If x contains any missing (NA), undefined (NaN) or
infinite (Inf, -Inf) values, they will be removed prior to
performing the estimation.
A tolerance interval for some population is an interval on the real line constructed so as to
contain \(100 \beta \%\) of the population (i.e., \(100 \beta \%\) of all
future observations), where \(0 < \beta < 1\). The quantity \(100 \beta \%\) is called
the coverage.
There are two kinds of tolerance intervals (Guttman, 1970):
A \(\beta\)-content tolerance interval with confidence level \(100(1-\alpha)\%\) is
constructed so that it contains at least \(100 \beta \%\) of the population (i.e., the
coverage is at least \(100 \beta \%\)) with probability \(100(1-\alpha)\%\), where
\(0 < \alpha < 1\). The quantity \(100(1-\alpha)\%\) is called the confidence level or
confidence coefficient associated with the tolerance interval.
A \(\beta\)-expectation tolerance interval is constructed so that the average coverage of
the interval is \(100 \beta \%\).
Note: A \(\beta\)-expectation tolerance interval with coverage \(100 \beta \%\) is
equivalent to a prediction interval for one future observation with associated confidence level
\(100 \beta \%\). Note that there is no explicit confidence level associated with a
\(\beta\)-expectation tolerance interval. If a \(\beta\)-expectation tolerance interval is
treated as a \(\beta\)-content tolerance interval, the confidence level associated with this
tolerance interval is usually around 50% (e.g., Guttman, 1970, Table 4.2, p.76).
Because of the discrete nature of the Poisson distribution,
even true tolerance intervals (tolerance intervals based on the true value of
\(\lambda\)) will usually not contain exactly \(\beta\%\) of the population.
For example, for the Poisson distribution with parameter lambda=2, the
interval [0, 4] contains 94.7% of this distribution and the interval [0, 5]
contains 98.3% of this distribution. Thus, no interval can contain exactly 95%
of this distribution.
\(\beta\)-Content Tolerance Intervals for a Poisson Distribution
Zacks (1970) showed that for monotone likelihood ratio (MLR) families of discrete
distributions, a uniformly most accurate upper \(\beta100\%\) \(\beta\)-content
tolerance interval with associated confidence level \((1-\alpha)100\%\) is
constructed by finding the upper \((1-\alpha)100\%\) confidence limit for the
parameter associated with the distribution, and then computing the \(\beta\)'th
quantile of the distribution assuming the true value of the parameter is equal to
the upper confidence limit. This idea can be extended to one-sided lower and
two-sided tolerance limits.
It can be shown that all distributions that are one parameter exponential families
have the MLR property, and the Poisson distribution is a one-parameter exponential
family, so the method of Zacks (1970) can be applied to a Poisson distribution.
Let \(X\) denote a Poisson random variable with parameter
lambda=\(\lambda\). Let \(x_{p|\lambda}\) denote the \(p\)'th quantile
of this distribution. That is,
$$Pr(X < x_{p|\lambda}) \le p \le Pr(X \le x_{p|\lambda}) \;\;\;\;\;\; (1)$$
Note that due to the discrete nature of the Poisson distribution, there will be
several values of \(p\) associated with one value of \(X\). For example, for
\(\lambda=2\), the value 1 is the \(p\)'th quantile for any value of \(p\)
between 0.140 and 0.406.
Let \(\underline{x}\) denote a vector of \(n\) observations from a
Poisson distribution with parameter lambda=\(\lambda\).
When ti.type="upper", the first step is to compute the one-sided upper
\((1-\alpha)100\%\) confidence limit for \(\lambda\) based on the observations
\(\underline{x}\) (see the help file for epois). Denote this upper
confidence limit by \(UCL\). The one-sided upper \(\beta100\%\) tolerance limit
is then given by:
$$[0, x_{\beta | \lambda = UCL}] \;\;\;\;\;\; (2)$$
Similarly, when ti.type="lower", the first step is to compute the one-sided
lower \((1-\alpha)100\%\) confidence limit for \(\lambda\) based on the
observations \(\underline{x}\). Denote this lower confidence limit by \(LCL\).
The one-sided lower \(\beta100\%\) tolerance limit is then given by:
$$[x_{1-\beta | \lambda = LCL}, \infty] \;\;\;\;\;\; (3)$$
Finally, when ti.type="two-sided", the first step is to compute the two-sided
\((1-\alpha)100\%\) confidence limits for \(\lambda\) based on the
observations \(\underline{x}\). Denote these confidence limits by \(LCL\) and
\(UCL\). The two-sided \(\beta100\%\) tolerance limit is then given by:
$$[x_{\frac{1-\beta}{2} | \lambda = LCL}, x_{\frac{1+\beta}{2} | \lambda = UCL}] \;\;\;\;\;\; (4)$$
Note that the function tolIntPois uses the exact confidence limits for
\(\lambda\) when computing \(\beta\)-content tolerance limits (see
epois).
\(\beta\)-Expectation Tolerance Intervals for a Poisson Distribution
As stated above, a \(\beta\)-expectation tolerance interval with coverage
\(\beta100\%\) is equivalent to a prediction interval for one future observation
with associated confidence level \(\beta100\%\). This is because the probability
that any single future observation will fall into this interval is \(\beta100\%\),
so the distribution of the number of \(N\) future observations that will fall into
this interval is binomial with parameters
size=\(N\) and prob=\(\beta\). Hence the expected proportion of
future observations that fall into this interval is \(\beta100\%\) and is
independent of the value of \(N\). See the help file for predIntPois
for information on how these intervals are constructed.