# nonpartolint: Nonparametric Tolerance Intervals In tolerance: Statistical Tolerance Intervals and Regions

## Description

Provides 1-sided or 2-sided nonparametric (i.e., distribution-free) tolerance intervals for any continuous data set.

## Usage

 ```1 2 3``` ```nptol.int(x, alpha = 0.05, P = 0.99, side = 1, method = c("WILKS", "WALD", "HM", "YM"), upper = NULL, lower = NULL) ```

## Arguments

 `x` A vector of data which no distributional assumptions are made. The data is only assumed to come from a continuous distribution. `alpha` The level chosen such that `1-alpha` is the confidence level. `P` The proportion of the population to be covered by this tolerance interval. `side` Whether a 1-sided or 2-sided tolerance interval is required (determined by `side = 1` or `side = 2`, respectively). `method` The method for determining which indices of the ordered observations will be used for the tolerance intervals. `"WILKS"` is the Wilks method, which produces tolerance bounds symmetric about the observed center of the data by using the beta distribution. `"WALD"` is the Wald method, which produces (possibly) multiple tolerance bounds for `side = 2` (each having at least the specified confidence level), but is the same as `method = "WILKS"` for `side = 1`. `"HM"` is the Hahn-Meeker method, which is based on the binomial distribution, but the upper and lower bounds may exceed the minimum and maximum of the sample data. For `side = 2`, this method will yield two intervals if an odd number of observations are to be trimmed from each side. `"YM"` is the Young-Mathew method for performing interpolation or extrapolation based on the order statistics. See below for more information on this method. `upper` The upper bound of the data. When `NULL`, then the maximum of `x` is used. If `method = "YM"` and extrapolation is performed, then `upper` will be greater than the maximum. `lower` The lower bound of the data. When `NULL`, then the minimum of `x` is used. If `method = "YM"` and extrapolation is performed, then `lower` will be less than the minimum.

## Details

For the Young-Mathew method, interpolation or extrapolation is performed. When `side = 1`, two intervals are given: one based on linear interpolation/extrapolation of order statistics (`OS-Based`) and one based on fractional order statistics (`FOS-Based`). When `side = 2`, only an interval based on linear interpolation/extrapolation of order statistics is given.

## Value

`nptol.int` returns a data frame with items:

 `alpha` The specified significance level. `P` The proportion of the population covered by this tolerance interval. `1-sided.lower` The 1-sided lower tolerance bound. This is given only if `side = 1`. `1-sided.upper` The 1-sided upper tolerance bound. This is given only if `side = 1`. `2-sided.lower` The 2-sided lower tolerance bound. This is given only if `side = 2`. `2-sided.upper` The 2-sided upper tolerance bound. This is given only if `side = 2`.

## References

Bury, K. (1999), Statistical Distributions in Engineering, Cambridge University Press.

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

Wald, A. (1943), An Extension of Wilks' Method for Setting Tolerance Limits, The Annals of Mathematical Statistics, 14, 45–55.

Wilks, S. S. (1941), Determination of Sample Sizes for Setting Tolerance Limits, The Annals of Mathematical Statistics, 12, 91–96.

Young, D. S. and Mathew, T. (2014), Improved Nonparametric Tolerance Intervals Based on Interpolated and Extrapolated Order Statistics, Journal of Nonparametric Statistics, 26, 415–432.

`distfree.est`, `npregtol.int`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ``` ## 90%/95% 2-sided nonparametric tolerance intervals for a ## sample of size 20. set.seed(100) x <- rlogis(20, 5, 1) out <- nptol.int(x = x, alpha = 0.10, P = 0.95, side = 1, method = "WILKS", upper = NULL, lower = NULL) out plottol(out, x, plot.type = "both", side = "two", x.lab = "X") ```