| toleranceBound {TailRank} | 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 tau = 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 tau. This problem is addressed by constructing a statistical tolerance interval (more precisely, a one-sided tolerance bound) that contains a given fraction, psi, of the population with a given confidence level, gamma [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 gamma% of the time, exceeds 100 psi% of G values from a normal distribution is approximated by X_U = bar{X} + k_{gamma,psi}s, where
k_{gamma, psi} = {z_{psi} + sqrt{z_{psi}^2 - ab} over a},
a = 1-{z_{1-gamma}^2over 2N-2},
b = z_{psi}^2 - {z_{1-gamma}^2over 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_{gamma, psi} with the property that the psith quantile will be less than the estimate X_U = bar{X} + k_{gamma,psi}s (based on N data points) at least 100 gamma% of the time.
Lower tolerance bounds on quantiles with psi less than
one-half can be obtained as X_U = bar{X} - k_{gamma,1-psi}s,
Kevin R. Coombes <kcoombes@mdanderson.org>
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 tau <- mean(x) + sd(x)*tolerance.factor # lower 90 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))