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NAG[g01jdc] NAG[nag_prob_lin_chi_sq] - Computes lower tail probability for a linear combination of (central) variables

Calling Sequence

g01jdc(method, rlam, d, c, prob, 'n'=n, 'fail'=fail)

nag_prob_lin_chi_sq(. . .)

Parameters

method - String;

On entry: indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.

Pan's method is used.

Imhof's method is used.

Pan's method is used if , for are at least distinct and ; otherwise Imhof's method is used.

Constraint: "Nag_LCCPan", "Nag_LCCImhof" or "Nag_LCCDefault". .

rlam - Vector(1..n, datatype=float[8]);

On entry: the weights, , for , of the central variables.

Constraint: for at least one . If , then the , for , must be at least distinct; see Section [Further Comments]. .

d - float;

On entry: , the multiplier of the central variables.

Constraint: . .

c - float;

On entry: , the value of the constant.

prob - assignable;

Note: On exit the variable prob will have a value of type float.

On exit: the lower tail probability for the linear combination of central variables.

'n'=n - integer; (optional)

On entry: , the number of independent standard Normal variates, (central variates).

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_prob_lin_chi_sq (g01jdc) calculates the lower tail probability for a linear combination of (central) variables.

Description

Let be independent Normal variables with mean zero and unit variance, so that have independent -distributions with unit degrees of freedom. nag_prob_lin_chi_sq (g01jdc) evaluates the probability that

If this is equivalent to the probability that

(lambda[1]u[1]^2+lambda[2]u[2]^2+&ctdot;+lambda[n]u[n]^2)/(u[1]^2+u[2]^2+&ctdot;+u[n]^2)<d

Alternatively let

then nag_prob_lin_chi_sq (g01jdc) returns the probability that

Two methods are available. One due to Pan (1964) (see Farebrother (1980)) makes use of series approximations. The other method due to Imhof (1961) reduces the problem to a one-dimensional integral. If then a non-adaptive method is used to compute the value of the integral otherwise d01ajc (nag_1d_quad_gen) is used.

Pan's procedure can only be used if the are sufficiently distinct; nag_prob_lin_chi_sq (g01jdc) requires the to be at least distinct; see Section [Further Comments]. If the are at least distinct and , then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_INT"

On entry, . Constraint: .

"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"

On entry, . Constraint: .

"NE_REAL_ARRAY"

On entry, all values of .

"NE_REAL_ARRAY_ENUM"

On entry, but two successive values of were not 1 percent distinct.

	Accuracy
	On successful exit at least four decimal places of accuracy should be achieved.

Further Comments

Pan's procedure can only work if the are sufficiently distinct. nag_prob_lin_chi_sq (g01jdc) uses the check , where the are the ordered non-zero values of .

For the situation when all the are positive g01jcc (nag_prob_lin_non_central_chi_sq) may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by g01epc (nag_prob_durbin_watson).

Examples

>	method := "Nag_LCCPan": n := 10: d := 1: c := 0: rlam := Vector([-9, -7, -5, -3, -1, 2, 4, 6, 8, 10], datatype=float[8]): NAG:-g01jdc(method, rlam, d, c, prob, 'n' = n):