Two kurtosis measures in a simulation study Anna Maria Fiori - PowerPoint PPT Presentation

Two kurtosis measures in a simulation study Anna Maria Fiori Department of Quantitative Methods for Economics and Business Sciences University of Milano-Bicocca anna.fiori@unimib.it COMPSTAT 2010 CNAM, Paris August 23, 2010

Overview � The IF (SIF) and its role in kurtosis studies � From inequality (Lorenz) ordering to right/left kurtosis measures � Two kurtosis measures: SIF comparison / numerical experiments 2 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Background From Pearson (1905) onwards statistics textbooks have defined kurtosis operationally as the characteristic of a distribution measured by its standardized fourth moment . However… Excess/defect of frequency near the mean compared to a normal Non-Gaussian type of curve (Pearson, 1894-1905) symmetry (First Ed. ESS) Reverse tendency 4 − µ X   β = E   Tailedness only (Ali, 1974) toward bimodality σ 2   (Darlington, 1970) Peakedness + tailedness Dispersion around the two (Dyson, 1943 – Finucan, 1964) values µ +/ µ +/ µ +/− µ +/ − − σ − σ σ σ (Moors, 1986) 3 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis and the IF An early intuition of L. Faleschini Faleschini ( Statistica , 1948) Hampel (1968) – Ruppert (1987) � � Take a frequency distribution : Consider a probability distribution F Background & method { a r , f r ; Σ f r = P } and the functional β 2 = β 2 ( F ) “How does β 2 change if we throw in � � “To investigate the behaviour of β 2 when a frequency f r is altered, an additional observation at some point x ? ” we compute the partial derivative of β 2 wrt f r ” � Contaminated distribution: F ε = ( 1 − ε ) F + ε δ x (0 < ε < 1) β − β � � � � � � ( ) ε �� �� �� �� � � � � ∂ β ( ) � � = �   ��� � ( ) � � � � � β , � = − β − β β − − γ � � � � ε Main result ε ↓   � � � � � � � � � ∂   � � ( ) � � = − β − β β − − γ � � � � � � � � � � − µ µ � � � = γ = � � − µ where: = � � � σ � σ � with: σ 4 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis and the IF Faleschini’s derivative – computational details µ s = 2, 4 � β =   � � ( ) � ∑ � � � µ = −   ⋅ ⋅ µ µ � �   − � � � � �   = � � i -th power of µ Raw moment of order s − i � � � � − ∑ � � ∑ = ⋅ µ = ⋅ � � � � � − � � � � � � � � = = � � � � ) ( ) � ∂ µ ∂ � � � ) ( ) − � � ∑ ∑ − − = µ − ⋅ − � � ∑ ∑ � � � � � � � � � = − ⋅ � � � � ( � � � � ( � � � � ∂ � ∑ ∂ � � ∑ � � � � � � � ( ) � � ( ) − � � − = µ − µ � � � � = − � � � − � � � � � ( ) { } � � ∂ µ   ∂ ⋅ µ � � ( ) ( ) � � ( ) = − µ − µ − − µ ⋅ ⋅ µ − ∑ � � � � � � � = −   �   − � � � � � ∂ ∂ � � � �   = � � � � 5 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis and the IF Kurtosis explained ( ) ( ) 2 β = − β − β β − IF x F z 2 ( ; , ) 1 In the symmetric case: 2 2 2 2 Quartic function with four real roots: IF = SIF ( ) = µ ± σ β ± β β − x 1 1 , 2 , 3 , 4 2 2 2 Unbounded C T T Local maximum: β 2 at x = µ F F = µ ± σ β Minima: β 2 ( 1 − β 2 ) at x x 1 x 2 µ x 3 x 4 2 This IF suggests that β 2 is likely to be KURTOSIS = overestimated by sample kurtosis peakedness + tailedness for x distant from µ , but but β 2 is dominated by underestimated at tailweight intermediate values of x 6 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis and the IF The normal case and sample kurtosis [ ]   ∂ β − µ 4 − µ 2 ( ) a a     1 1 2 = − − = − + z �� ���������� r r 2       2 3 6 6 3 r ∂ σ σ f P P         r center flank flank Roots of the quartic: = µ − σ = µ + σ a a ( 1 ) ( 3 ) 2 , 334 0 , 742 r r tail tail = µ − σ = µ + σ a a ( 2 ) ( 4 ) 0 , 742 2 , 334 r r 7 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis and the IF The normal case and sample kurtosis [ ]   ∂ β − µ 4 − µ 2 ( ) a a     1 1 2 = − − = − + z �� ���������� r r 2       2 3 6 6 3 r ∂ σ σ f P P         r There are two intervals ( flanks ) of -6 -6 substantial probability (22% each) in which the IF has relatively large negative values (minimum = − 6 ) center These possibly correspond to smaller values of sample kurtosis 0,54 flank flank -> Consistent with underestimation of β 2 by sample kurtosis (on average) 0,22 0,22 tail and undercoverage of confidence tail intervals for β 2 8 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis by inequality Zenga (ESS, 2006); Fiori (Comm. Statist., 2008) ������������������ ����������� �������������������� ��������������� ������������� ������������������ ������ �������������� ��������������� ���������������� S = γ – X | X < γ D = X – γ | X ≥ γ ��������������������� with E ( S ) = δ S < ∞ with E ( D ) = δ D < ∞ ���������� γ �� δ � �� δ � 9 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis by inequality Zenga’s kurtosis ordering (ESS, 2006) � � � Left kurtosis � � � � � � Lorenz curve of S : � � 1 p � 1 ) ∫ = − L p F t dt ( ) ( � � S S δ � � 0 S L S ( p ) L D ( p ) Right kurtosis Zenga’s kurtosis curve Lorenz curve of D : Represents kurtosis by a mirror plot 1 p 1 ) ∫ = − L p F t dt ( ) ( of the Lorenz curves for D and S D D δ 0 D � Unified treatment of symmetric and asymmetric distributions � Kurtosis ordering defined via nested Lorenz curves � Liu, Parelius and Singh (Ann. Statist., 1999) in a multivariate setting 10 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis by inequality Zenga’s kurtosis measures (ESS, 2006) right left right left � 1 [ ] [ ] = + = � + K R D R S K C D C S ( ) ( ) ( ) ( ) � 1 2 with: with: δ 2 = − C D D [ ] � ( ) 1 = − R D ∫ p L p dp � ( ) ( ) E D 2 D ( ) � [ ] dp � δ = − 2 ∫ R S p L p � ( ) ( ) = − S C S S ( ) 1 � E S 2 ( ) (ratios of right/left scale functionals) (right & left Gini indexes) Normalized measures, with values between 0 (minimum kurtosis) and 1 (maximum kurtosis) 11 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis measures A look at the SIF � Symmetric distribution ( γ = 0): � Symmetric contaminant (Ruppert, 1987): First measure of (right) kurtosis 12 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis measures A look at the SIF Second measure of (right) kurtosis 13 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis measures SIF comparison at standard normal All the measures are increased by contamination in the tails and at the center and are decreased by contamination in the shoulders/flanks. Having unbounded SIF, they are sensitive to the location of tail outliers as well as their frequency. However, conventional kurtosis is much more sensitive (quartic SIF). The magnitude of min SIF is considerably larger for conventional kurtosis. 14 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis measures Monte Carlo experiment (N = 20,000; n = 20 to 640) Relative bias Tukey λ = -0.089 Normal Laplace higher peak heavier tails ������������ 15 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010

Kurtosis measures Small/medium sample behaviour Bootstrap confidence intervals at 95% level (percentile method) B = 2,000 bootstrap resamples; N = 10,000 replications Empirical coverage Average length � K 2 is the measure which is likely to be estimated with the highest accuracy � The sample performance of K 2 improves as the parent distribution becomes more peaked (Laplace) 16 Anna M. Fiori Two kurtosis measures… COMPSTAT 2010