NAG Library Function Document

nag_normal_scores_var (g01dcc)

+− 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

1 Purpose

nag_normal_scores_var (g01dcc) computes an approximation to the variance-covariance matrix of an ordered set of independent observations from a Normal distribution with mean

0.0

and standard deviation

1.0

2 Specification

#include <nag.h>

#include <nagg01.h>

void	nag_normal_scores_var (Integer n, double exp1, double exp2, double sumssq, double vec[], NagError *fail)

3 Description

nag_normal_scores_var (g01dcc) is an adaptation of the Applied Statistics Algorithm AS 128, see Davis and Stephens (1978). An approximation to the variance-covariance matrix,

V

, using a Taylor series expansion of the Normal distribution function is discussed in David and Johnson (1954).

However, convergence is slow for extreme variances and covariances. The present function uses the David–Johnson approximation to provide an initial approximation and improves upon it by use of the following identities for the matrix.

For a sample of size

n

, let

m_{i}

be the expected value of the

i

th largest order statistic, then:

(a)	for any $i = 1, 2, \dots, n$ , $\sum_{j = 1}^{n} V_{i j} = 1$
(b)	$V_{12} = V_{11} + m_{n}^{2} - m_{n} m_{n - 1} - 1$
(c)	the trace of $V$ is $t r (V) = n - \sum_{i = 1}^{n} m_{i}^{2}$
(d)	$V_{i j} = V_{j i} = V_{r s} = V_{s r}$ where $r = n + 1 - i$ , $s = n + 1 - j$ and $i, j = 1, 2, \dots, n$ . Note that only the upper triangle of the matrix is calculated and returned column-wise in vector form.

4 References

David F N and Johnson N L (1954) Statistical treatment of censored data, Part 1. Fundamental formulae Biometrika 41 228–240

Davis C S and Stephens M A (1978) Algorithm AS 128: approximating the covariance matrix of Normal order statistics Appl. Statist. 27 206–212

5 Arguments

1: n – IntegerInput: On entry: $n$ , the sample size.
Constraint: $n > 0$ .
2: exp1 – doubleInput: On entry: the expected value of the largest Normal order statistic, $m_{n}$ , from a sample of size $n$ .
3: exp2 – doubleInput: On entry: the expected value of the second largest Normal order statistic, $m_{n - 1}$ , from a sample of size $n$ .
4: sumssq – doubleInput: On entry: the sum of squares of the expected values of the Normal order statistics from a sample of size $n$ .
5: vec[ $n \times (n + 1) / 2$ ] – doubleOutput: On exit: the upper triangle of the $n$ by $n$ variance-covariance matrix packed by column. Thus element $V_{i j}$ is stored in $vec [i + j \times (j - 1) / 2 - 1]$ , for $1 \leq i \leq j \leq n$ .
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
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.

7 Accuracy

For

n \leq 20

, where comparison with the exact values can be made, the maximum error is less than

0.0001

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_normal_scores_var (g01dcc) is approximately proportional to

n^{2}

The arguments

exp1

(

= m_{n}

exp2

(

= m_{n - 1}

) and

sumssq

(

= \sum_{j = 1}^{n} m_{j}^{2}

) may be found from the expected values of the Normal order statistics obtained from nag_normal_scores_exact (g01dac) .

10 Example

A program to compute the variance-covariance matrix for a sample of size

6

. nag_normal_scores_exact (g01dac) is called to provide values for exp1, exp2 and sumssq.

10.1 Program Text

Program Text (g01dcce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g01dcce.r)

nag_normal_scores_var (g01dcc) (PDF version)

g01 Chapter Contents

g01 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_normal_scores_var (g01dcc)