle count X_{s}has a Poisson distribution with mean and the instrument background noise count X_{n} has
a Poisson distribution with mean .
Since background is measured in a separate experiment,
we assume that X_{s} and X_{n} are independent random variables.
Our parameter of interest is
,
which is known
to be nonnegative.
We constructed an approximate
confidence interval for
where is the normal quantile. This background corrected confidence interval for the contamination level is easy to compute. Compared to the uncorrected one sample Poisson confidence interval, it is reasonably shifted to correct for the background. And when there is no background noise, the proposed confidence interval degenerates to the standard one sample Poisson confidence interval.
We ran simulations to check the coverage probability for the
proposed interval at
.
The values of
that we looked at are the integers
from 1 to 30,
and
are all positive integers less than or
equal to .
For each pair of
,
we simulated 10,000 pairs of
Poisson variables (X_{s},X_{n}), and constructed 10,000 confidence intervals.
The simulated coverage probability is the coverage frequency of these
10,000 confidence intervals.
The following figure summarizes the simulation results.
Figure 22: Simulated coverage probability for the proposed 95% confidence interval.
Date created: 7/20/2001 