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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_normal (g01fa)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_stat_inv_cdf_normal (g01fa) returns the deviate associated with the given probability of the standard Normal distribution.

Syntax

[result, ifail] = g01fa(p, 'tail', tail)

[result, ifail] = nag_stat_inv_cdf_normal(p, 'tail', tail)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:

tail was made optional (default 'L')

Description

The deviate,

x_{p}

associated with the lower tail probability,

p

, for the standard Normal distribution is defined as the solution to

P (X \leq x_{p}) = p = \int_{- \infty}^{x_{p}} Z (X) d X,

where

Z (X) = \frac{1}{\sqrt{2 π}} e^{- X^{2} / 2}, - \infty < X < \infty .

The method used is an extension of that of Wichura (1988).

p

is first replaced by

q = p - 0.5

(a)

|q| \leq 0.3

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = s \frac{A (s^{2})}{B (s^{2})},

where

s = \sqrt{2 π} q

and

A

B

are polynomials of degree

7

(b)

0.3 < |q| \leq 0.42

x_{p}

is computed by a rational Chebyshev approximation

x_{p} = sign q (\frac{C (t)}{D (t)}),

where

t = |q| - 0.3

and

C

D

are polynomials of degree

5

(c)

|q| > 0.42

x_{p}

is computed as

x_{p} = sign q [(\frac{E (u)}{F (u)}) + u],

where

u = \sqrt{- 2 \times \log (\min (p, 1 - p))}

and

E

F

are polynomials of degree

6

For the upper tail probability

- x_{p}

is returned, while for the two tail probabilities the value

x_{p^{*}}

is returned, where

p^{*}

is the required tail probability computed from the input value of

p

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

Parameters

Compulsory Input Parameters

1: $p$ – double scalar: $p$ , the probability from the standard Normal distribution as defined by tail.

Constraint: $0.0 < p < 1.0$ .

Optional Input Parameters

1: $tail$ – string (length ≥ 1)

Default:

'L'

Indicates which tail the supplied probability represents.

$tail ='L'$: The lower probability, i.e., $P (X \leq x_{p})$ .
$tail ='U'$: The upper probability, i.e., $P (X \geq x_{p})$ .
$tail ='S'$: The two tail (significance level) probability, i.e., $P (X \geq |x_{p}|) + P (X \leq - |x_{p}|)$ .
$tail ='C'$: The two tail (confidence interval) probability, i.e., $P (X \leq |x_{p}|) - P (X \leq - |x_{p}|)$ .

Constraint:

tail ='L'

'U'

'S'

'C'

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

If on exit

ifail \neq 0

, then nag_stat_inv_cdf_normal (g01fa) returns

0.0

$ifail = 1$

On entry,

tail \neq'L'

'U'

'S'

'C'

$ifail = 2$

On entry,	$p \leq 0.0$ ,
or	$p \geq 1.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy is mainly limited by the machine precision.

Further Comments

None.

Example

Four values of tail and p are input and the deviates calculated and printed.

Open in the MATLAB editor: g01fa_example

function g01fa_example


fprintf('g01fa example results\n\n');

p    = [ 0.975;    0.025;        0.950;   0.050];
tail = {'Lower'; 'Upper'; 'Confidence'; 'Significance'};

fprintf('  Tail    probability    deviate\n');
for j = 1:numel(p);

  [x, ifail] = g01fa( ...
                      p(j) ,'tail', tail{j});

  fprintf('%4s%14.3f%14.4f\n', tail{j}(1), p(j), x);
end

g01fa example results

  Tail    probability    deviate
   L         0.975        1.9600
   U         0.025        1.9600
   C         0.950        1.9600
   S         0.050        1.9600

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Chapter Contents

Chapter Introduction

NAG Toolbox