Download presentation

Presentation is loading. Please wait.

Published bySydney Flood Modified over 3 years ago

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 with Keming Yu

3
Set Starting point: simple symmetric g How 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: Generalised density of order statistic: (i,n integer) (a,b>0 real)

6
Roles of a and b a=b=1: f = g a=b: family of symmetric distributions ab: 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 & Faddys (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 Q: The (order statistics of the) logistic distribution generate the ??? A: 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 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 2 Parametrise 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
this (version of) DOUBLE KERNEL LOCAL LINEAR QUANTILE REGRESSION satisfies Writing and 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.

27
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?

28
References

Similar presentations

Presentation is loading. Please wait....

OK

Chapter 1: Introduction to Statistics

Chapter 1: Introduction to Statistics

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Cg ppt online application form 2012 Ppt on combination of resistances eve Ppt on as 14 amalgamation meaning Ppt on branches of physics Free ppt on swine flu Ppt on indian mathematicians and their contributions free download Ppt on financial system and economic development Ppt on buddhist architecture in india Ppt on industrial employment standing order act 1946 Pptx to ppt online free