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

naginterfaces.library.univar.robust_1var_trimmed(x, alpha)

robust_1var_trimmed calculates the trimmed and Winsorized means of a sample and estimates of the variances of the two means.

For full information please refer to the NAG Library document for g07dd

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g07/g07ddf.html

Parameters

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

The sample observations, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

alphafloat

$\alpha$, the proportion of observations to be trimmed at each end of the sorted sample.

Returns

tmeanfloat

The $\alpha$-trimmed mean, ${\overline{x}}_t$.

wmeanfloat

The $\alpha$-Winsorized mean, ${\overline{x}}_w$.

tvarfloat

Contains an estimate of the variance of the trimmed mean.

wvarfloat

Contains an estimate of the variance of the Winsorized mean.

kint

Contains the number of observations trimmed at each end, $k$.

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

Contains the sample observations sorted into ascending order.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\le 1$.

(errno $2$)

On entry, $alpha=⟨value⟩$.

Constraint: $0.0\le alpha<0.5$.

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.

robust_1var_trimmed calculates the $\alpha$-trimmed mean and $\alpha$-Winsorized mean for a given $\alpha$, as described below.

Let $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$ represent the $n$ sample observations sorted into ascending order. Let $k=\left[\alpha n\right]$ where $\left[y\right]$ represents the integer nearest to $y$; if $2k=n$ then $k$ is reduced by $1$.

Then the trimmed mean is defined as:

$${\overline{x}}_t=\frac{1}{n-2k}\sum _{i=k+1}^{n-k}{x}_i\text{,}$$

and the Winsorized mean is defined as:

$${\overline{x}}_w=\frac{1}{n}(\sum _{i=k+1}^{n-k}{x}_i+k{x}_{k+1}+k{x}_{n-k})\text{.}$$

robust_1var_trimmed then calculates the Winsorized variance about the trimmed and Winsorized means respectively and divides by $n$ to obtain estimates of the variances of the above two means.

Thus we have;

$$\text{Estimate of}var\left({\overline{x}}_t\right)=\frac{1}{n^2}(\sum _{i=k+1}^{n-k}{(x_i-{\overline{x}}_t)}^2+k{(x_{k+1}-{\overline{x}}_t)}^2+k{(x_{n-k}-{\overline{x}}_t)}^2)$$

and

$$\text{Estimate of}var\left({\overline{x}}_w\right)=\frac{1}{n^2}(\sum _{i=k+1}^{n-k}{(x_i-{\overline{x}}_w)}^2+k{(x_{k+1}-{\overline{x}}_w)}^2+k{(x_{n-k}-{\overline{x}}_w)}^2)\text{.}$$

References

Hampel, F R, Ronchetti, E M, Rousseeuw, P J and Stahel, W A, 1986, Robust Statistics. The Approach Based on Influence Functions, Wiley

Huber, P J, 1981, Robust Statistics, Wiley