Package 'tolerance'

Description

Statistical tolerance limits provide the limits between which we can expect to find a specified proportion of a sampled population with a given level of confidence. This package provides functions for estimating tolerance limits (intervals) for various univariate distributions (binomial, Cauchy, discrete Pareto, exponential, two-parameter exponential, extreme value, hypergeometric, Laplace, logistic, negative binomial, negative hypergeometric, normal, Pareto, Poisson-Lindley, Poisson, uniform, and Zipf-Mandelbrot), Bayesian normal tolerance limits, multivariate normal tolerance regions, nonparametric tolerance intervals, tolerance bands for regression settings (linear regression, nonlinear regression, nonparametric regression, and multivariate regression), and analysis of variance tolerance intervals. Visualizations are also available for most of these settings.

Details

Author(s)

Derek S. Young, Ph.D.

Maintainer: Derek S. Young <[email protected]>

References

Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals: A Guide for Practitioners, Wiley-Interscience.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Patel, J. K. (1986), Tolerance Intervals - A Review, Communications in Statistics - Theory and Methodology, 15, 2719–2762.

Young, D. S. (2010), tolerance: An R Package for Estimating Tolerance Intervals, Journal of Statistical Software, 36(5), 1–39.

Young, D. S. (2014), Computing Tolerance Intervals and Regions in R. In M. B. Rao and C. R. Rao (eds.), Handbook of Statistics, Volume 32: Computational Statistics with R, 309–338. North-Holland, Amsterdam.

See Also

confint

Description

Provides an upper bound on the number of acceptable rejects or nonconformities in a process. This is similar to a 1-sided upper tolerance bound for a hypergeometric random variable.

Usage

acc.samp(n, N, alpha = 0.05, P = 0.99, AQL = 0.01, RQL = 0.02)

Arguments

Value

acc.samp returns a matrix with the following quantities:

References

Montgomery, D. C. (2005), Introduction to Statistical Quality Control, Fifth Edition, John Wiley & Sons, Inc.

See Also

Hypergeometric

Examples

## A 90%/90% acceptance sampling plan for a sample of 450 
## drawn from a lot size of 960.

acc.samp(n = 450, N = 960, alpha = 0.10, P = 0.90, AQL = 0.07,
         RQL = 0.10)

Description

Tolerance intervals for each factor level in a balanced (or nearly-balanced) ANOVA.

Usage

anovatol.int(lm.out, data, alpha = 0.05, P = 0.99, side = 1,
             method = c("HE", "HE2", "WBE", "ELL", "KM", 
             "EXACT", "OCT"), m = 50)

Arguments

Value

anovatol.int returns a list where each element is a data frame corresponding to each main effect (i.e., factor) tested in the ANOVA and the rows of each data frame are the levels of that factor. The columns of each data frame report the following:

References

Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements, Journal of the American Statistical Association, 64, 610–620.

Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions, Technometrics, 2, 483–500.

See Also

K.factor, normtol.int, lm, anova

Examples

## 90%/95% 2-sided tolerance intervals for a 2-way ANOVA 
## using the "warpbreaks" data.

attach(warpbreaks)

lm.out <- lm(breaks ~ wool + tension)
out <- anovatol.int(lm.out, data = warpbreaks, alpha = 0.10,
                    P = 0.95, side = 2, method = "HE")
out

plottol(out, x = warpbreaks)

Description

Provides 1-sided or 2-sided Bayesian tolerance intervals under the conjugate prior for data distributed according to a normal distribution.

Usage

bayesnormtol.int(x = NULL, norm.stats = list(x.bar = NA, 
                 s = NA, n = NA), alpha = 0.05, P = 0.99, 
                 side = 1, method = c("HE", "HE2", "WBE", 
                 "ELL", "KM", "EXACT", "OCT"), m = 50,
                 hyper.par = list(mu.0 = NULL, 
                 sig2.0 = NULL, m.0 = NULL, n.0 = NULL))

Arguments

Details

Note that if one considers the non-informative prior distribution, then the Bayesian tolerance intervals are the same as the classical solution, which can be obtained by using normtol.int.

Value

bayesnormtol.int returns a data frame with items:

References

Aitchison, J. (1964), Bayesian Tolerance Regions, Journal of the Royal Statistical Society, Series B, 26, 161–175.

Guttman, I. (1970), Statistical Tolerance Regions: Classical and Bayesian, Charles Griffin and Company.

Young, D. S., Gordon, C. M., Zhu, S., and Olin, B. D. (2016), Sample Size Determination Strategies for Normal Tolerance Intervals Using Historical Data, Quality Engineering, 28, 337–351.

See Also

Normal, normtol.int, K.factor

Examples

## 95%/85% 2-sided Bayesian normal tolerance limits for 
## a sample of size 100. 

set.seed(100)
x <- rnorm(100)
out <- bayesnormtol.int(x = x, alpha = 0.05, P = 0.85, 
                        side = 2, method = "EXACT", 
                        hyper.par = list(mu.0 = 0, 
                        sig2.0 = 1, n.0 = 10, m.0 = 10))
out

plottol(out, x, plot.type = "both", side = "upper", 
        x.lab = "Normal Data")

Description

Provides 1-sided or 2-sided tolerance intervals for binomial random variables. From a statistical quality control perspective, these limits use the proportion of defective (or acceptable) items in a sample to bound the number of defective (or acceptable) items in future productions of a specified quantity.

Usage

bintol.int(x, n, m = NULL, alpha = 0.05, P = 0.99, side = 1, 
           method = c("LS", "WS", "AC", "JF", "CP", "AS", 
           "LO", "PR", "PO", "CL", "CC", "CWS"), 
           a1 = 0.5, a2 = 0.5)

Arguments

Value

bintol.int returns a data frame with items:

References

Brown, L. D., Cai, T. T., and DasGupta, A. (2001), Interval Estimation for a Binomial Proportion, Statistical Science, 16, 101–133.

Hahn, G. J. and Chandra, R. (1981), Tolerance Intervals for Poisson and Binomial Variables, Journal of Quality Technology, 13, 100–110.

Newcombe, R. G. (1998), Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods, Statistics in Medicine, 17, 857–872.

See Also

Binomial, umatol.int

Examples

## 85%/90% 2-sided binomial tolerance intervals for a future 
## lot of 2500 when a sample of 230 were drawn from a lot of 
## 1000.  All methods but Jeffreys' method are compared
## below.

bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "LS")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "WS")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "AC")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "CP")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "AS")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "LO")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "PR")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "PO")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "CL")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "CC")
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 2, method = "CWS")
           
## Using Jeffreys' method to construct the 85%/90% 1-sided 
## binomial tolerance limits.  The first calculation assumes 
## a prior on the proportion of defects which places greater
## density on values near 0.  The second calculation assumes
## a prior on the proportion of defects which places greater
## density on values near 1.

bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 1, method = "JF", a1 = 2, a2 = 10)
bintol.int(x = 230, n = 1000, m = 2500, alpha = 0.15, P = 0.90,
           side = 1, method = "JF", a1 = 5, a2 = 1)

Description

This function allows the user to control what proportion of the population is to be in the tails of the given distribution for a 2-sided tolerance interval. The result is a conservative approximation based on Bonferroni's inequality.

Usage

bonftol.int(fn, P1 = 0.005, P2 = 0.005, alpha = 0.05, ...)

Arguments

Value

The results for the 2-sided tolerance interval procedure are reported. See the corresponding help file for fn about specific output. Note that the (minimum) proportion of the population to be covered by this interval is 1 - (P1 + P2).

Note

This function can be used with any 2-sided tolerance interval function, including the regression tolerance interval functions.

References

Jensen, W. A. (2009), Approximations of Tolerance Intervals for Normally Distributed Data, Quality and Reliability Engineering International, 25, 571–580.

Patel, J. K. (1986), Tolerance Intervals - A Review, Communications in Statistics - Theory and Methodology, 15, 2719–2762.

Examples

## 95%/97% tolerance interval for normally distributed
## data controlling 1% of the data is in the lower tail
## and 2% of the data in the upper tail.

set.seed(100)
x <- rnorm(100, 0, 0.2)
bonftol.int(normtol.int, x = x, P1 = 0.01, P2 = 0.02,
            alpha = 0.05, method = "HE")

Description

Provides 1-sided or 2-sided tolerance intervals for Cauchy distributed data.

Usage

cautol.int(x, alpha = 0.05, P = 0.99, side = 1)

Arguments

Value

cautol.int returns a data.frame with items:

References

Bain, L. J. (1978), Statistical Analysis of Reliability and Life-Testing Models, Marcel Dekker, Inc.

See Also

Cauchy

Examples

## 95%/90% 2-sided Cauchy tolerance interval for a sample 
## of size 1000. 

set.seed(100)
x <- rcauchy(1000, 100000, 10)
out <- cautol.int(x = x, alpha = 0.05, P = 0.90, side = 2)
out

plottol(out, x, plot.type = "both", x.lab = "Cauchy Data")

Description

Provides 1-sided tolerance limits for the difference between two independent normal random variables. If the ratio of the variances is known, then an exact calculation is performed. Otherwise, approximation methods are implemented.

Usage

diffnormtol.int(x1, x2, var.ratio = NULL, alpha = 0.05, 
                P = 0.99, method = c("HALL", "GK", "RG"))

Arguments

Details

Satterthwaite's approximation for the degrees of freedom is used when the variance ratio is unknown.

Value

diffnormtol.int returns a data frame with items:

Note

Unlike other tolerance interval functions, the output from diffnormtol.int cannot be passed to plottol.

References

Guo, H. and Krishnamoorthy, K. (2004), New Approximate Inferential Methods for the Reliability Parameter in a Stress-Strength Model: The Normal Case, Communications in Statistics - Theory and Methods, 33, 1715–1731.

Hall, I. J. (1984), Approximate One-Sided Tolerance Limits for the Difference or Sum of Two Independent Normal Variates, Journal of Quality Technology, 16, 15–19.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Reiser, B. J. and Guttman, I. (1986), Statistical Inference for Pr(Y < X): The Normal Case, Technometrics, 28, 253–257.

See Also

Normal, K.factor, normtol.int

Examples

## 90%/99% tolerance limits for the difference between two
## simulated normal data sets.  This data is taken from
## Krishnamoorthy and Mathew (2009).  Note that there is a
## calculational error in their example, which yields different
## results with the output below. 

x1 <- c(10.166, 5.889, 8.258, 7.303, 8.757)
x2 <- c(-0.204, 2.578, 1.182, 1.892, 0.786, -0.517, 1.156,
        0.980, 0.323, 0.437, 0.397, 0.050, 0.812, 0.720)

diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "HALL")
diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "GK")
diffnormtol.int(x1, x2, alpha = 0.10, P = 0.99, method = "RG")
diffnormtol.int(x1, x2, var.ratio = 3.8, alpha = 0.10, P = 0.99)

Description

Density (mass), distribution function, quantile function, and random generation for the difference between two proportions. This is determined by taking the difference between two independent beta distributions.

Usage

ddiffprop(x, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5,
          log = FALSE, ...)
pdiffprop(q, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5,
          lower.tail = TRUE, log.p = FALSE, ...)
qdiffprop(p, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5,
          lower.tail = TRUE, log.p = FALSE, ...)
rdiffprop(n, k1, k2, n1, n2, a1 = 0.5, a2 = 0.5)

Arguments

Details

The difference between two proportions distribution has a fairly complicated functional form. Please see the article by Chen and Luo (2011), who corrected a typo in the article by Nadarajah and Kotz (2007), for the functional form of this distribution.

Value

ddiffprop gives the density (mass), pdiffprop gives the distribution function, qdiffprop gives the quantile function, and rdiffprop generates random deviates.

References

Chen, Y. and Luo, S. (2011), A Few Remarks on 'Statistical Distribution of the Difference of Two Proportions', Statistics in Medicine, 30, 1913–1915.

Nadarajah, S. and Kotz, S. (2007), Statistical Distribution of the Difference of Two Proportions, Statistics in Medicine, 26, 3518–3523.

See Also

runif and .Random.seed about random number generation.

Examples

## Randomly generated data from the difference between
## two proportions distribution.

set.seed(100)
x <- rdiffprop(n = 100, k1 = 2, k2 = 10, n1 = 17, n2 = 13)
hist(x, main = "Randomly Generated Data", prob = TRUE)

x.1 <- sort(x)
y <- ddiffprop(x = x.1, k1 = 2, k2 = 10, n1 = 17, n2 = 13)
lines(x.1, y, col = 2, lwd = 2)

plot(x.1, pdiffprop(q = x.1, k1 = 2, k2 = 10, n1 = 17, 
     n2 = 13), type = "l", xlab = "x", 
     ylab = "Cumulative Probabilities")

qdiffprop(p = 0.20, k1 = 2, k2 = 10, n1 = 17, n2 = 13, 
          lower.tail = FALSE)
qdiffprop(p = 0.80, k1 = 2, k2 = 10, n1 = 17, n2 = 13)

Description

Density (mass), distribution function, quantile function, and random generation for the discrete Pareto distribution.

Usage

ddpareto(x, theta, log = FALSE)
pdpareto(q, theta, lower.tail = TRUE, log.p = FALSE)
qdpareto(p, theta, lower.tail = TRUE, log.p = FALSE)
rdpareto(n, theta)

Arguments

Details

The discrete Pareto distribution has mass

$p(x) = \theta^{\log(1+x)}-\theta^{\log(2+x)},$

where $x=0,1,\ldots$ and $0<\theta<1$ is the shape parameter.

Value

ddpareto gives the density (mass), pdpareto gives the distribution function, qdpareto gives the quantile function, and rdpareto generates random deviates for the specified distribution.

References

Krishna, H. and Pundir, P. S. (2009), Discrete Burr and Discrete Pareto Distributions, Statistical Methodology, 6, 177–188.

Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2019), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, Statistica Neerlandica, 73, 4–21.

See Also

runif and .Random.seed about random number generation.

Examples

## Randomly generated data from the discrete Pareto
## distribution.

set.seed(100)
x <- rdpareto(n = 150, theta = 0.2)
hist(x, main = "Randomly Generated Data", prob = TRUE)

x.1 <- sort(x)
y <- ddpareto(x = x.1, theta = 0.2)
lines(x.1, y, col = 2, lwd = 2)

plot(x.1, pdpareto(q = x.1, theta = 0.2), type = "l", 
     xlab = "x", ylab = "Cumulative Probabilities")

qdpareto(p = 0.80, theta = 0.2, lower.tail = FALSE)
qdpareto(p = 0.95, theta = 0.2)

Description

When providing two of the three quantities n, alpha, and P, this function solves for the third quantity in the context of distribution-free tolerance intervals.

Usage

distfree.est(n = NULL, alpha = NULL, P = NULL, side = 1)

Arguments

Value

When providing two of the three quantities n, alpha, and P, distfree.est returns the third quantity. If more than one value of a certain quantity is specified, then a table will be returned.

References

Natrella, M. G. (1963), Experimental Statistics: National Bureau of Standards - Handbook No. 91, United States Government Printing Office, Washington, D.C.

See Also

nptol.int

Examples

## Solving for 1 minus the confidence level.

distfree.est(n = 59, P = 0.95, side = 1)

## Solving for the sample size.

distfree.est(alpha = 0.05, P = 0.95, side = 1)

## Solving for the proportion of the population to cover.

distfree.est(n = 59, alpha = 0.05, side = 1)

## Solving for sample sizes for many tolerance specifications.

distfree.est(alpha = seq(0.01, 0.05, 0.01), 
             P = seq(0.80, 0.99, 0.01), side = 2)

Description

Performs maximum likelihood estimation for the parameter of the discrete Pareto distribution.

Usage

dpareto.ll(x, theta = NULL, ...)

Arguments

Details

The discrete Pareto distribution is a discretized of the continuous Type II Pareto distribution (also called the Lomax distribution).

Value

See the help file for mle to see how the output is structured.

References

Krishna, H. and Pundir, P. S. (2009), Discrete Burr and Discrete Pareto Distributions, Statistical Methodology, 6, 177–188.

Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2019), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, Statistica Neerlandica, 73, 4–21.

See Also

mle, DiscretePareto

Examples

## Maximum likelihood estimation for randomly generated data
## from the discrete Pareto distribution. 

set.seed(100)

dp.data <- rdpareto(n = 500, theta = 0.2)
out.dp <- dpareto.ll(dp.data)
stats4::coef(out.dp)
stats4::vcov(out.dp)

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to the discrete Pareto distribution.

Usage

dparetotol.int(x, m = NULL, alpha = 0.05, P = 0.99, side = 1, 
                ...)

Arguments

Details

The discrete Pareto is a discretized of the continuous Type II Pareto distribution (also called the Lomax distribution). Discrete Pareto distributions are heavily right-skewed distributions and potentially good models for discrete lifetime data and extremes in count data. For most practical applications, one will typically be interested in 1-sided upper bounds.

Value

dparetotol.int returns a data frame with the following items:

References

Young, D. S., Naghizadeh Qomi, M., and Kiapour, A. (2019), Approximate Discrete Pareto Tolerance Limits for Characterizing Extremes in Count Data, Statistica Neerlandica, 73, 4–21.

See Also

DiscretePareto, dpareto.ll

Examples

## 95%/95% 1-sided tolerance intervals for data assuming 
## the discrete Pareto distribution.

set.seed(100)

x <- rdpareto(n = 500, theta = 0.5)
out <- dparetotol.int(x, alpha = 0.05, P = 0.95, side = 1)
out

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to a 2-parameter exponential distribution. Data with Type II censoring is permitted.

Usage

exp2tol.int(x, alpha = 0.05, P = 0.99, side = 1,
            method = c("GPU", "DUN", "KM"), type.2 = FALSE)

Arguments

Value

exp2tol.int returns a data frame with items:

References

Dunsmore, I. R. (1978), Some Approximations for Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 20, 317–318.

Engelhardt, M. and Bain, L. J. (1978), Tolerance Limits and Confidence Limits on Reliability for the Two-Parameter Exponential Distribution, Technometrics, 20, 37–39.

Guenther, W. C., Patil, S. A., and Uppuluri, V. R. R. (1976), One-Sided $\beta$-Content Tolerance Factors for the Two Parameter Exponential Distribution, Technometrics, 18, 333–340.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

See Also

TwoParExponential

Examples

## 95%/90% 1-sided 2-parameter exponential tolerance intervals
## for a sample of size 50. 

set.seed(100)
x <- r2exp(50, 6, shift = 55)
out <- exp2tol.int(x = x, alpha = 0.05, P = 0.90, side = 1,
                   method = "DUN", type.2 = FALSE)
out

plottol(out, x, plot.type = "both", side = "upper", 
        x.lab = "2-Parameter Exponential Data")

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to an exponential distribution. Data with Type II censoring is permitted.

Usage

exptol.int(x, alpha = 0.05, P = 0.99, side = 1, type.2 = FALSE)

Arguments

Value

exptol.int returns a data frame with items:

References

Blischke, W. R. and Murthy, D. N. P. (2000), Reliability: Modeling, Prediction, and Optimization, John Wiley & Sons, Inc.

See Also

Exponential

Examples

## 95%/99% 1-sided exponential tolerance intervals for a
## sample of size 50. 

set.seed(100)
x <- rexp(100, 0.004)
out <- exptol.int(x = x, alpha = 0.05, P = 0.99, side = 1,
                  type.2 = FALSE)
out

plottol(out, x, plot.type = "both", side = "lower", 
        x.lab = "Exponential Data")

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a Weibull distribution or extreme-value (also called Gumbel) distributions.

Usage

exttol.int(x, alpha = 0.05, P = 0.99, side = 1,
           dist = c("Weibull", "Gumbel"), ext = c("min", "max"), 
           NR.delta = 1e-8)

Arguments

Details

Recall that the relationship between the Weibull distribution and the extreme-value distribution for the minimum is that if the random variable $X$ is distributed according to a Weibull distribution, then the random variable $Y = ln(X)$ is distributed according to an extreme-value distribution for the minimum.

If dist = "Weibull", then the natural logarithm of the data are taken so that a Newton-Raphson algorithm can be employed to find the MLEs of the extreme-value distribution for the minimum and then the data and MLEs are transformed back appropriately. No transformation is performed if dist = "Gumbel". The Newton-Raphson algorithm is initialized by the method of moments estimators for the parameters.

Value

exttol.int returns a data frame with items:

References

Bain, L. J. and Engelhardt, M. (1981), Simple Approximate Distributional Results for Confidence and Tolerance Limits for the Weibull Distribution Based on Maximum Likelihood Estimators, Technometrics, 23, 15–20.

Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values, Springer.

See Also

Weibull

Examples

## 85%/90% 1-sided Weibull tolerance intervals for a sample
## of size 150. 

set.seed(100)
x <- rweibull(150, 3, 75)
out <- exttol.int(x = x, alpha = 0.15, P = 0.90, side = 1,
                  dist = "Weibull")
out

plottol(out, x, plot.type = "both", side = "lower", 
        x.lab = "Weibull Data")

Description

The Appell function of the first kind, which is a two variable extension of the hypergeometric distribution.

Usage

F1(a, b, b.prime, c, x, y, ...)

Arguments

Value

F1 returns the simple integral result for the Appell function of the first kind with the arguments specified above.

Note

This function is solved by using a simple integral representation for real numbers. While all four of the Appell functions can be extended to the complex plane, this is not an option for this code.

References

Bailey, W. N. (1935), Generalised Hypergeometric Series, Cambridge University Press.

See Also

DiffProp, integrate

Examples

## Sample calculation.

F1(a = 3, b = 4, b.prime = 5, c = 13, x = 0.2, y = 0.4)

Description

Provides 1-sided or 2-sided tolerance intervals for the function of two binomial proportions using fiducial quantities.

Usage

fidbintol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN, 
              alpha = 0.05, P = 0.99, side = 1, K = 1000, 
              B = 1000)

Arguments

Details

If $X$ is observed from a $Bin(n,p)$ distribution, then the fiducial quantity for $p$ is $Beta(X+0.5,n-X+0.5)$.

Value

fidbintol.int returns a list with two items. The first item (tol.limits) is a data frame with the following items:

The second item (fn) simply returns the functional form specified by FUN.

References

Clopper, C. J. and Pearson, E. S. (1934), The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial, Biometrika, 26, 404–413.

Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning and Inference, 140, 1182–1192.

Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.

See Also

fidnegbintol.int, fidpoistol.int

Examples

## 95%/99% 1-sided and 2-sided tolerance intervals for 
## the difference between binomial proportions.

set.seed(100)

p1 <- 0.2
p2 <- 0.4
n1 <- n2 <- 200
x1 <- rbinom(1, n1, p1)
x2 <- rbinom(1, n2, p2)
fun.ti <- function(x, y) x - y

fidbintol.int(x1, x2, n1, n2, m1 = 500, m2 = 500, FUN = fun.ti,
              alpha = 0.05, P = 0.99, side = 1)
fidbintol.int(x1, x2, n1, n2, m1 = 500, m2 = 500, FUN = fun.ti,
              alpha = 0.05, P = 0.99, side = 2)

Description

Provides 1-sided or 2-sided tolerance intervals for the function of two negative binomial proportions using fiducial quantities.

Usage

fidnegbintol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN, 
                 alpha = 0.05, P = 0.99, side = 1, K = 1000, 
                 B = 1000)

Arguments

Details

If $X$ is observed from a $NegBin(n,p)$ distribution, then the fiducial quantity for $p$ is $Beta(n,X+0.5)$.

Value

fidnegbintol.int returns a list with two items. The first item (tol.limits) is a data frame with the following items:

The second item (fn) simply returns the functional form specified by FUN.

References

Cai, Y. and Krishnamoorthy, K. (2005), A Simple Improved Inferential Method for Some Discrete Distributions, Computational Statistics and Data Analysis, 48, 605–621.

Clopper, C. J. and Pearson, E. S. (1934), The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial, Biometrika, 26, 404–413.

Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning and Inference, 140, 1182–1192.

Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.

See Also

fidbintol.int, fidpoistol.int

Examples

## 95%/99% 1-sided and 2-sided tolerance intervals for 
## the ratio of odds ratios for negative binomial proportions.

set.seed(100)

p1 <- 0.6
p2 <- 0.2
n1 <- n2 <- 50
x1 <- rnbinom(1, n1, p1)
x2 <- rnbinom(1, n2, p2)
fun.ti <- function(x, y) x * (1 - y) / (y * (1 - x))

fidnegbintol.int(x1, x2, n1, n2, m1 = 50, m2 = 50, FUN = fun.ti, 
                 alpha = 0.05, P = 0.99, side = 1)
fidnegbintol.int(x1, x2, n1, n2, m1 = 50, m2 = 50, FUN = fun.ti, 
                 alpha = 0.05, P = 0.99, side = 2)

Description

Provides 1-sided or 2-sided tolerance intervals for the function of two Poisson rates using fiducial quantities.

Usage

fidpoistol.int(x1, x2, n1, n2, m1 = NULL, m2 = NULL, FUN, 
               alpha = 0.05, P = 0.99, side = 1, K = 1000, 
               B = 1000)

Arguments

Details

If $X$ is observed from a $Poi(n*\lambda)$ distribution, then the fiducial quantity for $\lambda$ is $\chi^{2}_{2*x+1}/(2*n)$.

Value

fidpoistol.int returns a list with two items. The first item (tol.limits) is a data frame with the following items:

The second item (fn) simply returns the functional form specified by FUN.

References

Cox, D. R. (1953), Some Simple Approximate Tests for Poisson Variates, Biometrika, 40, 354–360.

Krishnamoorthy, K. and Lee, M. (2010), Inference for Functions of Parameters in Discrete Distributions Based on Fiducial Approach: Binomial and Poisson Cases, Journal of Statistical Planning and Inference, 140, 1182–1192.

Mathew, T. and Young, D. S. (2013), Fiducial-Based Tolerance Intervals for Some Discrete Distributions, Computational Statistics and Data Analysis, 61, 38–49.

See Also

fidbintol.int, fidnegbintol.int

Examples

## 95%/99% 1-sided and 2-sided tolerance intervals for 
## the ratio of two Poisson rates.

set.seed(100)

lambda1 <- 10
lambda2 <- 2
n1 <- 3000
n2 <- 3250
x1 <- rpois(1, n1 * lambda1)
x2 <- rpois(1, n2 * lambda2)
fun.ti <- function(x, y) x / y

fidpoistol.int(x1, x2, n1, n2, m1 = 2000, m2 = 2500, 
               FUN = fun.ti, alpha = 0.05, P = 0.99, side = 1)
fidpoistol.int(x1, x2, n1, n2, m1 = 2000, m2 = 2500, 
               FUN = fun.ti, alpha = 0.05, P = 0.99, side = 2)

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a gamma distribution or log-gamma distribution.

Usage

gamtol.int(x, alpha = 0.05, P = 0.99, side = 1, 
           method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT", 
           "OCT"), m = 50, log.gamma = FALSE)

Arguments

Details

Recall that if the random variable $X$ is distributed according to a log-gamma distribution, then the random variable $Y = ln(X)$ is distributed according to a gamma distribution.

Value

gamtol.int returns a data frame with items:

References

Krishnamoorthy, K., Mathew, T., and Mukherjee, S. (2008), Normal-Based Methods for a Gamma Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability, Technometrics, 50, 69–78.

See Also

GammaDist, K.factor

Examples

## 99%/99% 1-sided gamma tolerance intervals for a sample
## of size 50. 

set.seed(100)
x <- rgamma(50, 0.30, scale = 2)
out <- gamtol.int(x = x, alpha = 0.01, P = 0.99, side = 1)
out

plottol(out, x, plot.type = "both", side = "upper", 
        x.lab = "Gamma Data")

Description

Provides 1-sided or 2-sided tolerance intervals for hypergeometric random variables. From a sampling without replacement perspective, these limits use the proportion of units from group A (e.g., "black balls" in an urn) in a sample to bound the number of potential units drawn from group A in a future sample taken from the universe.

Usage

hypertol.int(x, n, N, m = NULL, alpha = 0.05, P = 0.99, 
             side = 1, method = c("EX", "LS", "CC"))

Arguments

Value

hypertol.int returns a data frame with items:

Note

As this methodology is built using large-sample theory, if the sampling rate is less than 0.05, then a warning is generated stating that the results are not reliable. Also, compare the functionality of this procedure with the acc.samp procedure, which is to determine a minimal acceptance limit for a particular sampling plan.

References

Brown, L. D., Cai, T. T., and DasGupta, A. (2001), Interval Estimation for a Binomial Proportion, Statistical Science, 16, 101–133.

Eichenberger, P., Hulliger, B., and Potterat, J. (2011), Two Measures for Sample Size Determination, Survey Research Methods, 5, 27–37.

Young, D. S. (2014), Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables, Sankhya: The Indian Journal of Statistics, Series B, 77(1), 114–140.

See Also

acc.samp, Hypergeometric

Examples

## 90%/95% 1-sided and 2-sided hypergeometric tolerance 
## intervals for a future sample of 30 when the universe
## is of size 100.

hypertol.int(x = 15, n = 50, N = 100, m = 30, alpha = 0.10, 
             P = 0.95, side = 1, method = "LS")
hypertol.int(x = 15, n = 50, N = 100, m = 30, alpha = 0.10, 
             P = 0.95, side = 1, method = "CC")
hypertol.int(x = 15, n = 50, N = 100, m = 30, alpha = 0.10, 
             P = 0.95, side = 2, method = "LS")
hypertol.int(x = 15, n = 50, N = 100, m = 30, alpha = 0.10, 
             P = 0.95, side = 2, method = "CC")

Description

Estimates k-factors for tolerance intervals based on normality.

Usage

K.factor(n, f = NULL, alpha = 0.05, P = 0.99, side = 1, 
         method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT", 
         "OCT"), m = 50)

Arguments

Value

K.factor returns the k-factor for tolerance intervals based on normality with the arguments specified above.

Note

For larger sample sizes, there may be some accuracy issues with the 1-sided calculation since it depends on the noncentral t-distribution. The code is primarily intended to be used for moderate values of the noncentrality parameter. It will not be highly accurate, especially in the tails, for large values. See TDist for further details.

References

Ellison, B. E. (1964), On Two-Sided Tolerance Intervals for a Normal Distribution, Annals of Mathematical Statistics, 35, 762–772.

Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements, Journal of the American Statistical Association, 64, 610–620.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Odeh, R. E. and Owen, D. B. (1980), Tables for Normal Tolerance Limits, Sampling Plans, and Screening, Marcel-Dekker.

Owen, D. B. (1964), Controls of Percentages in Both Tails of the Normal Distribution, Technometrics, 6, 377-387.

Wald, A. and Wolfowitz, J. (1946), Tolerance Limits for a Normal Distribution, Annals of the Mathematical Statistics, 17, 208–215.

Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions, Technometrics, 2, 483–500.

See Also

integrate, K.table, normtol.int, TDist

Examples

## Showing the k-factor under the Howe, Weissberg-Beatty, 
## and exact estimation methods.

K.factor(10, P = 0.95, side = 2, method = "HE")
K.factor(10, P = 0.95, side = 2, method = "WBE")
K.factor(10, P = 0.95, side = 2, method = "EXACT", m = 20)

Description

Estimates k-factors for simultaneous tolerance intervals based on normality.

Usage

K.factor.sim(n, l = NULL, alpha = 0.05, P = 0.99, side = 1, 
         method = c("EXACT", "BONF"), m = 50)

Arguments

Value

K.factor returns the k-factor for simultaneous tolerance intervals based on normality with the arguments specified above.

Note

For larger combinations of n and l when side = 2 and method = "EXACT", the calculation can be slow. For larger sample sizes when method = "BONF", there may be some accuracy issues with the 1-sided calculation since it depends on the noncentral t-distribution. The code is primarily intended to be used for moderate values of the noncentrality parameter. It will not be highly accurate, especially in the tails, for large values. See TDist for further details.

Thanks to Andrew Landgraf for providing the basic code for the method = "EXACT" procedure.

References

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Mee, R. W. (1990), Simultaneous Tolerance Intervals for Normal Populations with Common Variance, Technometrics, 32, 83-92.

See Also

integrate, K.factor

Examples

## Reproducing part of Table B5 from Krishnamoorthy and 
## Mathew (2009).

n_sizes <- c(2:20, seq(30, 100, 10))
l_sizes <- 2:10
KM_table <- sapply(1:length(l_sizes), function(i)
                   sapply(1:length(n_sizes), function(j)
                   round(K.factor.sim(n = n_sizes[j], 
                   l = l_sizes[i], side=1, alpha = 0.1, 
                   P = 0.9),3)))
dimnames(KM_table) <- list(n = n_sizes, l = l_sizes)
KM_table

Description

Tabulated summary of k-factors for tolerance intervals based on normality. The user can specify multiple values for each of the three inputs.

Usage

K.table(n, alpha, P, side = 1, f = NULL, method = c("HE", 
        "HE2", "WBE", "ELL", "KM", "EXACT", "OCT"), m = 50,
        by.arg = c("n", "alpha", "P"))

Arguments

Details

The method used for estimating the k-factors is that due to Howe as it is generally viewed as more accurate than the Weissberg-Beatty method.

Value

K.table returns a list with a structure determined by the argument by.arg described above.

References

Howe, W. G. (1969), Two-Sided Tolerance Limits for Normal Populations - Some Improvements, Journal of the American Statistical Association, 64, 610–620.

Weissberg, A. and Beatty, G. (1969), Tables of Tolerance Limit Factors for Normal Distributions, Technometrics, 2, 483–500.

See Also

K.factor

Examples

## Tables generated for each value of the sample size.

K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10), 
        P = c(0.90, 0.95, 0.99), by.arg = "n")

## Tables generated for each value of the confidence level.

K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10), 
        P = c(0.90, 0.95, 0.99), by.arg = "alpha")

## Tables generated for each value of the coverage proportion.

K.table(n = seq(50, 100, 10), alpha = c(0.01, 0.05, 0.10), 
        P = c(0.90, 0.95, 0.99), by.arg = "P")

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to a Laplace distribution.

Usage

laptol.int(x, alpha = 0.05, P = 0.99, side = 1)

Arguments

Value

laptol.int returns a data frame with items:

References

Bain, L. J. and Engelhardt, M. (1973), Interval Estimation for the Two Parameter Double Exponential Distribution, Technometrics, 15, 875–887.

Examples

## First generate data from a Laplace distribution with location
## parameter 70 and scale parameter 3.

set.seed(100)
tmp <- runif(40)
x <- rep(70, 40) - sign(tmp - 0.5)*rep(3, 40)*
              log(2*ifelse(tmp < 0.5, tmp, 1-tmp))

## 95%/90% 1-sided Laplace tolerance intervals for the sample
## of size 40 generated above. 

out <- laptol.int(x = x, alpha = 0.05, P = 0.90, side = 1) 
out

plottol(out, x, plot.type = "hist", side = "two", 
        x.lab = "Laplace Data")

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to a logistic or log-logistic distribution.

Usage

logistol.int(x, alpha = 0.05, P = 0.99, log.log = FALSE,
             side = 1, method = c("HALL", "BE"))

Arguments

Details

Recall that if the random variable $X$ is distributed according to a log-logistic distribution, then the random variable $Y = ln(X)$ is distributed according to a logistic distribution. For method = "HALL", the method due to Hall (1975) is implemented. This, however, can have numerical instabilities due to taking square roots of negative numbers in the calculation, thus leading to two-sided tolerance limits where the upper tolerance limit is smaller than the lower tolerance limit. method = "BE" calculates the limits using the method due to Bain and Englehardt (1991), which tends to be more reliable.

Value

logistol.int returns a data frame with items:

References

Bain, L. and Englehardt, M. (1991), Statistical Analysis of Reliability and Life Testing Models: Theory and Methods, Second Edition, Marcel Dekker, Inc.

Balakrishnan, N. (1992), Handbook of the Logistic Distribution, Marcel Dekker, Inc.

Hall, I. J. (1975), One-Sided Tolerance Limits for a Logistic Distribution Based on Censored Samples, Biometrics, 31, 873–880.

See Also

Logistic

Examples

## 90%/95% 1-sided logistic tolerance intervals for a sample
## of size 20. 

set.seed(100)
x <- rlogis(20, 5, 1)
out <- logistol.int(x = x, alpha = 0.10, P = 0.95, 
                    log.log = FALSE, side = 1) 
out

plottol(out, x, plot.type = "control", side = "two", 
        x.lab = "Logistic Data")

Description

Determines the appropriate tolerance factor for computing multivariate (multiple) linear regression tolerance regions based on Monte Carlo simulation.

Usage

mvregtol.region(mvreg, new.x = NULL, alpha = 0.05, P = 0.99, 
                B = 1000)

Arguments

Details

A basic sketch of how the algorithm works is as follows:

(1) Generate independent chi-square random variables and Wishart random matrices.

(2) Compute the eigenvalues of the randomly generated Wishart matrices.

(3) Iterate the above steps to generate a set of B sample values such that the 100(1-alpha)-th percentile is an approximate tolerance factor.

Value

mvregtol.region returns a matrix where the first column is the k-factor, the next q columns are the estimated responses from the least squares fit, and the final m columns are the predictor values. The first n rows of the matrix pertain to the raw data as specified by y and x. If values for new.x are specified, then there is one additional row appended to this output for each row in the matrix new.x.

Note

As of tolerance version 2.0.0, the arguments to this function have changed. This function no longer depends on inputted y and x matrices or an int argument. Instead, the function requires mvreg, which is of class "mlm", and provides all of the necessary components for the way the output is formatted. Also, new.x must now be a data frame with columns matching those from the data frame used in the mvreg fitted object.

References

Anderson, T. W. (2003) An Introduction to Multivariate Statistical Analysis, Third Edition, Wiley.

Krishnamoorthy, K. and Mathew, T. (2009), Statistical Tolerance Regions: Theory, Applications, and Computation, Wiley.

Krishnamoorthy, K. and Mondal, S. (2008), Tolerance Factors in Multiple and Multivariate Linear Regressions, Communications in Statistics - Simulation and Computation, 37, 546–559.

Examples

## 95%/95% multivariate regression tolerance factors using
## a fertilizer data set presented in Anderson (2003, p. 374). 

grain <- c(40, 17, 9, 15, 6, 12, 5, 9)
straw <- c(53, 19, 10, 29, 13, 27, 19, 30)
fert <- c(24, 11, 5, 12, 7, 14, 11, 18)
DF <- data.frame(grain,straw,fert)
new.x <- data.frame(fert = c(10, 15, 20))
mvreg <- lm(cbind(grain, straw) ~ fert + I(fert^2), data = DF)

set.seed(100)
out <- mvregtol.region(mvreg, new.x = new.x, alpha = 0.05,
                       P = 0.95, B = 5000)
out

Description

Determines the appropriate tolerance factor for computing multivariate normal tolerance regions based on Monte Carlo methods or other approximations.

Usage

mvtol.region(x, alpha = 0.05, P = 0.99, B = 1000, M = 1000,
             method = c("KM", "AM", "GM", "HM", "MHM", "V11", 
             "HM.V11", "MC"))

Arguments

Details

All of the methods are outlined in the references that we provided. In practice, we recommend using the Krishnamoorthy-Mondal approach. A basic sketch of how the Krishnamoorthy-Mondal algorithm works is as follows:

(1) Generate independent chi-square random variables and Wishart random matrices.

(2) Compute the eigenvalues of the randomly generated Wishart matrices.

(3) Iterate the above steps to generate a set of B sample values such that the 100(1-alpha)-th percentile is an approximate tolerance factor.

Value

mvtol.region returns a matrix where the rows pertain to each confidence level 1-alpha specified and the columns pertain to each proportion level P specified.

References

Krishnamoorthy, K. and Mathew, T. (1999), Comparison of Approximation Methods for Computing Tolerance Factors for a Multivariate Normal Population, Technometrics, 41, 234–249.

Krishnamoorthy, K. and Mondal, S. (2006), Improved Tolerance Factors for Multivariate Normal Distributions, Communications in Statistics - Simulation and Computation, 35, 461–478.

Examples

## 90%/90% bivariate normal tolerance region. 

set.seed(100)
x1 <- rnorm(100, 0, 0.2)
x2 <- rnorm(100, 0, 0.5)
x <- cbind(x1, x2)

out1 <- mvtol.region(x = x, alpha = 0.10, P = 0.90, B = 1000,
                     method = "KM")
out1
plottol(out1, x)

## 90%/90% trivariate normal tolerance region. 

set.seed(100)
x1 <- rnorm(100, 0, 0.2)
x2 <- rnorm(100, 0, 0.5)
x3 <- rnorm(100, 5, 1)
x <- cbind(x1, x2, x3)
mvtol.region(x = x, alpha = c(0.10, 0.05, 0.01), 
             P = c(0.90, 0.95, 0.99), B = 1000, method = "KM") 

out2 <- mvtol.region(x = x, alpha = 0.10, P = 0.90, B = 1000, 
                     method = "KM")
out2
plottol(out2, x)

Description

Provides 1-sided or 2-sided tolerance intervals for negative binomial random variables. From a statistical quality control perspective, these limits use the number of failures that occur to reach n successes to bound the number of failures for a specified amount of future successes (m).

Usage

negbintol.int(x, n, m = NULL, alpha = 0.05, P = 0.99, 
              side = 1, method = c("LS", "WU", "CB", 
              "CS", "SC", "LR", "SP", "CC"))

Arguments

Details

This function takes the approach for Poisson and binomial random variables developed in Hahn and Chandra (1981) and applies it to the negative binomial case.

Value

negbintol.int returns a data frame with items:

Note

Recall that the geometric distribution is the negative binomial distribution where the size is 1. Therefore, the case when n = m = 1 will provide tolerance limits for a geometric distribution.

References

Casella, G. and Berger, R. L. (1990), Statistical Inference, Duxbury Press.

Hahn, G. J. and Chandra, R. (1981), Tolerance Intervals for Poisson and Binomial Variables, Journal of Quality Technology, 13, 100–110.

Tian, M., Tang, M. L., Ng, H. K. T., and Chan, P. S. (2009), A Comparative Study of Confidence Intervals for Negative Binomial Proportions, Journal of Statistical Computation and Simulation, 79, 241–249.

Young, D. S. (2014), A Procedure for Approximate Negative Binomial Tolerance Intervals, Journal of Statistical Computation and Simulation, 84, 438–450.

See Also

NegBinomial, umatol.int

Examples

## Comparison of 95%/99% 1-sided tolerance limits with
## 50 failures before 10 successes are reached.

negbintol.int(x = 50, n = 10, side = 1, method = "LS")
negbintol.int(x = 50, n = 10, side = 1, method = "WU")
negbintol.int(x = 50, n = 10, side = 1, method = "CB")
negbintol.int(x = 50, n = 10, side = 1, method = "CS")
negbintol.int(x = 50, n = 10, side = 1, method = "SC")
negbintol.int(x = 50, n = 10, side = 1, method = "LR")
negbintol.int(x = 50, n = 10, side = 1, method = "SP")
negbintol.int(x = 50, n = 10, side = 1, method = "CC")

## 95%/99% 1-sided tolerance limits and 2-sided tolerance 
## interval for the same setting above, but when we are 
## interested in a future experiment that requires 20 successes 
## be reached for each trial.

negbintol.int(x = 50, n = 10, m = 20, side = 1)
negbintol.int(x = 50, n = 10, m = 20, side = 2)

Description

Density, distribution function, quantile function, and random generation for the negative hypergeometric distribution.

Usage

dnhyper(x, m, n, k, log = FALSE)
pnhyper(q, m, n, k, lower.tail = TRUE, log.p = FALSE)
qnhyper(p, m, n, k, lower.tail = TRUE, log.p = FALSE)
rnhyper(nn, m, n, k)

Arguments

Details

A negative hypergeometric distribution (sometimes called the inverse hypergeometric distribution) models the total number of trials until k successes occur. Compare this to the negative binomial distribution, which models the number of failures that occur until a specified number of successes has been reached. The negative hypergeometric distribution has density

$p(x) = \frac{{x-1 \choose k-1}{n-x \choose m-k}}{{n \choose m}}$

for $x=k,k+1,...,n-m+k$.

Value

dnhyper gives the density, pnhyper gives the distribution function, qnhyper gives the quantile function, and rnhyper generates random deviates.

Invalid arguments will return value NaN, with a warning.

References

Wilks, S. S. (1963), Mathematical Statistics, Wiley.

See Also

runif and .Random.seed about random number generation.

Examples

## Randomly generated data from the negative hypergeometric 
## distribution.

set.seed(100)
x <- rnhyper(nn = 1000, m = 15, n = 40, k = 10)
hist(x, main = "Randomly Generated Data", prob = TRUE)

x.1 = sort(x)
y <- dnhyper(x = x.1, m = 15, n = 40, k = 10)
lines(x.1, y, col = 2, lwd = 2)

plot(x.1, pnhyper(q = x.1, m = 15, n = 40, k = 10),             
     type = "l", xlab = "x", ylab = "Cumulative Probabilities")

qnhyper(p = 0.20, m = 15, n = 40, k = 10, lower.tail = FALSE)
qnhyper(p = 0.80, m = 15, n = 40, k = 10)

Description

Provides 1-sided or 2-sided tolerance intervals for negative hypergeometric random variables. When sampling without replacement, these limits are on the total number of expected draws in a future sample in order to achieve a certain number from group A (e.g., "black balls" in an urn).

Usage

neghypertol.int(x, n, N, m = NULL, alpha = 0.05, P = 0.99,
                side = 1, method = c("EX", "LS", "CC"))

Arguments

Value

neghypertol.int returns a data frame with items:

Note

As this methodology is built using large-sample theory, if the sampling rate is less than 0.05, then a warning is generated stating that the results are not reliable.

References

Khan, R. A. (1994), A Note on the Generating Function of a Negative Hypergeometric Distribution, Sankhya: The Indian Journal of Statistics, Series B, 56, 309–313.

Young, D. S. (2014), Tolerance Intervals for Hypergeometric and Negative Hypergeometric Variables, Sankhya: The Indian Journal of Statistics, Series B, 77(1), 114–140.

See Also

acc.samp, NegHypergeometric

Examples

## 90%/95% 2-sided negative hypergeometric tolerance
## intervals for a future number of 20 successes when
## the universe is of size 100.  The estimates are 
## based on having drawn 50 in another sample to achieve 
## 20 successes.

neghypertol.int(50, 20, 100, m = 20, alpha = 0.05, 
                P = 0.95, side = 2, method = "LS")

Description

Provides 1-sided or 2-sided nonlinear regression tolerance bounds.

Usage

nlregtol.int(formula, xy.data = data.frame(), x.new = NULL, 
             side = 1, alpha = 0.05, P = 0.99, maxiter = 50, 
             new = FALSE, ...)

Arguments

Details

It is highly recommended that the user specify starting values for the nls routine.

Value

npregtol.int2 returns a list with items:

References

Wallis, W. A. (1951), Tolerance Intervals for Linear Regression, in Second Berkeley Symposium on Mathematical Statistics and Probability, ed. J. Neyman, Berkeley: University of CA Press, 43–51.

Young, D. S. (2013), Regression Tolerance Intervals, Communications in Statistics - Simulation and Computation, 42, 2040–2055.

See Also

nls, nlregtol.int

Examples

## 95%/95% 2-sided nonlinear regression tolerance bounds
## for a sample of size 50.
set.seed(100)
x <- runif(50, 5, 45)
f1 <- function(x, b1, b2) b1 + (0.49 - b1)*exp(-b2*(x - 8)) +
  rnorm(50, sd = 0.01)
y <- f1(x, 0.39, 0.11)
formula <- as.formula(y ~ b1 + (0.49 - b1)*exp(-b2*(x - 8)))
out1 <- nlregtol.int(formula = formula,
                     xy.data = data.frame(cbind(y, x)),
                     x.new=c(10,20), side = 2,
                     alpha = 0.05, P = 0.95 , new = TRUE)
out1
#########
set.seed(100)
x1 <- runif(50, 5, 45)
x2 <- rnorm(50, 0, 10)
f1 <- function(x1, x2, b1, b2) {(0.49 - b1)*exp(-b2*(x1 + x2 - 8)) +
    rnorm(50, sd = 0.01)}
y <- f1(x1 , x2 , 0.25 , 0.39)
formula <- as.formula(y ~ (0.49 - b1)*exp(-b2*(x1 + x2 - 8)))
out2 <- nlregtol.int(formula = formula,
                     xy.data = data.frame(cbind(y, x1 , x2)),
                     x.new=cbind(c(10,20) , c(47 , 53)), side = 2,
                     alpha = 0.05, P = 0.95 , new = TRUE)
out2

Description

Provides OC-type curves to illustrate how values of the k-factors for normal tolerance intervals, confidence levels, and content levels change as a function of the sample size.

Usage

norm.OC(k = NULL, alpha = NULL, P = NULL, n, side = 1,
        method = c("HE", "HE2", "WBE", "ELL", "KM", "EXACT", 
        "OCT"), m = 50)

Arguments

Value

norm.OC returns a figure with the OC curves constructed using the specifications in the arguments.

References

Young, D. S. (2016), Normal Tolerance Interval Procedures in the tolerance Package, The R Journal, 8, 200–212.

See Also

K.factor, normtol.int

Examples

## The three types of OC-curves that can be constructed
## with the norm.OC function.
 
norm.OC(k = 4, alpha = NULL, P = c(0.90, 0.95, 0.99), 
        n = 10:20, side = 1)

norm.OC(k = 4, alpha = c(0.01, 0.05, 0.10), P = NULL, 
        n = 10:20, side = 1)

norm.OC(k = NULL, P = c(0.90, 0.95, 0.99), 
        alpha=c(0.01,0.05,0.10), n = 10:20, side = 1)

Description

Provides minimum sample sizes for a future sample size when constructing normal tolerance intervals. Various strategies are available for determining the sample size, including strategies that incorporate known specification limits.

Usage

norm.ss(x = NULL, alpha = 0.05, P = 0.99, delta = NULL,
        P.prime = NULL, side = 1, m = 50, spec = c(NA, NA),
        hyper.par = list(mu.0 = NULL, sig2.0 = NULL, 
        m.0 = NULL, n.0 = NULL), method = c("DIR", 
        "FW", "YGZO"))

Arguments

Value

norm.ss returns a data frame with items:

References

Faulkenberry, G. D. and Weeks, D. L. (1968), Sample Size Determination for Tolerance Limits, Technometrics, 10, 343–348.

Young, D. S., Gordon, C. M., Zhu, S., and Olin, B. D. (2016), Sample Size Determination Strategies for Normal Tolerance Intervals Using Historical Data, Quality Engineering, 28, 337–351.

See Also

bayesnormtol.int, Normal, normtol.int

Examples

## Sample size determination for 95%/95% 2-sided normal 
## tolerance intervals using the direct method.
 
set.seed(100)
norm.ss(alpha = 0.05, P = 0.95, side = 2, spec = c(-3, 3), 
        method = "DIR", hyper.par = list(mu.0 = 0, 
        sig2.0 = 1))

Description

Provides 1-sided or 2-sided tolerance intervals for data distributed according to either a normal distribution or log-normal distribution.

Usage

normtol.int(x, alpha = 0.05, P = 0.99, side = 1,
            method = c("HE", "HE2", "WBE", "ELL", "KM", 
            "EXACT", "OCT"), m = 50, log.norm = FALSE)

Arguments