NAG Library Routine Document

g07gbf (outlier_peirce_2var)


    1  Purpose
    7  Accuracy


g07gbf returns a flag indicating whether a single data point is an outlier as defined by Peirce's criterion.


Fortran Interface
Function g07gbf ( n, e, var1, var2, x, lx, ux, ifail)
Logical:: g07gbf
Integer, Intent (In):: n
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: e, var1, var2
Real (Kind=nag_wp), Intent (Out):: x, lx, ux
C Header Interface
#include nagmk26.h
Nag_Boolean  g07gbf_ (const Integer *n, const double *e, const double *var1, const double *var2, double *x, double *lx, double *ux, Integer *ifail)


g07gbf tests a potential outlying value using Peirce's criterion. Let
Peirce's method flags e~ as a potential outlier if e~x, where x=σ2z and z is obtained from the solution of
R = λ 1-n n-1 n-1 nn (1)
R = 2 exp z2 - 1 2 1- Φz (2)
and Φ is the cumulative distribution function for the standard Normal distribution.
Unlike g07gaf, both σ2 and σ~2 must be supplied and therefore no assumptions are made about the nature of the relationship between these two quantities. Only a single potential outlier is tested for at a time.
This routine uses an algorithm described in e04abf/e04aba to refine a lower, l, and upper, u, limit for x. This refinement stops when e~<l or e~>u.


Gould B A (1855) On Peirce's criterion for the rejection of doubtful observations, with tables for facilitating its application The Astronomical Journal 45
Peirce B (1852) Criterion for the rejection of doubtful observations The Astronomical Journal 45


1:     n – IntegerInput
On entry: n, the number of observations.
Constraint: n3.
2:     e – Real (Kind=nag_wp)Input
On entry: e~, the value being tested.
3:     var1 – Real (Kind=nag_wp)Input
On entry: σ2, the residual variance on fitting model M to y.
Constraint: var1>0.0.
4:     var2 – Real (Kind=nag_wp)Input
On entry: σ~2, the residual variance on fitting model M to y~.
  • var2>0.0;
  • var2<var1.
5:     x – Real (Kind=nag_wp)Output
On exit: an estimated value of x, the cutoff that indicates an outlier.
6:     lx – Real (Kind=nag_wp)Output
On exit: l, the lower limit for x.
7:     ux – Real (Kind=nag_wp)Output
On exit: u, the upper limit for x.
8:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n3.
On entry, var1=value.
Constraint: var1>0.0.
On entry, var1=value, var2=value.
Constraint: var2<var1.
On entry, var2=value.
Constraint: var2>0.0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


Not applicable.

Parallelism and Performance

g07gbf is not threaded in any implementation.

Further Comments



This example reads in a series of values and variances and checks whether each is a potential outlier.
The dataset used is from Peirce's original paper and consists of fifteen observations on the vertical semidiameter of Venus. Each subsequent line in the dataset, after the first, is the result of dropping the observation with the highest absolute value from the previous data and recalculating the variance.

Program Text

Program Text (g07gbfe.f90)

Program Data

Program Data (g07gbfe.d)

Program Results

Program Results (g07gbfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017