NAG Library Routine Document

G07DDF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

G07DDF calculates the trimmed and Winsorized means of a sample and estimates of the variances of the two means.

2 Specification

SUBROUTINE G07DDF (

N, X, ALPHA, TMEAN, WMEAN, TVAR, WVAR, K, SX, IFAIL)

INTEGER	N, K, IFAIL
REAL (KIND=nag_wp)	X(N), ALPHA, TMEAN, WMEAN, TVAR, WVAR, SX(N)

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 Parameters

1: $N$ – INTEGERInput: On entry: $n$ , the number of observations.

Constraint: $N \geq 2$ .
2: $X (N)$ – REAL (KIND=nag_wp) arrayInput: 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: $K$ – INTEGEROutput: On exit: contains the number of observations trimmed at each end, $k$ .
9: $SX (N)$ – REAL (KIND=nag_wp) arrayOutput: On exit: contains the sample observations sorted into ascending order.
10: $IFAIL$ – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . 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 \leq 1

$IFAIL = 2$

On entry,	$ALPHA < 0.0$ ,
or	$ALPHA \geq 0.5$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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.

NAG Library Routine DocumentG07DDF