Outlier Tests

Generalized ESD (Rosner 1983)

Iterative test that identifies up to k outliers simultaneously, avoiding masking.

Statusvalidated

Version1.0.0

Minitab equivalent—

Description

Rosner's (1983) Generalized Extreme Studentized Deviate test extends Grubbs to multiple outliers. Up to k candidates are extracted in decreasing order of their maximum studentised deviation; the largest iteration i for which Rᵢ > λᵢ defines the number of outliers. The λᵢ values rest on the t-distribution under Bonferroni correction and remove the masking effect.

Formulas

Rᵢ

R_{i} = \frac{max _{j} ∣ x _{j} - x ˉ _{i} ∣}{s _{i}}

Maximum standardised deviation in iteration i

λᵢ

λ_{i} = \frac{( n - i ) t _{p, n - i - 1}}{( n - i - 1 + t ^{2} ) ( n - i + 1 )}, p = 1 - \frac{α}{2 ( n - i + 1 )}

Critical value at iteration i (Bonferroni)

Assumptions

Data approximately normal apart from the outliers
Recommended for n ≥ 15

Limitations

Maximum k must be set in advance (typically ⌊n/4⌋, default cap 10)
Unreliable on heavily skewed data — transform first or use robust methods

References

Rosner, B. (1983), Percentage points for a generalized ESD many-outlier procedure, Technometrics 25(2), 165–172
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.17