g01jd calculates the lower tail probability for a linear combination of (central)

χ^{2}

variables.

Syntax

C#
public static void g01jd( string method, int n, double[] rlam, double d, double c, out double prob, out int ifail )

Visual Basic
Public Shared Sub g01jd ( _ method As String, _ n As Integer, _ rlam As Double(), _ d As Double, _ c As Double, _ <OutAttribute> ByRef prob As Double, _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g01jd( String^ method, int n, array<double>^ rlam, double d, double c, [OutAttribute] double% prob, [OutAttribute] int% ifail )

F#
static member g01jd : method : string * n : int * rlam : float[] * d : float * c : float * prob : float byref * ifail : int byref -> unit

Parameters

method

Type: System..::..String

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

$method = "P"$: Pan's method is used.
$method = "I"$: Imhof's method is used.
$method = "D"$: Pan's method is used if $λ_{i}^{*}$ , for $i = 1, 2, \dots, n$ are at least $1 %$ distinct and $n \leq 60$ ; otherwise Imhof's method is used.

Constraint:

method = "P"

"I"

"D"

n: Type: System..::..Int32
On entry: $n$ , the number of independent standard Normal variates, (central $χ^{2}$ variates).

Constraint: $n \geq 1$ .

rlam: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: the weights, $λ_{i}$ , for $i = 1, 2, \dots, n$ , of the central $χ^{2}$ variables.

Constraint: $rlam [i - 1] \neq d$ for at least one $i$ . If $method = "P"$ , then the $λ_{i}^{*}$ must be at least $1 %$ distinct; see [Further Comments], for $i = 1, 2, \dots, n$ .

d: Type: System..::..Double
On entry: $d$ , the multiplier of the central $χ^{2}$ variables.

Constraint: $d \geq 0.0$ .

c: Type: System..::..Double
On entry: $c$ , the value of the constant.

prob: Type: System..::..Double%
On exit: the lower tail probability for the linear combination of central $χ^{2}$ variables.

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

Let

u_{1}, u_{2}, \dots, u_{n}

be independent Normal variables with mean zero and unit variance, so that

u_{1}^{2}, u_{2}^{2}, \dots, u_{n}^{2}

have independent

χ^{2}

-distributions with unit degrees of freedom. g01jd evaluates the probability that

λ_{1} u_{1}^{2} + λ_{2} u_{2}^{2} + \dots + λ_{n} u_{n}^{2} < d (u_{1}^{2} + u_{2}^{2} + \dots + u_{n}^{2}) + c .

c = 0.0

this is equivalent to the probability that

\frac{λ_{1} u_{1}^{2} + λ_{2} u_{2}^{2} + \dots + λ_{n} u_{n}^{2}}{u_{1}^{2} + u_{2}^{2} + \dots + u_{n}^{2}} < d .

Alternatively let

λ_{i}^{*} = λ_{i} - d,   ​ i = 1, 2, \dots, n,

then g01jd returns the probability that

λ_{1}^{*} u_{1}^{2} + λ_{2}^{*} u_{2}^{2} + \dots + λ_{n}^{*} u_{n}^{2} < c .

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

n \geq 6

then a non-adaptive method described in d01bd is used to compute the value of the integral otherwise d01aj is used.

Pan's procedure can only be used if the

λ_{i}^{*}

are sufficiently distinct; g01jd requires the

λ_{i}^{*}

to be at least

1 %

distinct; see [Further Comments]. If the

λ_{i}^{*}

are at least

1 %

distinct and

n \leq 60

, then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.

References

Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227

Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426

Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$n < 1$ ,
or	$d < 0.0$ ,
or	$method \neq "P"$ , $"I"$ or $"D"$ .

$ifail = 2$: On entry, $rlam [i - 1] = d$ for all values of $i$ , for $i = 1, 2, \dots, n$ .

$ifail = 3$: On entry, $method = "P"$ yet two successive values of the ordered $λ_{i}^{*}$ , for $i = 1, 2, \dots, n$ , were not at least $1 %$ distinct.

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

Accuracy

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

Parallelism and Performance

None.

Further Comments

Pan's procedure can only work if the

λ_{i}^{*}

are sufficiently distinct. g01jd uses the check

|w_{j} - w_{j - 1}| \geq 0.01 \times \max (|w_{j}|, |w_{j - 1}|)

, where the

w_{j}

are the ordered nonzero values of

λ_{i}^{*}

For the situation when all the

λ_{i}

are positive g01jc may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by g01ep.