Probability bounds analysis: Difference between revisions

Revision as of 05:40, 28 July 2018

Revision as of 08:11, 28 July 2018

Line 66:

:<math> \mathbb{D} = \{ D | D: \R \to [0,1], D(x) \leq D(y) \text{ for all } x < y \}.</math>

Then aA p-box is a quintuple

Let <math>\mathbb{I}</math> denote the set of real [[Interval (mathematics)|intervals]], i.e.,

:<math>\mathbb{I} = \left \{ \left [i_1, i_2 \right ] | i_1 \leq i_2, i_1, i_2 \in \R \right \}.</math>

Then a p-box is a quintuple

:<math>\left \{ \overline{F}, \underline{F}, m, v, \mathbf{F} \right \},</math>

where <math>\overline{F}, \underline{F} \in \mathbb{D}, m,</math> vare \inreal \mathbb{I}intervals,</math> and <math>\mathbf{F} \subset \mathbb{D}.</math> This quintuple denotes the set of distribution functions <math>F \in \mathbf{F} \subset \mathbb{D}</math> such that:

:<math>\begin{align}
\forall x \in \R: \qquad &\overline{F}(x) \leq F(x) \leq \underline{F}(x),</math> \\[6pt]

&\int_\R x dF(x) \in m && \text{expectation condition} \\

&\int_\R x^2 dF(x) - \left ( \int_\R x dF(x) \right )^2 \in v && \text{variance condition}

\end{align}</math>

If ''F'' is a [[Cumulative distribution function|distribution function]]satisfies and ''B'' is a [[probability box|p-box]],all the notationconditions ''F''above ∈ ''B'' means that ''F''it is ansaid elementto ofbe ''Binside'' =the {''B''₁, ''B''₂, [''m''₁,''m''₂], [''v''₁, ''v''₂], '''B'''}, that is, ''B''₂(''x'') ≤ ''F''(''x'') ≤ ''B''₁(''x''), for all ''x'' ∈ ℝ, [[Expected value|E]](''F'') ∈ [''m''₁,''m''₂], [[Variance|V]](''F'') ∈ [''v''₁,''v''₂], and ''F'' ∈ '''B'''. We sometimes say ''F'' is ''inside'' ''B''p-box. In some cases, there may be no information about the moments or distribution family other than what is encoded in the two distribution functions that constitute the edges of the p-box. Then the quintuple representing the p-box <math>\{B_1, B_2, [-\infty, \infty], [0, \infty], \mathbb{D}\}</math> can be denoted more compactly as [''B''₁, ''B''₂]. This notation harkens to that of intervals on the real line, except that the endpoints are distributions rather than points.

and the mean and variance of ''F'' are in the intervals ''m'' and ''v'' respectively.

If ''F'' is a [[Cumulative distribution function|distribution function]] and ''B'' is a [[probability box|p-box]], the notation ''F'' ∈ ''B'' means that ''F'' is an element of ''B'' = {''B''₁, ''B''₂, [''m''₁,''m''₂], [''v''₁, ''v''₂], '''B'''}, that is, ''B''₂(''x'') ≤ ''F''(''x'') ≤ ''B''₁(''x''), for all ''x'' ∈ ℝ, [[Expected value|E]](''F'') ∈ [''m''₁,''m''₂], [[Variance|V]](''F'') ∈ [''v''₁,''v''₂], and ''F'' ∈ '''B'''. We sometimes say ''F'' is ''inside'' ''B''. In some cases, there may be no information about the moments or distribution family other than what is encoded in the two distribution functions that constitute the edges of the p-box. Then the quintuple representing the p-box <math>\{B_1, B_2, [-\infty, \infty], [0, \infty], \mathbb{D}\}</math> can be denoted more compactly as [''B''₁, ''B''₂]. This notation harkens to that of intervals on the real line, except that the endpoints are distributions rather than points.

The notation <math>X \sim F</math> denotes the fact that <math>X \in \R</math> is a random variable governed by the distribution function ''F'', that is,

Line 93 ⟶ 91:

This operation is called a [[convolution]] on ''F'' and ''G''. The analogous operation on p-boxes is straightforward for sums. Suppose

:''X'' ~ ''A'' = [''A''₁, ''A''₂] and

:<math>X \sim A = [A_1, A_2], \quad \text{and} \quad Y \sim B = [B_1, B_2].</math>

:''Y'' ~ ''B'' = [''B''₁, ''B''₂].

If ''X'' and ''Y'' are stochastically independent, then the distribution of ''Z'' = ''X'' + ''Y'' is inside the p-box

:<math> \left [A_1 * B_1, A_2 * B_2 \right ].</math>

Line 107 ⟶ 106:

The convolution under the intermediate assumption that ''X'' and ''Y'' have [[positive quadrant dependence|positive dependence]] is likewise easy to compute, as is the convolution under the extreme assumptions of [[Comonotonicity|perfect positive]] or [[countermonotonicity|perfect negative]] dependency between ''X'' and ''Y''.<ref name=Fersonetal04 />

Generalized convolutions for other operations such as subtraction, multiplication, division, etc., can be derived using transformations. For instance, p-box subtraction ''A'' − ''B'' can be defined as ''A'' + (−''B''), where the negative of a p-box ''B'' = [''B''₁, ''B''₂] is [''B''₂(−''x''), ''B''₁(−''x'')].

[''B''₂(−''x''), ''B''₁(−''x'')].

==Logical expressions==