g01ge returns the probability associated with the lower tail of the noncentral beta distribution.

Syntax

C#
public static double g01ge( double x, double a, double b, double rlamda, double tol, int maxit, out int ifail )

Visual Basic
Public Shared Function g01ge ( _ x As Double, _ a As Double, _ b As Double, _ rlamda As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ge( double x, double a, double b, double rlamda, double tol, int maxit, [OutAttribute] int% ifail )

F#
static member g01ge : x : float * a : float * b : float * rlamda : float * tol : float * maxit : int * ifail : int byref -> float

Parameters

x: Type: System..::..Double
On entry: $β$ , the deviate from the beta distribution, for which the probability $P (B \leq β : a, b; λ)$ is to be found.

Constraint: $0.0 \leq x \leq 1.0$ .

a: Type: System..::..Double
On entry: $a$ , the first parameter of the required beta distribution.

Constraint: $0.0 < a \leq 10^{6}$ .

b: Type: System..::..Double
On entry: $b$ , the second parameter of the required beta distribution.

Constraint: $0.0 < b \leq 10^{6}$ .

rlamda: Type: System..::..Double
On entry: $λ$ , the noncentrality parameter of the required beta distribution.

Constraint: $0.0 \leq rlamda \leq - 2.0 \log (U)$ , where $U$ is the safe range parameter as defined by x02am.

tol: Type: System..::..Double
On entry: the relative accuracy required by you in the results. If g01ge is entered with tol greater than or equal to $1.0$ or less than $10 \times machine precision$ (see x02aj), then the value of $10 \times machine precision$ is used instead.
See [Accuracy] for the relationship between tol and maxit.

maxit: Type: System..::..Int32
On entry: the maximum number of iterations that the algorithm should use.
See [Accuracy] for suggestions as to suitable values for maxit for different values of the parameters.

Suggested value: $500$ .

Constraint: $maxit \geq 1$ .

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]).

Return Value

g01ge returns the probability associated with the lower tail of the noncentral beta distribution.

Description

The lower tail probability for the noncentral beta distribution with parameters

a

and

b

and noncentrality parameter

λ

P (B \leq β : a, b; λ)

, is defined by

P (B \leq β : a, b; λ) = \sum_{j = 0}^{\infty} e^{- λ / 2} \frac{(λ / 2)}{j!} P (B \leq β : a, b; 0),

(1)

where

P (B \leq β : a, b; 0) = \frac{Γ (a + b)}{Γ (a) Γ (b)} \int_{0}^{β} B^{a - 1} {(1 - B)}^{b - 1} d B,

which is the central beta probability function or incomplete beta function.

Recurrence relationships given in Abramowitz and Stegun (1972) are used to compute the values of

P (B \leq β : a, b; 0)

for each step of the summation (1).

The algorithm is discussed in Lenth (1987).

References

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

Lenth R V (1987) Algorithm AS 226: Computing noncentral beta probabilities Appl. Statist. 36 241–244

Error Indicators and Warnings

Note: g01ge may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

$ifail = 1$

On entry,	$a \leq 0.0$ ,
or	$a > 10^{6}$ ,
or	$b \leq 0.0$ ,
or	$b > 10^{6}$ ,
or	$rlamda < 0.0$ ,
or	$rlamda > - 2.0 \log (U)$ , where $U =$ safe range parameter as defined by x02am,
or	$x < 0.0$ ,
or	$x > 1.0$ ,
or	$maxit < 1$ .

If on exit

ifail = 1

then g01ge returns zero.

$ifail = 2$: The solution has failed to converge in maxit iterations. You should try a larger value of maxit or tol. The returned value will be an approximation to the correct value.

$ifail = 3$: The probability is too close to $0.0$ or $1.0$ for the algorithm to be able to calculate the required probability. g01ge will return $0.0$ or $1.0$ as appropriate, this should be a reasonable approximation.

$ifail = 4$: The required accuracy was not achieved when calculating the initial value of $P (B \leq β : a, b; λ)$ . You should try a larger value of tol. The returned value will be an approximation to the correct value.

$ifail = -9000$: An error occured, see message report.

Accuracy

Convergence is theoretically guaranteed whenever

P (Y > maxit) \leq tol

where

Y

has a Poisson distribution with mean

λ / 2

. Excessive round-off errors are possible when the number of iterations used is high and tol is close to machine precision. See Lenth (1987) for further comments on the error bound.