g01da computes a set of Normal scores, i.e., the expected values of an ordered set of independent observations from a Normal distribution with mean and standard deviation .
Public Shared Sub g01da ( _ n As Integer, _ pp As Double(), _ etol As Double, _ <OutAttribute> ByRef errest As Double, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g01da( int n, array<double>^ pp, double etol, [OutAttribute] double% errest, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: , the size of the set.Constraint: .
- Type: array<System..::..Double>()An array of size [n]On exit: the Normal scores. contains the value , for .
- Type: System..::..DoubleOn entry: the maximum value for the estimated absolute error in the computed scores.Constraint: .
- Type: System..::..Double%On exit: a computed estimate of the maximum error in the computed scores (see [Accuracy]).
If a sample of observations from any distribution (which may be denoted by ), is sorted into ascending order, the th smallest value in the sample is often referred to as the th ‘order statistic’, sometimes denoted by (see Kendall and Stuart (1969)).
The order statistics therefore have the property
(If , is the sample median.)
For samples originating from a known distribution, the distribution of each order statistic in a sample of given size may be determined. In particular, the expected values of the order statistics may be found by integration. If the sample arises from a Normal distribution, the expected values of the order statistics are referred to as the ‘Normal scores’. The Normal scores provide a set of reference values against which the order statistics of an actual data sample of the same size may be compared, to provide an indication of Normality for the sample (see (G01AHF not in this release)). Normal scores have other applications; for instance, they are sometimes used as alternatives to ranks in nonparametric testing procedures.
g01da computes the th Normal score for a given sample size as
and denotes the complete beta function.
The method attempts to evaluate the scores so that the estimated error in each score is less than the value etol specified by you. All integrations are performed in parallel and arranged so as to give good speed and reasonable accuracy.
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
Errors or warnings detected by the method:
On entry, . On entry, .
Errors are introduced by evaluation of the functions and errors in the numerical integration process. Errors are also introduced by the approximation of the true infinite range of integration by a finite range but and are chosen so that this effect is of lower order than that of the other two factors. In order to estimate the maximum error the functions are also integrated over the range . g01da returns the estimated maximum error as
The program below generates the Normal scores for samples of size , , , and prints the scores and the computed error estimates.