void	g07ddc (Integer n, const double x[], double alpha, double tmean, double wmean, double tvar, double wvar, Integer k, double sx[], NagError fail)

The function may be called by the names: g07ddc, nag_univar_robust_1var_trimmed or nag_robust_trimmed_1var.

3 Description

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}) .

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

Constraint: $n \geq 2$ .
2: $x [n]$ – const double Input: On entry: the sample observations, $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: $alpha$ – double Input: 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: $sx [n]$ – double Output: On exit: contains the sample observations sorted into ascending order.
10: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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

Background information to multithreading can be found in the Multithreading documentation.

g07ddc is not threaded in any implementation.

9 Further Comments

The time taken by 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.

g07dd: FL CL CPP AD PY MB

NAG CL Interfaceg07ddc (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 CL Interface
g07ddc (robust_1var_trimmed)