A Short Note on the Equivalence of the Ontic and the Epistemic View - - PowerPoint PPT Presentation

A Short Note on the Equivalence of the Ontic and the Epistemic View on Data Imprecision for the Case of Stochastic Dominance for Interval-Valued Data

Georg Schollmeyer

Department of Statistics, Ludwig-Maximilians University Munich (LMU)

Our Working Group

Working Group Methodological Foundations of Statistics and their Applications:

• Thomas Augustin
• Eva Endres
• Cornelia Fuetterer
• Malte Nalenz
• Aziz Omar
• Patrick Schwaferts
• Georg Schollmeyer

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A Short Note on the Equivalence of the Ontic and the Epistemic View on Data Imprecision for the Case of Stochastic Dominance for Interval-Valued Data

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Ontic vs Epistemic Data Imprecision

• Epistemic view: A set-valued data point represents an imprecise
• bservation of a precise, but not directlyobservable data point
• f interest.
• Ontic view: A set-valued data point is understood as a precise
• bservation of somethinmg that is ’imprecise’ in nature.

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First Order Stochastic Dominance 1a: univariate case

amssymb amsmath

FY FX X ≤SD Y ⇐ ⇒ ∀c ∈ R : FX(c) ≥ FY (c) ⇐ ⇒ ∀c ∈ R : P(X ≥ c) ≤ P(Y ≥ c)

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First Order Stochastic Dominance 1b: univariate case, sample analogue

amsmath amssymb

FY FX X ≤ ˆ

SD Y ⇐

⇒ ∀c ∈ R : ˆ FX(c) ≥ ˆ FY (c) ⇐ ⇒ ∀c ∈ R : ˆ P(X ≥ c) ≤ ˆ P(Y ≥ c)

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First Order Stochastic Dominance 2: bivariate case

amsmath amssymb

b b b b b b b b b b b

X ≤ ˆ

SD Y ⇐

⇒ ∀U ⊆ R2 upset : ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U)

(A set U ⊆ R2 is called upset iff ∀x ∈ U, y ∈ R2 s.t. yi ≥ xi(i = 1, 2) = ⇒ y ∈ U)

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First Order Stochastic Dominance 2: bivariate case

amsmath amssymb

b b b b b b b b b b b

X ≤ ˆ

SD Y ⇐

⇒ ∀U ⊆ R2 upset : ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U)

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First Order Stochastic Dominance 3: general poset-valued case

• Given a poset (V, ≤), a subset U ⊆ V is called upset iff

x ∈ U , y ≥ x = ⇒ y ∈ U.

• X ≤ ˆ

SD Y ⇐

⇒ ∀U ⊆ R2 upset : ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U)

b b b

U

b b b b b b

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First Order Stochastic Dominance for Interval Data: univariate case, Epistemic view

amsmath amssymb

c X ≤ ˆ

SD Y ⇐

⇒ ∀x ∈ x, y ∈ y : ∀c ∈ R : |{i : xi ≥ c}| ≤ |{i : yi ≥ c}|

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First Order Stochastic Dominance for Interval Data: bivariate case, Epistemic View

amsmath amssymb

b b b b b b b b b b b

X ≤ ˆ

SD Y ⇐

⇒ ∀X ∈ X, Y ∈ Y : ∀U ⊆ R2 upset : ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U)

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First Order Stochastic Dominance for Interval Data: bivariate case, Epistemic View

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First Order Stochastic Dominance for Interval Data: bivariate case, Ontic View

amsmath amssymb

c X ≤ ˆ

SD Y ⇐

⇒ ∀U uspet w.r.t. ⊑: ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U) x ⊑ y : ⇐ ⇒ ∀x ∈ x, y ∈ y : x ≤ y

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First Order Stochastic Dominance for Interval Data: bivariate case, Ontic View

amsmath amssymb

x ⊑ y : ⇐ ⇒ ∀x ∈ x, y ∈ y : xi ≤ yi; i = 1, 2 ∀U upset w.r.t. ⊑: ˆ P(X ∈ U) ≤ ˆ P(Y ∈ U)

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Simple Fact

Epistemic and Ontic view lead to the same results w.r.t. the presence

• f stochatsic dominance.

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