The skew-normal distribution and related multivariate families. Adelchi Azzalini. Università di Padova. NordStat 2004: Jyväskylä, 6 10 June 2004

Size: px

Start display at page:

Download "The skew-normal distribution and related multivariate families. Adelchi Azzalini. Università di Padova. NordStat 2004: Jyväskylä, 6 10 June 2004"

Russell Page
5 years ago
Views:

1 The skew-normal distribution and related multivariate families Adelchi Azzalini Università di Padova NordStat 2004: Jyväskylä, 6 10 June 2004 A. Azzalini, NordStat 2004 p.1/53

2 Overview Background & motivation Skew-normal distribution (scalar/multivariate case, extensions) More general families skew-elliptical families, and some special cases a general formulation semi-parametric families Connection with other areas selective sampling stochastic frontier models forms of robust likelihood applications in mathematical geology, finance, etc. Some open problems A. Azzalini, NordStat 2004 p.2/53

3 Some general remarks A. Azzalini, NordStat 2004 p.3/53

4 Motivation Context: parametric families of pdf s for continuous variates Want to build flexible parametric classes extend normal family, in scalar and multivariate case construct families of arbitrary flexibility bridge between parametric and non-parametric approach normal class is central to multivariate data analysis,... but who has ever seen a multivariate normal sample? [Barnett (1979)] A. Azzalini, NordStat 2004 p.4/53

5 An old history (bivariate) Pearson family marginal transformation approach, including: Johnson family transformations and Edgeworth expansions mixtures and compound dimensional case: some approaches are specific of copulae specification via conditional distributions others... [Reviews: Pretorious (1930), Johnson & Kotz ( ), Hutchinson & Lay (1990), Joe (1997), Patil et al. (1976), Kotz et al. (2000 ),... ] A. Azzalini, NordStat 2004 p.5/53

6 Desiderata high flexibility in shape with wide range for indices of skewness and kurtosis mathematically tractable: simple form of density, moments, cdf, etc. closed under marginalization and conditioning retain nice properties of original family parameters should be directly linked to some aspects of pdf association with physical mechanism(s) to support statistical modeling for random number generation allow easy and reliable inference A. Azzalini, NordStat 2004 p.6/53

7 Expanding the Normal family: an approach A. Azzalini, NordStat 2004 p.7/53

8 $) A simple yet useful result Lemma (1D case) symmetric about 0, then are pdf s on and Suppose. is a density for any odd function are independent, and Proof. If ] &(' #%$ " [Azzalini (SJS, 1985) for case A. Azzalini, NordStat 2004 p.8/53

9 ) The skew-normal distribution ( Use lemma to skew the N(0,1) density:. Write for Some basic facts:, obtain N(0,1) if, then if for its cdf Extended version: A. Azzalini, NordStat 2004 p.9/53

10 Some SN pdf s ( ) α = α = pdf SN(0,1, 5) pdf SN(0,1,5) α = 0 α = A. Azzalini, NordStat 2004 p.10/53 pdf SN(0,1,0) pdf SN(0,1,Inf)

11 Forms of genesis, correlation and marginals N(0,1) with Conditioning: if then For Sum of Normal+HN: if iid, then Order statistics: A. Azzalini, NordStat 2004 p.11/53

12 A constructive example group above conditional mean SN Normal W Density H W W>E{W H} separate out group with W above average (given H): A. Azzalini, NordStat 2004 p.12/53

13 Some nice properties is Owen s function,, independent of hence even moments are the same of HN representation odd moments from N, or use (scaled) etc... [Azzalini (1985), Henze (1986), Chiogna (1998)] A. Azzalini, NordStat 2004 p.13/53

14 Early (independent) appearances Conditioning: Birnbaum (1950) obtains pdf of and some properties of first two moments. Motivation: selection process of personnel, in education, etc. Sum of Norm+HN: in Technom. (1964), discussion with contributions by Weinstein, Lipow, Mantel, Wilkinson and Nelson in Bayesian context, O Hagan & Leonard (1976) obtain ESN from a two-stage prior construction Order statistics: Roberts (1966) obtains SN pdf, and its property A. Azzalini, NordStat 2004 p.14/53

15 Other ways of skewing the normal pdf, use previous lemma with many options... [Nadarajah & Kotz (2003)] constant connnection with order statistics of N(0,1) [Balakrishnan (in discussion on Arnold & Beaver, 2002)] or take entirely different routes: two-pieces normal, etc. A. Azzalini, NordStat 2004 p.15/53

16 Skewing non-normal distributions ) exponential power dist n (or GED or Normal for heavy (or light) tails SEP [Azzalini (1986), DiCiccio & Monti (2004)] skew-laplace [Balakrishnan and Ambalagaspitya (1994)] skew-cauchy [Arnold & Beaver (2000b), include multivariate version] skew-logistic [Wahed & Ali (2000)] the wrapped SN on the circle (related to above form but not quite the same) [Pewsey (2000b, 2003)] A. Azzalini, NordStat 2004 p.16/53

17 Statistical aspects A. Azzalini, NordStat 2004 p.17/53

18 An old example: Johannsen s beans data Breadth of Phaseolus Vulgaris Johannsen s data -value d. f. Distribution normal log normal skew normal Normal Log-normal Edgeworth density 1st order nd order Skew-normal Note: data re-grouped from 20 into 16 bins [Charlier (1931), Cramér (1946, p )] breadth (mm) n=12000, source: Charlier, 1931 A. Azzalini, NordStat 2004 p.18/53

19 Likelihood function is regular, but... : always a stationarity point of Fisher expected information matrix becomes singular at [connection with non-standard cases of Rotnitzky et al. (2000)] Both can be avoided by suitable reparametrization for small MLE can occur at some solutions proposed MLE can have more than one local maximum small here is larger than elsewere [Azzalini (1985), Liseo (1990), Azzalini & Capitanio (1999), Pewsey (2000a), Monti (2003), Sartori (2003+) ] A. Azzalini, NordStat 2004 p.19/53

20 An example of profile log-likelihood dataset: otis dataset: otis lambda gamma omega Profile relative 2(logLikelihood) sigma Profile relative 2(logLikelihood) use of direct vs centred parameterization A. Azzalini, NordStat 2004 p.20/53

21 Bayesian inference ESN arises from two-stage prior [O Hagan & Leonard (1976)] Bayesian approach to estimation of parameters can avoid. [Liseo (1990), see also Liseo & Loperfido (2002, to appear)] linear Bayesian estimation for model [Mukhopadhyay & Vidakovic (1995)] A. Azzalini, NordStat 2004 p.21/53

22 Multivariate SN distribution A. Azzalini, NordStat 2004 p.22/53

23 Multivariate versions of 1D genesis Conditioning: marginals function Sum of Norm+HN: function The two mechanisms generate the same class of pdf s [Azzalini & Dalla Valle (1996)] A. Azzalini, NordStat 2004 p.23/53

24 Multivariate SN distribution is if pdf at Standard SN: write., get. If Add location/scale parameters: A. Azzalini, NordStat 2004 p.24/53

Some 0 1 2 3 z 1 0 1 2 3 z 1 pdf s α 1 = 2 α 2 = 6 Ω 12 =

7 0 1 2 3 z2 z2 0 1 2 3 z 1 0 1 2 3 z 1 A.

25 Some z z 1 pdf s α 1 = 2 α 2 = 6 Ω 12 = α 1 = 2 α 2 = Ω 12 = 0.7 α 1 = 2 α 2 = 6 Ω 12 = z2 z z z 1 A. Azzalini, NordStat 2004 p.25/53 α 1 = 2 α 2 = 6 Ω 12 = z z2

26 Some formal properties I are variation-independent Parameters Formal properties depend on e. g. var moment generating function & cumulants: -th order cumulant: constant A. Azzalini, NordStat 2004 p.26/53

27 Some formal properties II Random numbers: any of variate require a, namely computed via cdf of cdf of the class is closed under affine transformations hence also closed under marginalisation A. Azzalini, NordStat 2004 p.27/53

28 A characterisation property A parallel of a well-known property of the Normal class.. and var ) with be a r. v. (in. If Let Put., then such that for any [Gupta & Huang (2002)] A. Azzalini, NordStat 2004 p.28/53

29 Quadratic forms leads to Healy-type diagnostic plot, then and if, such that more general: for an even function for all, [Azzalini & Capitanio (1999, 2003), Genton et al. (2001), Loperfido (2001), Genton & Loperfido (to appear)] A. Azzalini, NordStat 2004 p.29/53

30 A numerical example: AIS data ) Australian Institute of Sport data ( observed Mahalanobis distances chi.square quantiles BMI scatterplot and SN fit using MLE LBM Healy s QQ plot for SN fit A. Azzalini, NordStat 2004 p.30/53

31 ESN and graphical models with pdf Extended SN: const is closed under conditioning ESN class [Arnold & Beaver (2000a)] Graphical models: can build a conditional independence graph for (with some restrictions) relationship between the conditional independence graphs of and of log-likelihood allows a parameter based factorization [Capitanio, Azzalini & Stanghellini (SJS, 2003)] A. Azzalini, NordStat 2004 p.31/53

32 Stochastic processes spatial processes with SN components, application to rainfall data, Bayesian inference via MCMC [Kim & Mallick (2004)] model for non-gaussian noise in signals processing [Gualtierotti (2004)] A. Azzalini, NordStat 2004 p.32/53

33 Multiple constraints where Consider Various cases: terms [Azzalini (1985), Arnold & Beaver (2000a) ] product of term, involves a case [Sahu, Dey & Branco (2004)] Hierarchical SN [Liseo & Loperfido (2003)] Fundamental SN [Arellano-Valle & Genton (to appear)] General SN ' [mentioned in Gupta, González-Farías & Domínguez-Molina (2004), with focus on case ] ' A. Azzalini, NordStat 2004 p.33/53

34 Perturbation of symmetric multivariate distributions A. Azzalini, NordStat 2004 p.34/53

Extend basic result to : (central) symmetry about 0 in N.B. includes elliptically contoured densities are pdf s symmetric about 0 in Lemma. Suppose and and, respectively, then, i. e. is a density for any real-valued odd function for.

35 Extend basic result to : (central) symmetry about 0 in N.B. includes elliptically contoured densities are pdf s symmetric about 0 in Lemma. Suppose and and, respectively, then, i. e. is a density for any real-valued odd function for.. Proof. Essentially the same of case, and other directions. -dimensional Can be extended to [Azzalini & Capitanio (2003), Genton & Loperfido (to appear), see also Arellano-Valle, del Pino & San Martín (2002) for another formulation] A. Azzalini, NordStat 2004 p.35/53

36 A simple stochastic representation, independent. Then, Suppose if if has density,, with and Corollary. If then. for any even function A. Azzalini, NordStat 2004 p.36/53

37 Can obtain high flexibility an example Consider a symmetric Beta pdf, scaled in : perturbed by where are additional parameters. There is more than skewing in the skew-symmetric class A. Azzalini, NordStat 2004 p.37/53

38 Perturbed symmetric Beta pdf s (a,b,p,q) = ( 2, 3, ( 3, 3 ), ( 0, 0 )) (a,b,p,q) = ( 2, 3, ( 8, 8 ), ( 0, 0 )) (a,b,p,q) = ( 3, 1, ( 1, 3 ), ( 2, 1 )) pdf pdf y y y x x x (a,b,p,q) = ( 3, 1.5, ( 3, 1 ), ( 2.5, 1 )) (a,b,p,q) = ( 3, 2, ( 2, 3 ), ( 2, 4 )) (a,b,p,q) = ( 3, 3, ( 1, 1 ), ( 3, 3 )) pdf pdf y y y x x x A. Azzalini, NordStat 2004 p.38/53 pdf pdf

39 Skew-ellipical densities and Route A: use previous lemma, with some [Azzalini & Capitanio (1999)] Route B: apply a random scale factor to a SN variate [Branco & Dey (2001)] Good news: for appropriate choice of the ingredients, essentially the same thing. [Azzalini & Capitanio (2003)] A. Azzalini, NordStat 2004 p.39/53

40 Analogies of SN and S-elliptical families the two stochastic representations for SN variates still hold, replacing normal with elliptical variates, via conditioning: via addition:, also representation via order statistics for case the distribution of quadratic forms is the same for skew-elliptical and elliptical variates with common symmetric component [Azzalini & Capitanio (2003), Fang (2003)] A. Azzalini, NordStat 2004 p.40/53

41 A noteworthy case: skew-, then and If has pdf where Unlimited range of indices of skewness and kurtosis. Suitable for robust inference. for QQ- and PP-plot A. Azzalini, NordStat 2004 p.41/53

42 Example: fiber glass strength ) data on breaking strength of fiber glass ( PP plot for skew t distribution PP plot for normal distribution Glass fiber data: nonparametric, SN and St fit glass glass nonparametric estimate, SN fit, ST fit Probability density function glass PP-plot for Normal model and ST model A. Azzalini, NordStat 2004 p.42/53

43 Example: MM returns ) MM excess returns vs CRSP index ( Histogram of res Density residuals CRSP linear regression with Normal and ST fit Martin Marietta returns histogram of residuals and fitted ST dist n A. Azzalini, NordStat 2004 p.43/53

44 & # & # Example: MM returns dataset: m.marietta Profile deviance PP plot for skew t distribution PP plot for normal distribution log(df) alpha m.marietta m.marietta ) for relative L PP-plot for Normal model and ST model profile A. Azzalini, NordStat 2004 p.44/53

Building expansions of arbitrary pdf s SkSy class: can be factorised as An arbitrary pdf is odd. where is symmetric around 0, and The factorisation depends on the chosen.

45 Building expansions of arbitrary pdf s SkSy class: can be factorised as An arbitrary pdf is odd. where is symmetric around 0, and The factorisation depends on the chosen. denote odd polynomial of degree. The class Let is dense in SkSy-class (under some weak conditions). [ Wang, Boyer & Genton (2004), Ma & Genton (SJS, 2004) ] A. Azzalini, NordStat 2004 p.45/53

46 Some applications A. Azzalini, NordStat 2004 p.46/53

47 Use as a mathematical tool not really an application, still useful to approximate a distribution (see above) local departures from normality, local efficiency,... [Salvan (1986), Durio & Nikitin (2003), Muliere & Nikitin (2004)] to produce alternatives in simulation studies [DiCiccio et al. (1997), Chu et al. (2001), etc.] skew link for dicotomous response analysis [Chen, Dey & Qi (1999)] A. Azzalini, NordStat 2004 p.47/53

48 Connected areas Selected (non-random) sampling and Heckman model response eqn: selection eqn: corr type ESN [Copas & Li (1997)] stochastic frontier model [Aigner, Lowell & Schmidt (1977),..., Tancredi (2003+)] A. Azzalini, NordStat 2004 p.48/53

49 Compositional data i.e. frequent sort of data in geology, among others areas, popular approach: Aitchison ALR transformation to enjoyes a set of convenient formal properties s then apply methods for multivariate Normal data to preserves closure of suitable : and use of in place of transformations of on subcompositions [Mateu-Figueras (Ph.D thesis, 2003), Aitchison, Mateu-Figueras & Ng (2003)] A. Azzalini, NordStat 2004 p.49/53

50 Applications in finance multivariate SN fits well into portfolio selection theory use of skew- allows fat tails [Adcock & Shutes (1999), Adcock & Mead (2003), Harvey et al. (to appear)] incorporating skewness/kurtosis in components of ARCH, GARCH and SV models [Goria (thesis, 1999), Pietrobon (thesis, 2003), De Luca & Loperfido (2004), Cappuccio et al. (2004), Liseo & Loperfido (to appear)] A. Azzalini, NordStat 2004 p.50/53

51 nearly there... A. Azzalini, NordStat 2004 p.51/53

52 Some open problems a multitude of proposals (emphasis on probability side, less about statistics; what about their properties, connections, suitability for data analysis, etc.?) various issues in inference: avoid some peculiarities of MLE alternatives to MLE extend various methods from Normal to non-normal case can approximate any distribution, but what is best type of approximation? how to get approximation with prescribed properties? is all of this really useful? A. Azzalini, NordStat 2004 p.52/53

53 Resources References, some papers, software, etc. at A. Azzalini, NordStat 2004 p.53/53

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew distribution

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew distribution ADELCHI AZZALINI Dipartimento di Scienze Statistiche Università di Padova azzalini@stat.unipd.it ANTONELLA