# pmsd: Median scaled difference probabilities and quantiles In metRology: Support for Metrological Applications

## Description

Cumulative lower tail probability and quantile for median of scaled differences.

## Usage

 1 2 3 4 5 6 7 dmsd(q, n, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) pmsd(q, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199) qmsd(p, n, lower.tail = TRUE, method=c('fast', 'exact', 'even', 'asymp'), max.odd=199)

## Arguments

 q Vector of quantiles. p Vector of probabilities. n Number of observations from which msd was calculated. Unused (and can be missing) for method="asymp" lower.tail logical; if TRUE (the default), probabilities are P[X <= x]; otherwise, P[X > x]. method Calculation method. See details. max.odd Highest odd n for which exact values are calculated.

## Details

pmsd, dmsd and qmsd return probabilities, densities and quantiles, respectively, for the median scaled difference applied to a single observation in a standard normal distribution, where otehr values are also IID normal.

n is the number of observations in the data set of interest and not the degrees of freedom or number of differences (msd for a value x[i] in a set of n observations involves n-1 scaled differences).

n, p and q are recycled to the length of the longest, as necessary.

method determines the method of calculation. For method="fast", probabilities are calculated using monotonic spline interpolation on precalculated probabilities. qmsd with method="fast" is obtained by root-finding on the corresponding spline function using uniroot, and densities are estimated from the first derivative of the interpolating spline. This provides fast calculation, and values for most practical probabilities are within 10^-6 of exact calculations. For high probabilites and for low quantiles (below 0.48) at high n, fast quantile accuracy is poorer due to the very low function gradients in this regions, but is still guaranteed monotonic with p.

For method="exact", probabilities and densities are calculated using quadrature integration for an order statistic. For odd n, this requires a double integral. Values for odd n accordingly take about an order of magnitude longer to obtain than for even n. This can be slow (seconds for a vector of several hundred values of q on an Intel x86 machine running at 1-2GHz). qmsd with method="exact" is obtained by root-finding from pmsd(..., method="excat") using uniroot, and is over an order of magnitude slower than pmsd pmsd.

For method="exact", asymptotic (large n) probabilities, densities and quantiles are returned. n is unused and can be missing.

For method="exact", odd n above max.odd are replaced with the next lower even value. This provides a fair approximation for n above 30 (though the fast method is better) and a good approximation above the default of 199. Values of max.odd above 199 are not recommended as integration can become unstable at high odd n; a warning is issued if max.odd > 199.

For method="even", an exact calculation is performed with any odd n replaced with the next lower even value. This is equivalent to setting method="exact" and max.odd=0. This is provided for interest only; the method="fast" method provides a substantially better approximation for odd n than method="even" and is faster.

Note that these functions are appropriate for the distribution of single values. If seeking an outlier test in a data set of size N, either adjust p for N comparisons before applying qmsd to find a critical value, or adjust the returned p-values using, for example, Holm adjustment.

## Value

A vector of length length(p) or length(q) (or, if longer, length(n)) of cumulative probabilities, densities or quantiles respectively.

## Author(s)

S Ellison s.ellison@lgc.co.uk