Talk:Probability box

Three formulations of the constraints expressed by a p-box

There are different ways (shown below) to formulate the displayed constraints in the section 'Mathematical definition'. Each seems to have disadvantages. A formulation based on Riemann-Stieltjes integrals introduces the notation dF(x), which will likely be foreign to a lot of readers. We could formulate the integrals in terms of the quasi-inverse of the distribution function, but then we'd need to explain quasi-inverses. Or, we could use a formulation suggested by the entry Variance#Calculation from the CDF, but that requires identifying the minimum possible value of the random x-value. Currently, the entry uses the first of these options. Does anyone have suggestions about how to express the constraints?

Scwarebang (talk) 05:00, 31 March 2011 (UTC)

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F (x) ≤ F(x) ≤ F(x),

∫ ^∞
_-∞ x dF(x) ∈ m,

∫^∞
_-∞ x²dF(x)) – (∫ ^∞
_-∞ x dF(x))² ∈ v, and

F ∈ F.

These Riemann-Stieltjes integrals do not depend on the differentiability of F.

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F (x) ≤ F(x) ≤ F(x),

∫ ¹
₀ F ^-1(u)du ∈ m,

∫ ¹
₀ F ^-1(u)²du – (∫ ¹
₀ F ^-1(u)du)² ∈ v, and

F ∈ F

where F ^-1 is the quasi-inverse of F. These formulations do not depend on the invertibility of F but only on its monotonicity.

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F (x) ≤ F(x) ≤ F(x),

∫ ^∞
_a (1 – F(x – a))dx ∈ m,

2∫^∞
_a (x – a)(1 – F(x – a))dx – (∫ ^∞
_a (1 –F(x – a))dx)² ∈ v, and

F ∈ F

where a = sup_{F(x)=0 x is the smallest possible value of the random variable.}

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Adjective to describe bounds that really do bound

Because of the commonness of ideas such as confidence intervals, unmodified words like 'interval' and 'bounds' do not necessarily convey to a reader the idea that all possible values are within a specified range. So some adjective is required to indicate when bounds or intervals are assumed or constructed to enclose all possible values.

Different adjectives have been used for this purpose to modify 'bounds' and 'interval' in different fields, including

• sure
• true — Preceding unsigned comment added by 71.125.50.9 (talk) 22:23, 23 August 2011 (UTC)
• rigorous
• absolute
• exact
• crisp
• strict
• almost sure
• 100% confidence

The article currently uses the word 'rigorous', but this may be misleading to many readers. Can we come to a consensus about the best adjective to use?

The difference between strict intervals and statistical intervals is not addressed in the Wikipedia entry on bounds. Both kinds of bounds represent an implicit guarantee. But in the case of statistical intervals, the guarantee is in terms of some statistical performance in which the interval encloses the value(s) with some specified surety.

Scwarebang (talk) 21:42, 3 April 2011 (UTC)