Fat-Tailed Distributions
Data, Diagnostics and Dependence, Volume 1
Roger M. Cooke ; Daan Nieboer ; Jolanta Misiewicz
- This
- title
- is
- written
- for
- the
- numerate
- nonspecialist,
- and
- hopes
- to
- serve
- three
- purposes.
- First
- it
- gathers
- mathematical
- material
- from
- diverse
- but
- related
- fields
- of
- order
- statistics,
- records,
- extreme
- value
- theory,
- majorization,
- regular
- variation
- and
- subexponentiality.
- All
- of
- these
- are
- relevant
- for
- understanding
- fat
- tails,
- but
- they
- are
- not,
- to
- our
- knowledge,
- brought
- together
- in
- a
- single
- source
- for
- the
- target
- readership.
- Proofs
- that
- give
- insight
- are
- included,
- but
- for
- most
- fussy
- calculations
- the
- reader
- is
- referred
- to
- the
- excellent
- sources
- referenced
- in
- the
- text.
- Multivariate
- extremes
- are
- not
- treated.
- This
- allows
- us
- to
- present
- material
- spread
- over
- hundreds
- of
- pages
- in
- specialist
- texts
- in
- twenty
- pages.
- Chapter
- 5
- develops
- new
- material
- on
- heavy
- tail
- diagnostics
- and
- gives
- more
- mathematical
- detail.
- Since
- variances
- and
- covariances
- may
- not
- exist
- for
- heavy
- tailed
- joint
- distributions,
- Chapter
- 6
- reviews
- dependence
- concepts
- for
- certain
- classes
- of
- heavy
- tailed
- joint
- distributions,
- with
- a
- view
- to
- regressing
- heavy
- tailed
- variables.Second,
- it
- presents
- a
- new
- measure
- of
- obesity.
- The
- most
- popular
- definitions
- in
- terms
- of
- regular
- variation
- and
- subexponentiality
- invoke
- putative
- properties
- that
- hold
- at
- infinity,
- and
- this
- complicates
- any
- empirical
- estimate.
- Each
- definition
- captures
- some
- but
- not
- all
- of
- the
- intuitions
- associated
- with
- tail
- heaviness.
- Chapter
- 5
- studies
- two
- candidate
- indices
- of
- tail
- heaviness
- based
- on
- the
- tendency
- of
- the
- mean
- excess
- plot
- to
- collapse
- as
- data
- are
- aggregated.
- The
- probability
- that
- the
- largest
- value
- is
- more
- than
- twice
- the
- second
- largest
- has
- intuitive
- appeal
- but
- its
- estimator
- has
- very
- poor
- accuracy.
- The
- Obesity
- index
- is
- defined
- for
- a
- positive
- random
- variable
- X
- as:Ob(X)
- =
- P
- (X1
- +X4
- >
- X2
- +X3|X1
- =
- X2
- =
- X3
- =
- X4),
- Xi
- independent
- copies
- of
- X.For
- empirical
- distributions,
- obesity
- is
- defined
- by
- bootstrapping.
- This
- index
- reasonably
- captures
- intuitions
- of
- tail
- heaviness.
- Among
- its
- properties,
- if
- a
- >
- 1
- then
- Ob(X)
- <
- Ob(Xa).
- However,
- it
- does
- not
- completely
- mimic
- the
- tail
- index
- of
- regularly
- varying
- distributions,
- or
- the
- extreme
- value
- index.
- A
- Weibull
- distribution
- with
- shape
- 1/4
- is
- more
- obese
- than
- a
- Pareto
- distribution
- with
- tail
- index
- 1,
- even
- though
- this
- Pareto
- has
- infinite
- mean
- and
- the
- Weibull’s
- moments
- are
- all
- finite.
- Chapter
- 5
- explores
- properties
- of
- the
- Obesity
- index.Third
- and
- most
- important,
- we
- hope
- to
- convince
- the
- reader
- that
- fat
- tail
- phenomena
- pose
- real
- problems;
- they
- are
- really
- out
- there
- and
- they
- seriously
- challenge
- our
- usual
- ways
- of
- thinking
- about
- historical
- averages,
- outliers,
- trends,
- regression
- coefficients
- and
- confidence
- bounds
- among
- many
- other
- things.
- Data
- on
- flood
- insurance
- claims,
- crop
- loss
- claims,
- hospital
- discharge
- bills,
- precipitation
- and
- damages
- and
- fatalities
- from
- natural
- catastrophes
- drive
- this
- point
- home.
- While
- most
- fat
- tailed
- distributions
- are
- ”bad”,
- research
- in
- fat
- tails
- is
- one
- distribution
- whose
- tail
- will
- hopefully
- get fatter.
pro-mbooks3 : libris
