NAG Library Function Document

nag_robust_trimmed_1var (g07ddc)

void	nag_robust_trimmed_1var (Integer n, const double x[], double alpha, double tmean, double wmean, double tvar, double wvar, Integer k, double sx[], NagError fail)

3 Description

nag_robust_trimmed_1var (g07ddc) 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}) .

nag_robust_trimmed_1var (g07ddc) 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 – IntegerInput: On entry: the number of observations, $n$ .
Constraint: $n \geq 2$ .
2: x[n] – const doubleInput: On entry: the sample observations, $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: alpha – doubleInput: On entry: the proportion of observations to be trimmed at each end of the sorted sample, $α$ .
Constraint: $0.0 \leq alpha < 0.5$ .
4: tmean – double *Output: On exit: the $α$ -trimmed mean, ${\bar{x}}_{t}$ .
5: wmean – double *Output: On exit: the $α$ -Winsorized mean, ${\bar{x}}_{w}$ .
6: tvar – double *Output: On exit: contains an estimate of the variance of the trimmed mean.
7: wvar – double *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] – doubleOutput: On exit: contains the sample observations sorted into ascending order.
10: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_INT_ARG_LT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL_ARG_GE: On entry, alpha must not be greater than or equal to 0.5: $alpha = ⟨value⟩$ .
NE_REAL_ARG_LT: On entry, alpha must not be less than 0.0: $alpha = ⟨value⟩$ .

7 Accuracy

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

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_robust_trimmed_1var (g07ddc) 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 Function Documentnag_robust_trimmed_1var (g07ddc)

+− Contents

1 Purpose

2 Specification