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>

void	g01dac (Integer n, double pp[], double etol, double errest, NagError fail)

The function may be called by the names: g01dac, nag_stat_normal_scores_exact or nag_normal_scores_exact.

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.

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: $n$ – Integer Input: On entry: $n$ , the size of the set.

Constraint: $n > 0$ .
2: $pp [n]$ – double Output: On exit: the Normal scores. $pp [i - 1]$ contains the value $E (x_{(i)})$ , for $i = 1, 2, \dots, n$ .
3: $etol$ – double Input: 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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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. The best result obtained is returned together with the associated value of errest.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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]

. 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

g01dac is not threaded in any implementation.

9 Further Comments

The time taken by 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 g01dac and the sample quantiles obtained by sorting the observed data using m01cac. A reference line at