Probability That at Least One Future Observation Falls Outside a Prediction Interval for a Lognormal Distribution
Compute the probability that at least one out of k future observations (or geometric means) falls outside a prediction interval for k future observations (or geometric means) for a normal distribution.
predIntLnormAltTestPower(n, df = n - 1, n.geomean = 1, k = 1,
ratio.of.means = 1, cv = 1, pi.type = "upper", conf.level = 0.95)n |
vector of positive integers greater than 2 indicating the sample size upon which the prediction interval is based. |
df |
vector of positive integers indicating the degrees of freedom associated with
the sample size. The default value is |
n.geomean |
positive integer specifying the sample size associated with the future
geometric means. The default value is |
k |
vector of positive integers specifying the number of future observations that the
prediction interval should contain with confidence level |
ratio.of.means |
numeric vector specifying the ratio of the mean of the population that will be
sampled to produce the future observations vs. the mean of the population that
was sampled to construct the prediction interval. See the DETAILS section below
for more information. The default value is |
cv |
numeric vector of positive values specifying the coefficient of variation for
both the population that was sampled to construct the prediction interval and
the population that will be sampled to produce the future observations. The
default value is |
pi.type |
character string indicating what kind of prediction interval to compute.
The possible values are |
conf.level |
numeric vector of values between 0 and 1 indicating the confidence level of the
prediction interval. The default value is |
A prediction interval for some population is an interval on the real line
constructed so that it will contain k future observations or averages
from that population with some specified probability (1-α)100\%,
where 0 < α < 1 and k is some pre-specified positive integer.
The quantity (1-α)100\% is call the confidence coefficient or
confidence level associated with the prediction interval. The function
predIntNorm computes a standard prediction interval based on a
sample from a normal distribution.
The function predIntNormTestPower computes the probability that at
least one out of k future observations or averages will not be contained in
a prediction interval based on the assumption of normally distributed observations,
where the population mean for the future observations is allowed to differ from
the population mean for the observations used to construct the prediction interval.
The function predIntLnormAltTestPower assumes all observations are
from a lognormal distribution. The observations used to
construct the prediction interval are assumed to come from a lognormal distribution
with mean θ_2 and coefficient of variation τ. The future
observations are assumed to come from a lognormal distribution with mean
θ_1 and coefficient of variation τ; that is, the means are
allowed to differ between the two populations, but not the coefficient of variation.
The function predIntLnormAltTestPower calls the function
predIntNormTestPower, with the argument delta.over.sigma
given by:
\frac{δ}{σ} = \frac{log(R)}{√{log(τ^2 + 1)}} \;\;\;\;\;\; (1)
where R is given by:
R = \frac{θ_1}{θ_2} \;\;\;\;\;\; (2)
and corresponds to the argument ratio.of.means for the function
predIntLnormAltTestPower, and τ corresponds to the argument
cv.
vector of numbers between 0 and 1 equal to the probability that at least one of k future observations or geometric means will fall outside the prediction interval.
See the help files for predIntNormTestPower.
Steven P. Millard (EnvStats@ProbStatInfo.com)
See the help files for predIntNormTestPower and
tTestLnormAltPower.
# Show how the power increases as ratio.of.means increases. Assume a # 95% upper prediction interval. predIntLnormAltTestPower(n = 4, ratio.of.means = 1:3) #[1] 0.0500000 0.1459516 0.2367793 #---------- # Look at how the power increases with sample size for an upper one-sided # prediction interval with k=3, ratio.of.means=4, and a confidence level of 95%. predIntLnormAltTestPower(n = c(4, 8), k = 3, ratio.of.means = 4) #[1] 0.2860952 0.4533567 #---------- # Show how the power for an upper 95% prediction limit increases as the # number of future observations k increases. Here, we'll use n=20 and # ratio.of.means=2. predIntLnormAltTestPower(n = 20, k = 1:3, ratio.of.means = 2) #[1] 0.1945886 0.2189538 0.2321562
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