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

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

Working Group Methodological Foundations of Statistics and their Applications: • Thomas Augustin • Eva Endres • Cornelia Fuetterer • Malte Nalenz • Aziz Omar • Patrick Schwaferts • Georg Schollmeyer 1 Our Working Group

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 2

• Epistemic view: A set-valued data point represents an imprecise observation of a precise, but not directlyobservable data point of interest. • Ontic view: A set-valued data point is understood as a precise observation of somethinmg that is ’imprecise’ in nature. 3 Ontic vs Epistemic Data Imprecision

4 First Order Stochastic Dominance 1a: univariate case amssymb amsmath X ≤ SD Y ⇐ ⇒ ∀ c ∈ R : F X ( c ) ≥ F Y ( c ) ⇐ ⇒ ∀ c ∈ R : P ( X ≥ c ) ≤ P ( Y ≥ c ) F X F Y

5 First Order Stochastic Dominance 1b: univariate case, sample analogue amsmath amssymb ⇒ ∀ c ∈ R : ˆ F X ( c ) ≥ ˆ F Y ( c ) X ≤ ˆ SD Y ⇐ ⇒ ∀ c ∈ R : ˆ P ( X ≥ c ) ≤ ˆ ⇐ P ( Y ≥ c ) F X F Y

6 b (A set b b b b b b b b b b First Order Stochastic Dominance 2: bivariate case amsmath amssymb ⇒ ∀ U ⊆ R 2 upset : ˆ P ( X ∈ U ) ≤ ˆ X ≤ ˆ SD Y ⇐ P ( Y ∈ U ) U ⊆ R 2 is called upset iff ∀ x ∈ U , y ∈ R 2 s.t. y i ≥ x i ( i = 1 , 2 ) = ⇒ y ∈ U )

7 b b b b b b b b b b b First Order Stochastic Dominance 2: bivariate case amsmath amssymb ⇒ ∀ U ⊆ R 2 upset : ˆ P ( X ∈ U ) ≤ ˆ X ≤ ˆ SD Y ⇐ P ( Y ∈ U )

8 b b b b b b b b b 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 . ⇒ ∀ U ⊆ R 2 upset : ˆ P ( X ∈ U ) ≤ ˆ P ( Y ∈ U ) • X ≤ ˆ SD Y ⇐ U

9 First Order Stochastic Dominance for Interval Data: univariate case, Epistemic view amsmath amssymb X ≤ ˆ SD Y ⇐ ⇒ ∀ x ∈ x , y ∈ y : ∀ c ∈ R : |{ i : x i ≥ c }| ≤ |{ i : y i ≥ c }| c

10 b b b b b b b b b b b First Order Stochastic Dominance for Interval Data: bivariate case, Epistemic View amsmath amssymb X ≤ ˆ SD Y ⇐ ⇒ ∀ X ∈ X , Y ∈ Y : ∀ U ⊆ R 2 upset : ˆ P ( X ∈ U ) ≤ ˆ P ( Y ∈ U )

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

12 First Order Stochastic Dominance for Interval Data: bivariate case, Ontic View amsmath amssymb x ⊑ y : ⇐ ⇒ ∀ x ∈ x , y ∈ y : x ≤ y ⇒ ∀ U uspet w.r.t. ⊑ : X ≤ ˆ SD Y ⇐ P ( X ∈ U ) ≤ ˆ ˆ P ( Y ∈ U ) c

13 First Order Stochastic Dominance for Interval Data: bivariate case, Ontic View amsmath amssymb x ⊑ y : ⇐ ⇒ ∀ x ∈ x , y ∈ y : x i ≤ y i ; i = 1 , 2 ∀ U upset w.r.t. ⊑ : ˆ P ( X ∈ U ) ≤ ˆ P ( Y ∈ U )

of stochatsic dominance. Epistemic and Ontic view lead to the same results w.r.t. the presence 14 Simple Fact