toleranceBound | R Documentation |
The function toleranceBound
computes theoretical upper tolerance
bounds on the quantiles of the standard normal distribution. These can
be used to produce reliable data-driven estimates of the quantiles in
any normal distribution.
toleranceBound(psi, gamma, N)
psi |
A real number between 0 and 1 giving the desired quantile |
gamma |
A real number between 0 and 1 giving the desired tolerance bound |
N |
An integer giving the number of observations used to estimate the quantile |
Suppose that we collect N observations from a normal distribution with unknown mean and variance, and wish to estimate the 95th percentile of the distribution. A simple point estimate is given by τ = \bar{X} + 1.68s. However, only the mean of the distribution is less than this value 95\% of the time. When N=40, for example, almost half of the time (43.5\%), fewer than 95\% of the observed values will be less than τ. This problem is addressed by constructing a statistical tolerance interval (more precisely, a one-sided tolerance bound) that contains a given fraction, ψ, of the population with a given confidence level, γ [Hahn and Meeker, 1991]. With enough samples, one can obtain distribution-free tolerance bounds [op.\ cit., Chapter 5]. For instance, one can use bootstrap or jackknife methods to estimate these bounds empirically.
Here, however, we assume that the measurements are normally distributed. We let \bar{X} denote the sample mean and let s denote the sample standard deviation. The upper tolerance bound that, 100 γ\% of the time, exceeds 100 ψ\% of G values from a normal distribution is approximated by X_U = \bar{X} + k_{γ,ψ}s, where
k_{γ, ψ} = {z_{ψ} + √{z_{ψ}^2 - ab} \over a},
a = 1-{z_{1-γ}^2\over 2N-2},
b = z_{ψ}^2 - {z_{1-γ}^2\over N},
and, for any π, z_π is the critical value of the normal distribution that is exceeded with probability π [Natrella, 1963].
Returns the value of k_{γ, ψ} with the property that the ψth quantile will be less than the estimate X_U = \bar{X} + k_{γ,ψ}s (based on N data points) at least 100 γ\% of the time.
Lower tolerance bounds on quantiles with psi
less than
one-half can be obtained as X_U = \bar{X} - k_{γ,1-ψ}s,
Kevin R. Coombes <krc@silicovore.com>
Natrella, M.G. (1963) Experimental Statistics. NBS Handbook 91, National Bureau of Standards, Washington DC.
Hahn, G.J. and Meeker, W.Q. (1991) Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, Inc., New York.
N <- 50 x <- rnorm(N) tolerance <- 0.90 quant <- 0.95 tolerance.factor <- toleranceBound(quant, tolerance, N) # upper 90% tolerance bound for 95th percentile tau <- mean(x) + sd(x)*tolerance.factor # lower 90% tolerance bound for 5th percentile rho <- mean(x) - sd(x)*tolerance.factor # behavior of the tolerance bound as N increases nn <- 10:100 plot(nn, toleranceBound(quant, tolerance, nn)) # behavior of the bound as the tolerance varies xx <- seq(0.5, 0.99, by=0.01) plot(xx, toleranceBound(quant, xx, N))
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