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A Cantelli-type inequality for constructing non-parametric p-boxes - - PowerPoint PPT Presentation
A Cantelli-type inequality for constructing non-parametric p-boxes - - PowerPoint PPT Presentation
A Cantelli-type inequality for constructing non-parametric p-boxes based on exchangeability Matthias C. M. Troffaes Tathagata Basu Department of Mathematical Sciences Durham University, UK July 2019 Outline 1 Cantellis inequality and
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Cantelli’s inequality & induced p-box
- random variable X, known mean µ and variance σ2
- Cantelli’s inequality [1]:
0 ≤ P X − µ σ ≤ λ
- ≤
1 1 + λ2 if λ ≤ 0, (1a) λ2 1 + λ2 ≤ P X − µ σ ≤ λ
- ≤ 1
if λ ≤ 0. (1b)
- induces a p-box (lower & upper cdf, bounding a set of
probability measures) Issue What if only sample mean and sample standard deviation are known?
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Contributions: Problem Statement
Assumptions X1,. . . ,Xn, Xn+1 is a finite sequence of discrete exchangeable random variables. Notation
- X := 1
n
n
j=1 Xj
- S2 :=
n
j=1(Xj−X)2
(n−1)
Objective Find functions f and f such that f (λ, n) ≤ P Xn+1 − X S ≤ λ
- ≤ f (λ, n)
(2)
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Contributions: Cantelli-type inequality
Theorem For every λ ≥ 0, 1 n + 1 (n + 1)λ2
n
λ2
n + 1
- ≤ P
- Xn+1 − X
S + ∆n
√n
< λ
- ≤ 1
(3) where λn :=
n √ n2−1λ and ∆n :=
- n+1
n−1(max Xj − min Xj).
Similarly, for every λ ≤ 0, 0 ≤ P
- Xn+1 − X
S + ∆n
√n
≤ λ
- ≤
1 n + 1 n + 1 λ2
n + 1
- .
(4) Here, ⌊x⌋ := max{n ∈ Z: n ≤ x} and ⌈x⌉ := −⌊−x⌋.
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Contributions: Cantelli-type inequality
Plotting left and right hand sides from inequalities in theorem:
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Contributions: P-box and Prediction Interval
Inequalities from theorem induce the following: Non-parametric p-box . . . on the random variable Zn+1 = Xn+1−X
S+ ∆n √n
. Not a p-box on Xn+1 directly! Prediction interval . . . on the random variable Xn+1. For any ℓ1 and ℓ2, we can calculate α1 and α2 such that α1 ≤ P(X − ℓ1Sn < Xn+1 ≤ X + ℓ2Sn) ≤ α2 (5) where, Sn := S + ∆n
√n.
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Conclusions I
- novel Cantelli-type inequality
- induces non-parametric p-box and prediction interval
- only assumes exchangeability (rather than conditional
independence)
- only uses sample mean and sample standard deviation
- similar to Saw [2] (but Saw does not induce a p-box)
- useful for modelling when only sample mean and sample
standard deviation are known (e.g. measurement problems)
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Conclusions II
- p-box only on Zn+1 and not on Xn+1
- use of prediction interval not entirely clear
- p-box on Xn+1 , conditional on X and S, using
exchangeability, remains and open problem (might be impossible as pointed out by a kind reviewer)
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Thank you for your attention! We look forward to seeing you at our poster!
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References
- F. P. Cantelli. “Sui confini della probabilit`
a”. In: Atti del Congresso Internazional del Matematici 6 (1928). Bologna,
- pp. 47–59.