Deviates for the Normal distribution

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NAG[g01fac] NAG[nag_deviates_normal] - Deviates for the Normal distribution

Calling Sequence

g01fac(tail, p, 'fail'=fail)

nag_deviates_normal(. . .)

Parameters

tail - String;

On entry: indicates which tail the supplied probability represents.

The lower tail probability, i.e., .

The upper tail probability, i.e., .

The two tail (significance level) probability, i.e., .

The two tail (confidence interval) probability, i.e., .

Constraint: "Nag_UpperTail", "Nag_LowerTail", "Nag_TwoTailSignif" or "Nag_TwoTailConfid". .

p - float;

On entry: the probability, , from the standard Normal distribution as defined by tail.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_deviates_normal (g01fac) returns the deviate associated with the given probability of the standard Normal distribution.

Description

The deviate, associated with the lower tail probability, , for the standard Normal distribution is defined as the solution to:

P(X,<=,x[p])=p=(∫)[-infinity]^(x[p])Z(X)dX

where

Z(X) = e^(-(1/2)*X^2)/sqrt(2*Pi), -infinity < X and X < infinity

The method used is an extension of that of Beasley and Springer (1977). is first replaced by .

a. if , is computed by a rational Chebyshev approximation

x[p] = s*A(s^2)/B(s^2)

where and , are polynomials of degree 7.

b. if , is computed by a rational Chebyshev approximation

where and , are polynomials of degree 5.

c. if , is computed as

where and , are polynomials of degree 6.

For the upper tail probability is returned while for the two tail probabilities the value is returned where is the required tail probability computed from the input value of .

	Error Indicators and Warnings
	"NE_BAD_PARAM" On entry, argument tail had an illegal value. "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 consult NAG for assistance. "NE_REAL_ARG_GE" On entry, p must not be greater than or equal to : . "NE_REAL_ARG_LE" On entry, p must not be less than or equal to : .

	Accuracy
	Accuracy is mainly limited by the machine precision.

	Further Comments
	None.

Examples

>	tail := "Nag_LowerTail": p := 0.975: NAG:-g01fac(tail, p);

	See Also
	Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications Beasley J D and Springer S G (1977) Algorithm AS111. The percentage points of the normal distribution Appl. Statist. 26 118–120 Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworths g01 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.