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3.3.5 Background Corrected Confidence Intervals For Particle Contamination Levels

Hung-kung 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 particle-free 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 Xshas a Poisson distribution with mean $\lambda_s$and the instrument background noise count Xn has a Poisson distribution with mean $\lambda_n$. Since background is measured in a separate experiment, we assume that Xs and Xn are independent random variables. Our parameter of interest is $\lambda=\lambda_s-\lambda_n$, which is known to be non-negative. We constructed an approximate $1-\alpha$ confidence interval for $\lambda$

\begin{displaymath}(X_s-X_n)+\frac{1}{2} q_{\alpha/2}^2\pm
q_{\alpha/2}\sqrt{(X_s+X_n)+\frac{1}{4} q_{\alpha/2}^2}

where $q_{\alpha/2}$ is the $(1-\alpha/2)$ 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 $\alpha=.05$. The values of $\lambda_s$ that we looked at are the integers from 1 to 30, and $\lambda_n$ are all positive integers less than or equal to $\lambda_s$. For each pair of $(\lambda_s,\lambda_n)$, we simulated 10,000 pairs of Poisson variables (Xs,Xn), 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.

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Date created: 7/20/2001
Last updated: 7/20/2001
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