g01dc computes an approximation to the variance-covariance matrix of an ordered set of independent observations from a Normal distribution with mean and standard deviation .
- Type: System..::..Int32On entry: , the sample size.Constraint: .
- Type: System..::..DoubleOn entry: the expected value of the largest Normal order statistic, , from a sample of size .
- Type: System..::..DoubleOn entry: the expected value of the second largest Normal order statistic, , from a sample of size .
- Type: System..::..DoubleOn entry: the sum of squares of the expected values of the Normal order statistics from a sample of size .
- Type: array<System..::..Double>()An array of size On exit: the upper triangle of the by variance-covariance matrix packed by column. Thus element is stored in , for .
g01dc is an adaptation of the Applied Statistics Algorithm AS 128, see Davis and Stephens (1978). An approximation to the variance-covariance matrix, , using a Taylor series expansion of the Normal distribution function is discussed in David and Johnson (1954).
However, convergence is slow for extreme variances and covariances. The present method uses the David–Johnson approximation to provide an initial approximation and improves upon it by use of the following identities for the matrix.
For a sample of size , let be the expected value of the th largest order statistic, then:
|(a)||for any ,|
|(c)||the trace of is|
|(d)||where , and . Note that only the upper triangle of the matrix is calculated and returned column-wise in vector form.|
David F N and Johnson N L (1954) Statistical treatment of censored data, Part 1. Fundamental formulae Biometrika 41 228–240
Davis C S and Stephens M A (1978) Algorithm AS 128: approximating the covariance matrix of Normal order statistics Appl. Statist. 27 206–212
Errors or warnings detected by the method:
On entry, .
For , where comparison with the exact values can be made, the maximum error is less than .
The time taken by g01dc is approximately proportional to .