dphat {spc}

Percent defective for normal samples

Description

Density, distribution function and quantile function for the sample percent defective calculated on normal samples with mean equal to mu and standard deviation equal to sigma.

Usage

dphat(x, n, mu=0, sigma=1, type="known", LSL=-3, USL=3, nodes=30)

pphat(q, n, mu=0, sigma=1, type="known", LSL=-3, USL=3, nodes=30)

qphat(p, n, mu=0, sigma=1, type="known", LSL=-3, USL=3, nodes=30)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

sample size.

mu, sigma

parameters of the underlying normal distribution.

type

choose whether the standard deviation is given and fixed ("known") or estimated and potententially monitored ("estimated").

LSL, USL

lower and upper specification limit, respectively.

nodes

number of quadrature nodes needed for type="estimated".

Details

Bruhn-Suhr/Krumbholz (1990) derived the cumulative distribution function of the sample percent defective calculated on normal samples to applying them for a new variables sampling plan. These results were heavily used in Krumbholz/Zöller (1995) for Shewhart and in Knoth/Steinmetz (2013) for EWMA control charts. For algorithmic details see, essentially, Bruhn-Suhr/Krumbholz (1990). Two design variants are treated: The simple case, type="known", with known normal variance and the presumably much more relevant and considerably intricate case, type="estimated", where both parameters of the normal distribution are unknown. Basically, given lower and upper specification limits and the normal distribution, one estimates the expected yield based on a normal sample of size n.

Value

Returns vector of pdf, cdf or qf values for the statistic phat.

Author(s)

Sven Knoth

References

M. Bruhn-Suhr and W. Krumbholz (1990), A new variables sampling plan for normally distributed lots with unknown standard deviation and double specification limits, Statistical Papers 31(1), 195-207.

W. Krumbholz and A. Zöller (1995), p-Karten vom Shewhartschen Typ für die messende Prüfung, Allgemeines Statistisches Archiv 79, 347-360.

S. Knoth and S. Steinmetz (2013), EWMA p charts under sampling by variables, International Journal of Production Research 51(13), 3795-3807.

See Also

phat.ewma.arl for routines using the herewith considered phat statistic.

Examples

# Figures 1 (c) and (d) from Knoth/Steinmetz (2013)
n      <-  5
LSL    <- -3
USL    <-  3

par(mar=c(5, 5, 1, 1) + 0.1)

p.star <- 2*pnorm( (LSL-USL)/2 ) # for p <= p.star pdf and cdf vanish

p_ <- seq(p.star+1e-10, 0.07, 0.0001) # define support of Figure 1

# Figure 1 (c)
pp_ <- pphat(p_, n)
plot(p_, pp_, type="l", xlab="p", ylab=expression(P( hat(p) <= p )),
     xlim=c(0, 0.06), ylim=c(0,1), lwd=2)
abline(h=0:1, v=p.star, col="grey")

# Figure 1 (d)
dp_ <- dphat(p_, n)
plot(p_, dp_, type="l", xlab="p", ylab="f(p)", xlim=c(0, 0.06),
     ylim=c(0,50), lwd=2)
abline(h=0, v=p.star, col="grey")