g01jc returns the lower tail probability of a distribution of a positive linear combination of random variables.
Public Shared Sub g01jc ( _ a As Double(), _ mult As Integer(), _ rlamda As Double(), _ n As Integer, _ c As Double, _ <OutAttribute> ByRef p As Double, _ <OutAttribute> ByRef pdf As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g01jc( array<double>^ a, array<int>^ mult, array<double>^ rlamda, int n, double c, [OutAttribute] double% p, [OutAttribute] double% pdf, double tol, int maxit, [OutAttribute] int% ifail )
- Type: array<System..::..Double>()On entry: the weights, .Constraint: , for .
- Type: array<System..::..Int32>()On entry: the degrees of freedom, .Constraint: , for .
- Type: array<System..::..Double>()On entry: the noncentrality parameters, .Constraint: , for .
- Type: System..::..Int32On entry: , the number of random variables in the combination, i.e., the number of terms in equation (1).Constraint: .
- Type: System..::..DoubleOn entry: , the point for which the lower tail probability is to be evaluated.Constraint: .
- Type: System..::..Double%On exit: the lower tail probability associated with the linear combination of random variables with degrees of freedom, and noncentrality parameters , for .
- Type: System..::..Double%On exit: the value of the probability density function of the linear combination of variables.
- Type: System..::..Double
- Type: System..::..Int32On entry: the maximum number of terms that should be used during the summation.Suggested value: .Constraint: .
For a linear combination of noncentral random variables with integer degrees of freedom the lower tail probability is
where and are positive constants and where represents an independent random variable with degrees of freedom and noncentrality parameter . The linear combination may arise from considering a quadratic form in Normal variables.
Ruben's method as described in Farebrother (1984) is used. Ruben has shown that (1) may be expanded as an infinite series of the form
where , i.e., the probability that a central is less than .
The value of is set at
unless , in which case
is used, where and , for .
Farebrother R W (1984) The distribution of a positive linear combination of random variables Appl. Statist. 33(3)
Note: g01jc may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If on exit or , then g01jc returns .
On entry, , or , or .
- The central calculation has failed to converge. This is an unlikely exit. A larger value of tol should be tried.
- The solution appears to be too close to or for accurate calculation. The value returned is or as appropriate.
The number of variables is read along with their coefficients, degrees of freedom and noncentrality parameters. The lower tail probability is then computed and printed.