# NAG Library Function Document

## 1Purpose

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 $1.0$.

## 2Specification

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

## 3Description

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 $i=1,2,\dots ,n$, $\sum _{j=1}^{n}{V}_{ij}=1$ (b) ${V}_{12}={V}_{11}+{m}_{n}^{2}-{m}_{n}{m}_{n-1}-1$ (c) the trace of $V$ is $tr\left(V\right)=n-\sum _{i=1}^{n}{m}_{i}^{2}$ (d) ${V}_{ij}={V}_{ji}={V}_{rs}={V}_{sr}$ 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.

## 4References

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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the sample size.
Constraint: ${\mathbf{n}}>0$.
2:    $\mathbf{exp1}$doubleInput
On entry: the expected value of the largest Normal order statistic, ${m}_{n}$, from a sample of size $n$.
3:    $\mathbf{exp2}$doubleInput
On entry: the expected value of the second largest Normal order statistic, ${m}_{n-1}$, from a sample of size $n$.
4:    $\mathbf{sumssq}$doubleInput
On entry: the sum of squares of the expected values of the Normal order statistics from a sample of size $n$.
5:    $\mathbf{vec}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$doubleOutput
On exit: the upper triangle of the $n$ by $n$ variance-covariance matrix packed by column. Thus element ${V}_{ij}$ is stored in ${\mathbf{vec}}\left[i+j×\left(j-1\right)/2-1\right]$, for $1\le i\le j\le n$.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For $n\le 20$, where comparison with the exact values can be made, the maximum error is less than $0.0001$.

## 8Parallelism and Performance

nag_normal_scores_var (g01dcc) is not threaded in any implementation.

The time taken by nag_normal_scores_var (g01dcc) is approximately proportional to ${n}^{2}$.
The arguments ${\mathbf{exp1}}$ ($={m}_{n}$), ${\mathbf{exp2}}$ ($={m}_{n-1}$) and ${\mathbf{sumssq}}$ ($=\sum _{j=1}^{n}{m}_{j}^{2}$) may be found from the expected values of the Normal order statistics obtained from nag_normal_scores_exact (g01dac) .

## 10Example

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.

### 10.1Program Text

Program Text (g01dcce.c)

None.

### 10.3Program Results

Program Results (g01dcce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017