Dixon {dixonTest}

Dixon distribution

Description

Density, distribution function, quantile function and random generation for Dixon's ratio statistics r_{j,i-1} for outlier detection.

Usage

qdixon(p, n, i = 1, j = 1, log.p = FALSE, lower.tail = TRUE)

pdixon(q, n, i = 1, j = 1, lower.tail = TRUE, log.p = FALSE)

ddixon(x, n, i = 1, j = 1, log = FALSE)

rdixon(n, i = 1, j = 1)

Arguments

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required

i

number of observations <= x_i

j

number of observations >= x_j

log.p

logical; if TRUE propabilities p are given as log(p)

lower.tail

logical; if TRUE (default), probabilities are P[X <= x] otherwise, P[X > x].

q

vector of quantiles

x

vector of quantiles.

log

logical; if TRUE (default), probabilities p are given as log(p).

Details

According to McBane (2006) the density of the statistics r_{j,i-1} of Dixon can be yield if x and v are integrated over the range (-\infty < x < \infty, 0 \le v < \infty)

\begin{array}{lcl} f(r) & = & \frac{n!}{\left(i-1\right)! \left(n-j-i-1\right)!\left(j-1\right)!} \\ & & \times \int_{-\infty}^{\infty} \int_{0}^{\infty} \left[\int_{-\infty}^{x-v} \phi(t)dt\right]^{i-1} \left[\int_{x-v}^{x-rv} \phi(t)dt \right]^{n-j-i-1} \\ & & \times \left[\int_{x-rv}^x \phi(t)dt \right]^{j-1} \phi(x-v)\phi(x-rv)\phi(x)v ~ dv ~ dx \\ \end{array}

where v is the Jacobian and \phi(.) is the density of the standard normal distribution. McBane (2006) has proposed a numerical solution using Gaussian quadratures (Gauss-Hermite quadrature and half-range Hermite quadrature) and coded a library in Fortran. These R functions are wrapper functions to use the respective Fortran code.

Value

ddixon gives the density function, pdixon gives the distribution function, qdixon gives the quantile function and rdixon generates random deviates.

Source

The R code is a wrapper to the Fortran code released under GPL >=2 in the electronic supplement of McBane (2006). The original files are ‘rfuncs.f’, ‘utility.f’ and ‘dixonr.fi’. They were slightly modified to comply with current CRAN policy and the R manual ‘Writing R Extensions’.

Note

The file ‘slowTest/d-p-q-r-tests.R.out.save’ that is included in this package contains some results for the assessment of the numerical accuracy.

The slight numerical differences between McBane's original Fortran output (see files ‘slowTests/test[1,2,4].ref.output.txt’) and this implementation are related to different floating point rounding algorithms between R (see ‘round to even’ in round) and Fortran's write(*,'F6.3') statement.

References

Dixon, W. J. (1950) Analysis of extreme values. Ann. Math. Stat. 21, 488–506. doi:10.1214/aoms/1177729747.

Dean, R. B., Dixon, W. J. (1951) Simplified statistics for small numbers of observation. Anal. Chem. 23, 636–638. doi:10.1021/ac60052a025.

McBane, G. C. (2006) Programs to compute distribution functions and critical values for extreme value ratios for outlier detection. J. Stat. Soft. 16. doi:10.18637/jss.v016.i03.

Examples

set.seed(123)
n <- 20
Rdixon <- rdixon(n, i = 3, j = 2)
Rdixon
pdixon(Rdixon, n = n, i = 3, j = 2)
ddixon(Rdixon, n = n, i = 3, j = 2)