Computing Interval Estimates for Components of Statistical Information with Respect to Judgements on...
-
Upload
amice-blake -
Category
Documents
-
view
224 -
download
5
Transcript of Computing Interval Estimates for Components of Statistical Information with Respect to Judgements on...
Computing Interval Estimates for Components of Statistical Information
with Respect to Judgements on Probability Density
Functions
Victor G. Krymsky
Ufa State Aviation Technical University, Russia;
E-mail: [email protected]
Kgs. Lyngby, Denmark, 2004
Imprecise Prevision Theory (IPT)
Starting Points: Fundamental Publications
[1], [2]
[1] Walley P., Statistical reasoning with imprecise probabilities, Chapman and Hall, New York, (1991);
[2] Kuznetsov V., Interval statistical models, Radio and Sviaz, Moscow, (1991) (in Russian).
Traditional Problem Formulation
in the Framework of IPT Constraints:
,0)( x
(1) .,...,2,1 ,)()(0
niadxxxfaT
iii
)()(inf0
)(
T
xdxxxg
It is necessary to find:
(2) )()(sup0)(T
xdxxxg
subject to constraints (1).
,1)(0 T
dxx and
as well as
Dual for the Initial Problem Statement
n
i
iiii
idiccadaccgM
10
,,0
)(sup
).()()(1
0 xgxfdccn
iiii
RdcRc ii , ,0subject to and for any x ≥ 0,
i=1,2,…,n:
n
iiiii
idiccadaccgM
10
,,0
)(inf)(
And
subject to RdcRc ii , ,0 and for any x ≥ 0,
i=1,2,…,n:
).()()(1
0 xgxfdccn
iiii
(3)
(4)
(5)
(6)
Let us find:
Important Conclusion Concerning Optimal
Solutions
(L. Utkin and I. Kozine, [3])
[3] Utkin L. and Kozine I. Different faces of the natural extension. In: Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, ISIPTA '01, 2001, pp. 316-323.
Optimal solutions belong to a family of DEGENERATE distributions
(such probability densities are composed of δ-functions)
Distribution of Probabilistic Masses
Masses are concentrated in the fixed points:
∞
Δ x→0
x
Density
0
Use of Additional Judgements
Additional judgement can be
reflected by inequality:
(x)K=const,(7)
where is such that RK .1KT
T0 x
K
ρ(x)
Main Goal(Theorem)
If there is no any finite interval for which function g(x) can be represented in the form
(8)
where then function ρ(x) providing solution of optimization problem mentioned above, belongs to class of step-functions with minimum value equal to 0 and maximum value equal to K.
, ],,0[],[ Tx
n
iii xfhhxg
10 ),()(
,,...,, 10 Rhhh n
Some Comments
To provide (8) the system
)()(
.............................
),()(
),()(
),()(
)(
1
)(
1
1
10
xgxfh
xgxfh
xgxfh
xgxfhh
nn
i
nii
n
iii
n
iii
n
iii
must have at least one solution which is independent on x in some interval ].[α,βx
Applying Methodology of the Calculus of VariationsThe inequalities
(9)
should be excluded from direct consideration in order to allow operating in the open domain with the values of the function.
The requirement (x)≥0 can be replaced by denoting).()( 2 xzx The requirement (x)≤K can be reflected by equality
,)()( 22 Kxvxz where v(x) is newly introduced function.
0(x)K
(10)
(11)
Modified Formulation of the Problem
We would like to estimate
and
subject to
T
xzdxxzxg
0
2
)()()(inf
T
xzdxxzxg
0
2
)()()(sup
,)()( 22 Kxvxz
,1)(0
2 T
dxxz
,)()(0
2i
T
i adxxzxf .,...,2,1 ,)()(
0
2 niadxxzxf i
T
i
(12)
(13)
(14)
(15)
(16)
Lagrange Approach
n
iii xzxfxz
xvxzxxzxgvzF
1
220
222
)()()(
)()()()()(),(
n
niii xzxf
2
1
2 )()(
)()(),( 2 xzxgvzF
;0),(),(
z
vzF
dx
d
z
vzF
.0),(),(
v
vzF
dx
d
v
vzF
Equations of Euler – Lagrange:
(17)
The Necessary Conditions of Optimality
The equations look here as
follows:
;0)()()()()(1
0
n
iinii xfxxgxz
.0)()( xvx
. ],,0[],[ Tx
Let us fix any interval
Case 1. 0)( xz
inside the interval.Kxv )( .0)( xThen and
(18)
Case 2. ,0)( xz so0)( xv and .)( Kxz
Practical Implementation
Optimal probability density:
x x x x x x … x0 1 2 3 4 5
K
ρ(x)
,)(),(1
1
jx
jx
jj dxxgxxG
,...,n.,idxxfxx
jx
jx
ijji 21 ,)(),(
1
1
Denote:
(19)
(20)
0
Reformulation of the Problem Statement
We would like to estimate
,),(min0
122,...1,0
m
jjj
xxxxGK
m
jjj
xxxxGK
0122
,...1,0
),(max
m
jjj xxK
0212 ;1)(
.,...,2,1 ,),(0
122 niaxxKa i
m
jjjii
subject to
(21)
(22)
(23)
(24)
Example 1
The information concerning a continuous random variable X
is where K, T are fixed positive numbers. What are the bounds for the expectation M(X)?
* * *
Let us choose m=0.
Objective function:
,)()()( ],0[ xIKxx T
.2
)(1
0
20
21
0
x
x
Txx
KxdxKdxxxJ
Solution of Optimization Problem
Lower and upper bounds of J
interval:
.2
1min
KJ
.2
1
2
1)/1(max
KT
KKTJ
1/2K 1/K T
ρ(x)
K
(T-1/K) (T-1/2K) T
0
0
x
x
ρ(x)
K
Example 2
We add the constraint:
where is the indicator function. Here also any finite interval of x values for which
cannot be found, so the theorem can be applied. Further analysis shows, that m=1 is the best choice for such situation.
T
bdxxxf0
1 ,)()(
)()(],[1 xIxf
aa
)()(],[10 xIccxxg
aa
Example 2(Continuation 1)
To provide
we have to set:
(i) if
;afor 0
;for
;1
for 0
;1
0for
)(
TxK
bK
baxaK
axK
bK
bxK
x
minJ
:1
K
ba
Example 2(Continuation 2)
(ii) if :1
K
ba
.1
for 0
;1
afor
;for 0
;0for
)(
TxK
bKaa
K
bKaaxK
axK
ba
K
baxK
x
K
bKab
KJ
)1(
2
1min
As the result
.
)(1)(
2
1min
K
bKaabKaKa
KJ
or
Acknowledgements
• The research was initiated by Dr. Igor Kozine of Risø National Laboratory, Denmark, whose kind attention to this work is gratefully acknow-ledged.
• The work was partially supported by the grant T02-3.2-346 of Russian Ministry for Education which is also acknowledged.