Quantile/expectile regression, and extreme data analysis

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LMS quantile regression LMS quantile regression III First method: the Cole-Green method Given ̂η, the 100α% quantile (e.g., α = 50 for median) is [ ̂µ(x) 1 + ̂λ(x) 1/ λ(x) ̂σ(x) Φ (α/100)] −1 b . (2) i.e., apply the inverse Box-Cox transformation to N(0, 1) quantiles. Easy! A problem with the LMS method is to find justification for the underlying method. © T. W. Yee (University of Auckland) Quantile/expectile regression, and extreme data analysis 18 July 2012 @ Cagliari 12/101/ 101
LMS quantile regression Using the gamma model avoids a range of problems: It has finite expectations of the required derivatives of the likelihood function (not so for the normal version, particularly when σ is small.) The off-diagonal elements of the W i are 0 (or ≈ 0) (relative to the diagonal elts). Unlike the normal case, the range of transformation does not depend on λ. Thus, in the gamma model Y ranges over (0, ∞) for all λ, µ and σ. © T. W. Yee (University of Auckland) Quantile/expectile regression, and extreme data analysis 18 July 2012 @ Cagliari 13/101/ 101 Second Method: The LMS Gamma Method† Lopatatzidis and Green (1998) proposed transforming Y to a gamma distribution—it has some theoretical and practical advantages. Then W = (Y /µ) λ is assumed gamma with unit mean and variance λ 2 σ 2 . The 100α percentile of Y at x is µ(x) Wα 1/λ(x) where W α is the equivalent deviate of size α for the gamma distribution with mean 1 and variance λ(x) 2 σ(x) 2 .
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Page 21 and 22: LMS quantile regression Example wit
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Page 31 and 32: Expectile regression Interpretation
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Page 35 and 36: Expectile regression Expectiles III
Page 37 and 38: Expectile regression Interrelations
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Page 45 and 46: Asymmetric MLE Asymmetric MLE II an
Page 47 and 48: Asymmetric MLE ALS notes I Here are
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Page 51 and 52: Asymmetric MLE Example II 60 ● 20
Page 53 and 54: Asymmetric MLE Estimation† An ite
Page 55 and 56: Asymmetric MLE AML Poisson example
Page 57 and 58: Asymmetric Laplace distribution Asy
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Asymmetric Laplace distribution Exa
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Extreme value data analysis Extreme
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Extreme value data analysis © T. W
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Extreme value data analysis Classic
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Extreme value data analysis GEV II
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Extreme value data analysis GEV IV
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Extreme value data analysis GPD I T
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Extreme value data analysis GPD III
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Extreme value data analysis GPD IX
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Extreme value data analysis GPD XI
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Extreme value data analysis The Blo
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Extreme value data analysis The Blo
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Extreme value data analysis 20 15 0
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Extreme value data analysis ● 180
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Extreme value data analysis Number
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Extreme value data analysis To chec
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Extreme value data analysis Finally
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Concluding remarks Concluding remar
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Quantile/expectile regression, and extreme data analysis

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