Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	k
Real (Kind=nag_wp), Intent (In)	::	x(n), alpha
Real (Kind=nag_wp), Intent (Out)	::	tmean, wmean, tvar, wvar, sx(n)

C Header Interface

#include <nag.h>

void	g07ddf_ (const Integer n, const double x[], const double alpha, double tmean, double wmean, double tvar, double wvar, Integer k, double sx[], Integer ifail)

The routine may be called by the names g07ddf or nagf_univar_robust_1var_trimmed.

3 Description

g07ddf calculates the

α

-trimmed mean and

α

-Winsorized mean for a given

α

, as described below.

Let

x_{i}

, for

i = 1, 2, \dots, n

represent the

n

sample observations sorted into ascending order. Let

k = [α n]

where

[y]

represents the integer nearest to

y

; if

2 k = n

then

k

is reduced by

1

Then the trimmed mean is defined as:

{\bar{x}}_{t} = \frac{1}{n - 2 k} \sum_{i = k + 1}^{n - k} x_{i},

and the Winsorized mean is defined as:

{\bar{x}}_{w} = \frac{1}{n} (\sum_{i = k + 1}^{n - k} x_{i} + k x_{k + 1} + k x_{n - k}) .

g07ddf 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;

Estimate of ​ var ({\bar{x}}_{t}) = \frac{1}{n^{2}} (\sum_{i = k + 1}^{n - k} {(x_{i} - {\bar{x}}_{t})}^{2} + k {(x_{k + 1} - {\bar{x}}_{t})}^{2} + k {(x_{n - k} - {\bar{x}}_{t})}^{2})

and

Estimate of ​ var ({\bar{x}}_{w}) = \frac{1}{n^{2}} (\sum_{i = k + 1}^{n - k} {(x_{i} - {\bar{x}}_{w})}^{2} + k {(x_{k + 1} - {\bar{x}}_{w})}^{2} + k {(x_{n - k} - {\bar{x}}_{w})}^{2}) .

4 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

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of observations.

Constraint: $n \geq 2$ .
2: $x (n)$ – Real (Kind=nag_wp) array Input: On entry: the sample observations, $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: $alpha$ – Real (Kind=nag_wp) Input: On entry: $α$ , the proportion of observations to be trimmed at each end of the sorted sample.

Constraint: $0.0 \leq alpha < 0.5$ .
4: $tmean$ – Real (Kind=nag_wp) Output: On exit: the $α$ -trimmed mean, ${\bar{x}}_{t}$ .
5: $wmean$ – Real (Kind=nag_wp) Output: On exit: the $α$ -Winsorized mean, ${\bar{x}}_{w}$ .
6: $tvar$ – Real (Kind=nag_wp) Output: On exit: contains an estimate of the variance of the trimmed mean.
7: $wvar$ – Real (Kind=nag_wp) Output: On exit: contains an estimate of the variance of the Winsorized mean.
8: $k$ – Integer Output: On exit: contains the number of observations trimmed at each end, $k$ .
9: $sx (n)$ – Real (Kind=nag_wp) array Output: On exit: contains the sample observations sorted into ascending order.
10: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = 〈value〉$ .
Constraint: $n \leq 1$ .

$ifail = 2$: On entry, $alpha = 〈value〉$ .
Constraint: $0.0 \leq alpha < 0.5$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The results should be accurate to within a small multiple of machine precision.

8 Parallelism and Performance

g07ddf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken is proportional to

n

10 Example

The following program finds the

α

-trimmed mean and

α

-Winsorized mean for a sample of

16

observations where

α = 0.15

. The estimates of the variances of the above two means are also calculated.

g07dd: FL CL CPP AD

NAG FL Interfaceg07ddf (robust_​1var_​trimmed)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g07ddf (robust_1var_trimmed)