g01db calculates an approximation to the 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

Syntax

C#
public static void g01db( int n, double[] pp, out int ifail )

Visual Basic
Public Shared Sub g01db ( _ n As Integer, _ pp As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g01db( int n, array<double>^ pp, [OutAttribute] int% ifail )

F#
static member g01db : n : int * pp : float[] * ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the size of the sample.

Constraint: $n \geq 1$ .

pp: Type: array<System..::..Double>[]()[][]
An array of size [n]
On exit: the Normal scores. $pp [i - 1]$ contains the value $E (x_{(i)})$ , for $i = 1, 2, \dots, n$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

g01db is an adaptation of the Applied Statistics Algorithm AS

177.3

, see Royston (1982). If you are particularly concerned with the accuracy with which g01db computes the expected values of the order statistics (see [Accuracy]), then g01da which is more accurate should be used instead at a cost of increased storage and computing time.

Let

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

be the order statistics from a random sample of size

n

from the standard Normal distribution. Defining

P_{r, n} = Φ (- E (x_{(r)}))

and

Q_{r, n} = \frac{r - ε}{n + γ}, r = 1, 2, \dots, n,

where

E (x_{(r)})

is the expected value of

x_{(r)}

, the current method approximates the Normal upper tail area corresponding to

E (x_{(r)})

as,

{\tilde{P}}_{r, n} = Q_{r, n} + \frac{δ_{1}}{n} Q_{r, n}^{λ} + \frac{δ_{2}}{n} Q_{r, n}^{2 λ} - C_{r, n} .

for

r = 1, 2, 3

, and

r \geq 4

. Estimates of

ε

γ

δ_{1}

δ_{2}

and

λ

are obtained. A small correction

C_{r, n}

{\tilde{P}}_{r, n}

is necessary when

r \leq 7

and

n \leq 20

The approximation to

E (X_{(r)})

is thus given by

E (x_{(r)}) = - Φ^{- 1} ({\tilde{P}}_{r, n}), r = 1, 2, \dots, n .

Values of the inverse Normal probability integral

Φ^{- 1}

are obtained from g01fa.

References

Royston J P (1982) Algorithm AS 177: expected normal order statistics (exact and approximate) Appl. Statist. 31 161–165

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,

n < 1

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

For

n \leq 2000

, the maximum error is

0.0001

, but g01db is usually accurate to

5

6

decimal places. For

n

up to

5000

, comparison with the exact scores calculated by g01da shows that the maximum error is

0.001

Parallelism and Performance

None.

Further Comments

The time taken by g01db is proportional to

n

Example

A program to calculate the expected values of the order statistics for a sample of size

10

Example program (C#): g01dbe.cs

Example program data: g01dbe.d

Example program results: g01dbe.r