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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum Likelihood
Maximum likelihood with Flexible Probabilities
• Topic: parametric estimation of the invariants distribution based onthe Maximum Likelihood approach
• We introduce the Maximum Likelihood with FlexibleProbabilities estimate
• We consider the special case of invariants whose distribution belongs toan exponential family
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Canonical Maximum Likelihood
Intuition behind Maximum Likelihood estimation
Suppose that the number of observations t is small, or the number of simul-taneous invariants ı is large.
A parametric assumption on the distribution of the invariants is
εt ∼ fε ∈ {fθ}θ∈Θ (2a.58)
where• θ ≡ (θ1, . . . , θl)
′, l × 1 vector
• Θ ⊆ Rl, discrete or continuous
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Canonical Maximum Likelihood
Given the realized time series of the invariants, the likelihood is
fθ (i) =∏tt=1fθ (εt) (2a.60)
where i represents the realization of the information I ≡ (ε1| . . . |εt| . . . |εt).
The Maximum Likelihood estimate is
fMLε ≡ f
θMLε
, θML
ε ≡ argmaxθ∈Θ
{1
t
∑tt=1 ln fθ (εt)} (2a.61)
The Maximum Likelihood distribution is unequivocally determinedby the corresponding Maximum Likelihood parameters
fMLε ⇔ θ
ML
ε (2a.62)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Canonical Maximum Likelihood
Given the realized time series of the invariants, the likelihood is
fθ (i) =∏tt=1fθ (εt) (2a.60)
where i represents the realization of the information I ≡ (ε1| . . . |εt| . . . |εt).
The Maximum Likelihood estimate is
fMLε ≡ f
θMLε
, θML
ε ≡ argmaxθ∈Θ
{1
t
∑tt=1 ln fθ (εt)} (2a.61)
The Maximum Likelihood distribution is unequivocally determinedby the corresponding Maximum Likelihood parameters
fMLε ⇔ θ
ML
ε (2a.62)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Properties of the ML estimation
Equivalently, the Maximum Likelihood parameters are defined as
θML
ε = argminθ∈Θ E(fHistε ||fθ) (2a.63)
where E(fHistε ||fθ) is the relative entropy (25.32) and fHist
ε the historicaldistribution (2a.22).
Consistency: the parametric assumption (2a.59) holding true, θML
ε
approximates the true, unknown, parameters θε ∈ Θ of the invariants(fε ≡ fθε), and
limt→∞ θML
ε = θε ⇔ limt→∞ fMLε = fε (2a.64)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Example 2a.17. Consistency of the maximum likelihoodparameter
• Invariants: εt ∼ t(3, 0, 2)
• Number of observations: 500
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Generalization with Flexible Probabilities
Consider Flexible Probabilities {pt}tt=1 rather than the equal probabilityweights pt ≡ 1/t.
The Maximum Likelihood with Flexible Probabilities (MLFP)estimate is
fMLFPε ≡ f
θMLFPε
, θMLFP
ε ≡ argmaxθ∈Θ
{∑tt=1pt ln fθ (εt)} (2a.65)
The MLFP distribution is unequivocally determined by the corre-sponding MLFP parameters
fMLFPε ⇔ θ
MLFP
ε (2a.66)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Properties of the MLFP estimation
Equivalently, the Maximum Likelihood with Flexible Probabilitiesparameters are defined as
θMLFP
ε = argminθ∈Θ E(fHFPε ||fθ) (2a.67)
where E(fHFPε ||fθ) is the relative entropy (25.32) and fHFP
ε the Historicalwith Flexible Probability distribution (2a.24).
Consistency: the parametric assumption (2a.58) holding true, andprovided the Effective Number of Scenarios (2a.21) is ens(p1, . . . , pt) ≈ t
θMLFP
ε ≈ θε ⇔ fMLFPε ≈ fε (2a.68)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Extracting PropertiesUnder the parametric assumption, a generic property sε ≡ S{εt} reads
sε = hS(θε) (2a.69)
for some function hS.How to estimate the properties of fε?
Maximum Likelihood with Flexible Probabilities (MLFP) es-timate
sMLFPε ≡ SMLFP{ε} ≡ hS(θ
MLFP
ε ) (2a.73)See also Example 2a.18
θMLFP
ε ⇔ fMLFPε ≈
ens(p)→∞θε ⇔ fε
hS
y yhS
sMLFPε ≈
ens(p)→∞sε
(2a.74)
where ens(p) is the Effective Number of Scenarios (2a.21).
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update
The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodFrom Maximum Likelihood to Flexible Probabilities
Extracting PropertiesUnder the parametric assumption, a generic property sε ≡ S{εt} reads
sε = hS(θε) (2a.69)
for some function hS.How to estimate the properties of fε?
Maximum Likelihood with Flexible Probabilities (MLFP) es-timate
sMLFPε ≡ SMLFP{ε} ≡ hS(θ
MLFP
ε ) (2a.73)See also Example 2a.18
θMLFP
ε ⇔ fMLFPε ≈
ens(p)→∞θε ⇔ fε
hS
y yhS
sMLFPε ≈
ens(p)→∞sε
(2a.74)
where ens(p) is the Effective Number of Scenarios (2a.21).ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update
The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodExponential family invariants
Exponential family invariants
Consider ML estimation from a sample {εt, pt}tt=1 in the exponential family(22.109)
fθ(εt) = h(εt)eθ′φ(εt)−ψ(θ) (2a.76)
The log-likelihood maximization (2a.64) becomes
θMLFP
≡ argmaxθ{θ′ηHFPε − ψ(θ) +
∑tt=1pt lnh(εt)} (2a.77)
where ηHFPε is the Historical with Flexible Probabilities (HFP) estimator
(2a.37) of the expected features (22.112)
ηHFPε ≡
∑tt=1ptφ(εt) (2a.78)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodExponential family invariants
Exponential family invariants
From an information geometry perspective, ηHFPε is the estimator of the
expectation parameters, or m-coordinates (25.57).
The estimators θMLFP
ε and ηHFPε are related as follows
θMLFP
ε ≡ (∇θψ)−1(ηHFPε ) (2a.79)
where ψ is the log-partition function (22.110).
See also Example 2a.20
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
Goal: Use Maximum Likelihood to estimate tails modeled via ExtremeValue Theory
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Extreme value theory quantile estimation
• Invariant: realized residual of GARCH(1, 1) (2.88)-(2.89) fit ofCSCO’s return
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail modeling: Extreme Value Theory (EVT)
zoom on left tail(losses)
1 Conditional excess distribution (CED)
fε−X|X≤ε (x) =fX(ε− x)
FX(ε)(3a.88)
2 Generalized Pareto distribution (GDP)
fξ,σ (x) ≡
1x≥0
σ1/ξ
(σ + ξx)1+1/ξif ξ ≥ 0
10≤x≤−σ/ξσ1/ξ
(σ + ξx)1+1/ξif ξ < 0
(3a.89)
fξ,σξ→0−→ exponential distr.
Theorem of Extreme Value Theory
fε−X|X≤ε (x) ≈ fξ∗,σ∗ (x) (3a.90)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail modeling: Extreme Value Theory (EVT)
zoom on left tail(losses)
1 Conditional excess distribution (CED)
fε−X|X≤ε (x) =fX(ε− x)
FX(ε)(3a.88)
2 Generalized Pareto distribution (GDP)
fξ,σ (x) ≡
1x≥0
σ1/ξ
(σ + ξx)1+1/ξif ξ ≥ 0
10≤x≤−σ/ξσ1/ξ
(σ + ξx)1+1/ξif ξ < 0
(3a.89)
fξ,σξ→0−→ exponential distr.
Theorem of Extreme Value Theory
fε−X|X≤ε (x) ≈ fξ∗,σ∗ (x) (3a.90)
ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Apr-05-2017 - Last update
The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail modeling: Extreme Value Theory (EVT)
zoom on left tail(losses)
1 Conditional excess distribution (CED)
fε−X|X≤ε (x) =fX(ε− x)
FX(ε)(3a.88)
2 Generalized Pareto distribution (GDP)
fξ,σ (x) ≡
1x≥0
σ1/ξ
(σ + ξx)1+1/ξif ξ ≥ 0
10≤x≤−σ/ξσ1/ξ
(σ + ξx)1+1/ξif ξ < 0
(3a.89)
fξ,σξ→0−→ exponential distr.
Theorem of Extreme Value Theory
fε−X|X≤ε (x) ≈ fξ∗,σ∗ (x) (3a.90)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail estimation: Maximum Likelihood with Flexible Probabilities (MLFP)
1 MLFP Pareto parameters
(ξMLFP , σMLFP ) ≡ argmaxσ>0,ξ
{∑εt≤ε
pt ln fξ,σ(ε− εt)} (3a.91)
2 MLFP cdf
FMLFPε (x) =
∫ x
−∞fMLFPε (x)dx (3a.92)
≡ fξMLFP ,σMLFP
3 MLFP quantile
qMLFPε (c) = (ε− σMLFP
ξMLFP(
(c
c
)−ξMLFP
− 1), c ≤ c ≡ FMLFPε (ε) (3a.93)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail estimation: Maximum Likelihood with Flexible Probabilities (MLFP)
1 MLFP Pareto parameters
(ξMLFP , σMLFP ) ≡ argmaxσ>0,ξ
{∑εt≤ε
pt ln fξ,σ(ε− εt)} (3a.91)
2 MLFP cdf
FMLFPε (x) =
∫ x
−∞fMLFPε (x)dx (3a.92)
≡ fξMLFP ,σMLFP
3 MLFP quantile
qMLFPε (c) = (ε− σMLFP
ξMLFP(
(c
c
)−ξMLFP
− 1), c ≤ c ≡ FMLFPε (ε) (3a.93)
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The “Checklist” > 3a. Estimation: Flexible Probabilities > Maximum LikelihoodTails: Extreme Value Theory
Tails: Extreme Value Theory
• Tail estimation: Maximum Likelihood with Flexible Probabilities (MLFP)
1 MLFP Pareto parameters
(ξMLFP , σMLFP ) ≡ argmaxσ>0,ξ
{∑εt≤ε
pt ln fξ,σ(ε− εt)} (3a.91)
2 MLFP cdf
FMLFPε (x) =
∫ x
−∞fMLFPε (x)dx (3a.92)
≡ fξMLFP ,σMLFP
3 MLFP quantile
qMLFPε (c) = (ε− σMLFP
ξMLFP(
(c
c
)−ξMLFP
− 1), c ≤ c ≡ FMLFPε (ε) (3a.93)
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