$e$ denote a vector of $n$ residuals with mean zero and variance $σ^{2}$ obtained from fitting some model $M$ to a series of data $y$ ,
$\tilde{e}$ denote the largest absolute residual in $e$ , i.e., $| \tilde{e} | \geq | e_{i} |$ for all $i$ , and let $\tilde{y}$ denote the data series $y$ with the observation corresponding to $\tilde{e}$ having been omitted,
${\tilde{σ}}^{2}$ denote the residual variance on fitting model $M$ to $\tilde{y}$ ,
$λ$ denote the ratio of $\tilde{σ}$ and $σ$ with $λ = \frac{\tilde{σ}}{σ}$ .

Peirce's method flags

\tilde{e}

as a potential outlier if

| \tilde{e} | \geq x

, where

x = σ^{2} z

and

z

is obtained from the solution of

R = λ^{1 - n} \frac{{(n - 1)}^{n - 1}}{n^{n}}

(1)

where

R = 2 \exp ((\frac{z^{2} - 1}{2}) (1 - Φ (z)))

(2)

and

Φ

is the cumulative distribution function for the standard Normal distribution.

Unlike g07gac, both

σ^{2}

and

{\tilde{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 function uses an algorithm described in e04abc to refine a lower,

l

, and upper,

u

, limit for

x

. This refinement stops when

| \tilde{e} | < l

| \tilde{e} | > u

4 References

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of observations.

Constraint:

n \geq 3

2: $e$ – double Input

On entry:

\tilde{e}

, the value being tested.

3: $var1$ – double Input

On entry:

σ^{2}

, the residual variance on fitting model

M

y

Constraint:

var1 > 0.0

4: $var2$ – double Input

On entry:

{\tilde{σ}}^{2}

, the residual variance on fitting model

M

\tilde{y}

Constraints:

$var2 > 0.0$ ;
$var2 < var1$ .

5: $x$ – double * Output

On exit: an estimated value of

x

, the cutoff that indicates an outlier.

6: $lx$ – double * Output

On exit:

l

, the lower limit for

x

7: $ux$ – double * Output

On exit:

u

, the upper limit for

x

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 3$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $var1 = ⟨ value ⟩$ .
Constraint: $var1 > 0.0$ .

On entry, $var2 = ⟨ value ⟩$ .
Constraint: $var2 > 0.0$ .
NE_REAL_2: On entry, $var1 = ⟨ value ⟩$ , $var2 = ⟨ value ⟩$ .
Constraint: $var2 < var1$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

g07gbc is not threaded in any implementation.

9 Further Comments

None.

10 Example

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.

g07gb: FL CL CPP AD

NAG CL Interfaceg07gbc (outlier_​peirce_​2var)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g07gbc (outlier_peirce_2var)