NAG Library Function Document

nag_normal_scores_exact (g01dac)

+− 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_exact (g01dac) computes a set of Normal scores, i.e., the expected values 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_exact (Integer n, double pp[], double etol, double errest, NagError fail)

3 Description

If a sample of

n

observations from any distribution (which may be denoted by

x_{1}, x_{2}, \dots, x_{n}

), is sorted into ascending order, the

r

th smallest value in the sample is often referred to as the

r

th ‘order statistic’, sometimes denoted by

x_{(r)}

(see Kendall and Stuart (1969)).

The order statistics therefore have the property

x_{(1)} \leq x_{(2)} \leq \dots \leq x_{(n)} .

(If

n = 2 r + 1

x_{r + 1}

is the sample median.)

For samples originating from a known distribution, the distribution of each order statistic in a sample of given size may be determined. In particular, the expected values of the order statistics may be found by integration. If the sample arises from a Normal distribution, the expected values of the order statistics are referred to as the ‘Normal scores’. The Normal scores provide a set of reference values against which the order statistics of an actual data sample of the same size may be compared, to provide an indication of Normality for the sample . A plot of the data against the scores gives a normal probability plot. Normal scores have other applications; for instance, they are sometimes used as alternatives to ranks in nonparametric testing procedures.

nag_normal_scores_exact (g01dac) computes the

r

th Normal score for a given sample size

n

E (x_{(r)}) = \int_{- \infty}^{\infty} x_{r} d G_{r},

where

d G_{r} = \frac{A_{r}^{r - 1} {(1 - A_{r})}^{n - r} d A_{r}}{β (r, n - r + 1)}, A_{r} = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x_{r}} e^{- t^{2} / 2} d t, r = 1, 2, \dots, n,

and

β

denotes the complete beta function.

The function attempts to evaluate the scores so that the estimated error in each score is less than the value etol specified by you. All integrations are performed in parallel and arranged so as to give good speed and reasonable accuracy.

4 References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

5 Arguments

1: n – IntegerInput: On entry: $n$ , the size of the set.
Constraint: $n > 0$ .
2: pp[n] – doubleOutput: On exit: the Normal scores. $pp [i - 1]$ contains the value $E (x_{(i)})$ , for $i = 1, 2, \dots, n$ .
3: etol – doubleInput: On entry: the maximum value for the estimated absolute error in the computed scores.
Constraint: $etol > 0.0$ .
4: errest – double *Output: On exit: a computed estimate of the maximum error in the computed scores (see Section 7).
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_ERROR_ESTIMATE: The function was unable to estimate the scores with estimated error less than etol.
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.
NE_REAL: On entry, $etol = ⟨value⟩$ .
Constraint: $etol > 0.0$ .

7 Accuracy

Errors are introduced by evaluation of the functions

d G_{r}

and errors in the numerical integration process. Errors are also introduced by the approximation of the true infinite range of integration by a finite range

[a, b]

but

a

and

b

are chosen so that this effect is of lower order than that of the other two factors. In order to estimate the maximum error the functions

d G_{r}

are also integrated over the range

[a, b]

. nag_normal_scores_exact (g01dac) returns the estimated maximum error as

errest = \max_{r} [\max (|a|, |b|) \times |\int_{a}^{b} d G_{r} - 1.0|] .

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_normal_scores_exact (g01dac) depends on etol and n. For a given value of etol the timing varies approximately linearly with n.

10 Example

The program below generates the Normal scores for samples of size

5

10

15

, and prints the scores and the computed error estimates.

10.1 Program Text

Program Text (g01dace.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g01dace.r)

This shows a Q-Q plot for a randomly generated set of data. The normal scores have been calculated using nag_normal_scores_exact (g01dac) and the sample quantiles obtained by sorting the observed data using nag_double_sort (m01cac). A reference line at