dixonTest (version 1.0.0)

Dixon: Dixon distribution

Description

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

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).

Value

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

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 range \((-\infty < x < \infty, 0 \le v < \infty)\)

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

References

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

Dean, R. B., Dixon, W. J. (1951) Simplified statistics for small numbers of observation. Anal. Chem. 23, 636--638. http://dx.doi.org/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. http://dx.doi.org/10.18637/jss.v016.i03.

Examples

# NOT RUN {
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)

# }