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Parametric Families of Distributions and Their Interaction with the Workshop Title

This workshop by Chris Jones from The Open University, U.K., explores the interaction of different families of distributions with a focus on skewness, kurtosis, and tail weights. It delves into various properties of order statistic distributions and provides tractable examples to understand these concepts better. The workshop also discusses the practical application of these distributions in robust estimation of location and scale using smooth (nonparametric) Quantile Regression (QR).

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Parametric Families of Distributions and Their Interaction with the Workshop Title

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  1. Parametric Families of Distributions and Their Interaction with the Workshop Title Chris Jones The Open University, U.K.

  2. How the talk will pan out … • it will start as a talk in distribution theory • concentrating on generating one family of distributions • then will continue as a talk in distribution theory • concentrating on generating a different family of distributions • but in this second part, the talk will metamorphose through links with kernels and quantiles … • … and finally get on to a more serious application to smooth (nonparametric) QR • the parts of the talk involving QR are joint withKeming Yu

  3. Set Starting point: simple symmetric gHow might we introduce (at most two) shape parameters a and b which will account for skewness and/or “kurtosis”/tailweight (while retaining unimodality)?Modelling data with such families of distributions will, inter alia, afford robust estimation of location (and maybe scale).

  4. FAMILY 1 g

  5. Actual density of order statistic: (i,n integer) Generalised density of order statistic: (a,b>0 real)

  6. Roles of a and b • a=b=1: f = g • a=b: family of symmetric distributions • a≠b: skew distributions • a controls left-hand tail weight, b controls right • the smaller a or b, the heavier the corresponding tail

  7. Properties of (Generalised) Order Statistic Distributions • Distribution function: • Tail behaviour. For large x>0: • power tails: • exponential tails: • Limiting distributions: • a and b large: normal distribution • one of a or b large, appropriate extreme value distribution Other properties such as moments and modality need to be examined on a case-by-case basis For more, see Jones (2004, Test)

  8. Tractable Example 1 Jones & Faddy’s (2003, JRSSB)skew t density When a=b, Student t density on 2a d.f.

  9. Some skew t densities

  10. … and with a and b swopped

  11. f = skew t density arises from ??? g

  12. Yes, the t distribution on 2 d.f.!

  13. Tractable Example 2 • : The (order statistics of the) logistic distribution generate the ??? • : Log F distribution • This has exponential tails

  14. These examples, seen before, are therefore log F distributions

  15. The log F distribution

  16. The simple exponential tail property is shared by: • the log F distribution • the asymmetric Laplace distribution • the hyperbolic distribution Is there a general form for such distributions?

  17. FAMILY 2: distributions with simple exponential tails Starting point: simple symmetric g with distribution function G and General form for density is:

  18. Special Cases • G is point mass at zero, G^[2]=xI(x>0) • f is asymmetric Laplace • G is logistic, G^[2]=log(1+exp(x)) • f is log F • G is t_2, G^[2]=½(x+√(1+x^2)) • f is hyperbolic • G is normal, G^[2]= xΦ(x)+φ(x) • G uniform, G^[2]=½(1+x)I(-1<x<1)+I(x>1)

  19. solid line: log F dashed line: hyperbolic dotted line: normal-based

  20. Practical Point 1 • the asymmetric Laplace is a three parameter distribution; other members of family have four; • fourth parameter is redundant in practice: (asymptotic) correlations between ML estimates of σ and either of a or b are very near 1; • reason: σ, a and b are all scale parameters, yet you only need two such parameters to describe main scale-related aspects of distribution [either (i) a left-scale and a right-scale or (ii) an overall scale and a left-right comparer]

  21. Practical Point 2Parametrise by μ, σ, a=1-p, b=p.Then, score equation for μ reads: This is kernel quantile estimation, with kernel G and bandwidth σ

  22. Includes bandwidth selection by choosing σ to solve the second score equation: But its simulation performance is variable:

  23. And so to Quantile Regression: The usual (regression) log-likelihood, is kernel localised to point x by

  24. Writing and this (version of) DOUBLE KERNEL LOCAL LINEAR QUANTILE REGRESSIONsatisfies Contrast this with Yu & Jones (1998, JASA) version of DKLLQR: where

  25. The ‘vertical’ bandwidth σ=σ(x) can also be estimated by ML: solve Compare 3 versions of DKLLQR: Yu & Jones (1998) including r-o-t σ and h; new version including r-o-t σ and h; new version including above σ and r-o-t h.

  26. Based on this limited evidence: • Clear recommendation: • replace Yu & Jones (1998) DKLLQR method by (gently but consistently improved) new version • Unclear non-recommendation: • use new bandwidth selection?

  27. References

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