Page - 434 - in Differential Geometrical Theory of Statistics
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Entropy2016,18, 375
AppendixA. ProofsofMonotonicity
LemmaA1. β(t)= [(
ν−a
ν−a+t )ν−a+t( b−ν
b−ν−t )b−ν−t] nb−a
is strictlydecreasing in t.
Proof. We show the equivalent statement that β˜(t) = ln [(
ν−a
ν−a+t )ν−a+t( b−ν
b−ν−t )b−ν−t]
is strictly
decreasing in t:
d
dt β˜(t)= d
dt (( ln(ν−a)− ln(ν−a+ t))(ν−a+ t)+(ln(b−ν)− ln(b−ν− t))(b−ν− t))
= ln(ν−a)− ln(ν−a+ t)− 1ν−a+t(ν−a+ t)− ln(b−ν)+ ln(b−ν− t)+ 1b−ν−t(b−ν− t)
= ln ( b−ν− t
b−ν︸
︷︷ ︸
<1 · ν−a
ν−a+
t︸
︷︷ ︸
<1 )
<0.
Hence, β˜(t)andthusβ(t)arestrictlydecreasing in t.
LemmaA2. Let t= t(γ,ν,a,b)bethesolutiontotheequationβ(t)=γ.Then,ν+ t is strictly increasing inν.
Proof. t is thesolutionof theequation
(ν−a+ t)ln
( ν−a
ν−a+ t )
+(b−ν− t)ln
( b−ν
b−ν− t )
= b−a
n lnγ. (A1)
The derivatives of the left-hand side of Equation (A1) w.r.t. ν and t exist and are continuous.
Furthermore, thederivativew.r.t. tdoesnotvanishforany t∈ (0,b−ν), cf. theproofofLemmaA1,
whence the derivative t′ = dtdν exists by the implicit function theorem. When differentiating
Equation(A1)withrespect toν, oneobtains
(1+ t′)ln
( ν−a
ν−a+ t )
+(ν−a+ t) (
1
ν−a− 1+ t′
ν−a+ t )
−(1+ t′)ln
( b−ν
b−ν− t )
+(b−ν− t) (
− 1
b−ν+ 1+ t′
b−ν− t )
=0,
orequivalently
(1+ t′) [
ln
( ν−a
ν−a+ t )
︸ ︷︷ ︸
<0 −ln
( b−ν
b−ν− t )
︸ ︷︷ ︸
>0 ]
= t(a−b)
(v−a)(b−v)<0,
whence1+ t′= ddν(ν+ t)>0finishes theproof.
LemmaA3. The function
ξ (
σ̂2 )
= [( 1− σ̂2
1−Vn )1−Vn( σ̂2
Vn )Vn]n
is strictlydecreasing in σ̂2∈ [Vn,1].
434
Differential Geometrical Theory of Statistics
- Title
- Differential Geometrical Theory of Statistics
- Authors
- Frédéric Barbaresco
- Frank Nielsen
- Editor
- MDPI
- Location
- Basel
- Date
- 2017
- Language
- English
- License
- CC BY-NC-ND 4.0
- ISBN
- 978-3-03842-425-3
- Size
- 17.0 x 24.4 cm
- Pages
- 476
- Keywords
- Entropy, Coding Theory, Maximum entropy, Information geometry, Computational Information Geometry, Hessian Geometry, Divergence Geometry, Information topology, Cohomology, Shape Space, Statistical physics, Thermodynamics
- Categories
- Naturwissenschaften Physik