Description Usage Arguments Details Value Author(s) See Also Examples

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

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`q` |
Vector of quantiles. |

`p` |
Vector of probabilities. |

`n` |
Number of observations from which msd was calculated. Unused (and can be missing)
for |

`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 |

`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.

A vector of length `length(p)`

or `length(q)`

(or, if longer, `length(n)`

) of
cumulative probabilities, densities or quantiles respectively.

S Ellison s.ellison@lgc.co.uk

`msd`

for calculation of MSD values, and `bootMSD`

for
a parametric bootstrap (MCS) method of obtaining *p*-values and quantiles
for the more general non-IID case.

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