NAG Library Routine Document

G07DDF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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
*double precision*	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 xi, for i=1,2,…,n represent the n sample observations sorted into ascending order. Let k=αn where y represents the integer nearest to y; if 2k=n then k is reduced by 1.

Then the trimmed mean is defined as:

x-t = 1 n-2k ∑ i=k+1 n-k xi ,

and the Winsorized mean is defined as:

x-w = 1n ∑ i=k+ 1 n-k xi + k,xk+ 1 + k,xn-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 x-t = 1n2 ∑ i=k+1 n-k xi - x-t 2 + k xk+1 - x-t 2 + k xn-k - x-t 2

and

Estimate of var x-w = 1 n2 ∑ i=k+ 1 n-k xi - x-w 2 + k xk+ 1 - x-w 2 + k xn-k - 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≥2.
2: X(N) – double precision arrayInput: On entry: the sample observations, xi, for i=1,2,…,n.
3: ALPHA – double precisionInput: On entry: α, the proportion of observations to be trimmed at each end of the sorted sample.
Constraint: 0.0≤ALPHA<0.5.
4: TMEAN – double precisionOutput: On exit: the α-trimmed mean, x-t.
5: WMEAN – double precisionOutput: On exit: the α-Winsorized mean, x-w.
6: TVAR – double precisionOutput: On exit: contains an estimate of the variance of the trimmed mean.
7: WVAR – double precisionOutput: 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) – double precision 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.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).
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.

6 Error Indicators and Warnings

If on entry IFAIL=0 or -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≤1.

IFAIL=2: On entry, ALPHA<0.0,
or ALPHA≥0.5.

7 Accuracy

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

8 Further Comments

The time taken is proportional to n.

9 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.

On entry,	ALPHA<0.0,
or	ALPHA≥0.5.

NAG Library Routine DocumentG07DDF

+− Contents

NAG Library Routine Document

G07DDF