naginterfaces.library.univar.robust_​1var_​median

naginterfaces.library.univar.robust_1var_median(x)

robust_1var_median finds the median, median absolute deviation, and a robust estimate of the standard deviation for a set of ungrouped data.

For full information please refer to the NAG Library document for g07da

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g07/g07daf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The vector of observations, $x_1,x_2,\dots ,x_n$.

Returns

yfloat, ndarray, shape $\left(n\right)$

The observations sorted into ascending order.

xmefloat

The median, ${\theta }_{med}$.

xmdfloat

The median absolute deviation, ${\sigma }_{med}$.

xsdfloat

The robust estimate of the standard deviation, ${\sigma }_{med}^{\prime }$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The data consists of a sample of size $n$, denoted by $x_1,x_2,\dots ,x_n$, drawn from a random variable $X$.

robust_1var_median first computes the median,

$${\theta }_{med}={med}_i\left\{x_i\right\}\text{,}$$

and from this the median absolute deviation can be computed,

$${\sigma }_{med}={med}_i\left\{|x_i-{\theta }_{med}|\right\}\text{.}$$

Finally, a robust estimate of the standard deviation is computed,

$${\sigma }_{med}^{\prime }={\sigma }_{med}/{\Phi }^{-1}\left(0.75\right)$$

where ${\Phi }^{-1}\left(0.75\right)$ is the value of the inverse standard Normal function at the point $0.75$.

robust_1var_median is based upon function LTMDDV within the ROBETH library, see Marazzi (1987).

References

Huber, P J, 1981, Robust Statistics, Wiley

Marazzi, A, 1987, Subroutines for robust estimation of location and scale in ROBETH, Cah. Rech. Doc. IUMSP, No. 3 ROB 1, Institut Universitaire de Médecine Sociale et Préventive, Lausanne