Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
Brownian confidence bands
(Statistical MTV at Warwick)
Wilfrid Kendall
w.s.kendall@warwick.ac.uk
Department of Statistics, University of Warwick
3 November 2005
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Introduction . . .
This is joint work with
Christian Robert and
Jean-Michel Marin:
informal convergence
diagnostics for Monte
Carlo leading to an
apparently novel
application of
Brownian local time
and thus to an
unresolved question.
Conclusion
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Monte Carlo
IID sampling from π gives an estimate and CLT:
n
1X
b
In =
h(Xi )
n
Z
−→
I=
h(x)π(d x)
i=1
(asymptotically Gaussian fluctuations . . . )
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Monte Carlo
IID sampling from π gives an estimate and CLT:
n
Z
1X
b
In =
h(Xi )
n
−→
I=
h(x)π(d x)
i=1
(asymptotically Gaussian fluctuations . . . )
Assess path of
sequential estimates.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Monte Carlo
IID sampling from π gives an estimate and CLT:
n
Z
1X
b
In =
h(Xi )
n
−→
I=
h(x)π(d x)
i=1
(asymptotically Gaussian fluctuations . . . )
Assess path of
sequential estimates.
Standard normal π
h(x) = exp(x 2 /4.01)
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Monte Carlo
IID sampling from π gives an estimate and CLT:
n
Z
1X
b
In =
h(Xi )
n
−→
I=
h(x)π(d x)
i=1
(asymptotically Gaussian fluctuations . . . )
Assess path of
sequential estimates.
Standard normal π
h(x) = exp(x 2 /4.01)
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Monte Carlo
IID sampling from π gives an estimate and CLT:
n
Z
1X
b
In =
h(Xi )
n
−→
I=
h(x)π(d x)
i=1
(asymptotically Gaussian fluctuations . . . )
Assess path of
sequential estimates.
Standard normal π
h(x) = exp(x 2 /4.01)
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Brownian band
We want to design a band likely to contain the (asymptotic)
Brownian path of sequential partial sums.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Brownian band
We want to design a band likely to contain the (asymptotic)
Brownian path of sequential partial sums.1
1
Actually not quite Brownian — but this can be fixed.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Brownian band
We want to design a band likely to contain the (asymptotic)
Brownian path of sequential partial sums.1
Fix α, solve minimum area problem
Z
min
u,v
1
(u(t) + v (t)) d t
subject to
0
P [−v (t) ≤ W (t) ≤ u(t) for all t ∈ (0, 1)] = 1 − α ,
with u, v ≥ 0 .
1
Actually not quite Brownian — but this can be fixed.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Brownian band
We want to design a band likely to contain the (asymptotic)
Brownian path of sequential partial sums.1
Fix α, solve minimum area problem
Z
min
u,v
1
(u(t) + v (t)) d t
subject to
0
P [−v (t) ≤ W (t) ≤ u(t) for all t ∈ (0, 1)] = 1 − α ,
with u, v ≥ 0 .
Numerical solutions, efficient methods of computation,
1
Actually not quite Brownian — but this can be fixed.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Brownian band
We want to design a band likely to contain the (asymptotic)
Brownian path of sequential partial sums.1
Fix α, solve minimum area problem
Z
min
u,v
1
(u(t) + v (t)) d t
subject to
0
P [−v (t) ≤ W (t) ≤ u(t) for all t ∈ (0, 1)] = 1 − α ,
with u, v ≥ 0 .
Numerical solutions, efficient methods of computation,
but hard to produce anything of theoretical value.
1
Actually not quite Brownian — but this can be fixed.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Dual problem
The dual problem is just as hard . . .
Conclusion
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Dual problem
The dual problem is just as hard . . .
(using symmetry to produce one-sided version)
min P [W (t) ≥ u(t) for some t ∈ (0, 1)]
u≥0
Z
subject to
1
u(t) d t = κ .
0
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
The “Kobayashi Maru” strategy (I)
Time to learn from Captain James T. Kirk:
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (I)
Time to learn from Captain James T. Kirk: when faced with
an unwinnable situation,
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (I)
Time to learn from Captain James T. Kirk: when faced with
an unwinnable situation, change the rules!
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (I)
Time to learn from Captain James T. Kirk: when faced with
an unwinnable situation, change the rules!
Use local time of W at u instead of exceedance probability.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (I)
Time to learn from Captain James T. Kirk: when faced with
an unwinnable situation, change the rules!
Use local time of W at u instead of exceedance probability.
min
u≥0
1
E [Lu (1)] =
2
Z
0
1
!
u(s)2 d s
√
exp −
2s
2πs
Z 1
subject to
u(t)dt = κ .
0
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (II)
There is an “explicit” solution u ∗ involving the ProductLog
or Lambert W function (W (z) solves WeW = z maximally),
(p
∗
u (s)
=
0
sW (κ2 s2 )
√
if κs ≤ 1/ e ,
otherwise,
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (II)
There is an “explicit” solution u ∗ involving the ProductLog
or Lambert W function (W (z) solves WeW = z maximally),
(p
∗
u (s)
=
sW (κ2 s2 )
0
which allows explicit approximation,
√
if κs ≤ 1/ e ,
otherwise,
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (II)
There is an “explicit” solution u ∗ involving the ProductLog
or Lambert W function (W (z) solves WeW = z maximally),
(p
∗
u (s)
=
0
sW (κ2 s2 )
√
if κs ≤ 1/ e ,
otherwise,
which allows explicit approximation, and agrees closely
with numerical solutions to the original problem.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (II)
There is an “explicit” solution u ∗ involving the ProductLog
or Lambert W function (W (z) solves WeW = z maximally),
(p
∗
u (s)
=
0
sW (κ2 s2 )
√
if κs ≤ 1/ e ,
otherwise,
which allows explicit approximation, and agrees closely
with numerical solutions to the original problem.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
The “Kobayashi Maru” strategy (II)
There is an “explicit” solution u ∗ involving the ProductLog
or Lambert W function (W (z) solves WeW = z maximally),
(p
∗
u (s)
=
0
sW (κ2 s2 )
√
if κs ≤ 1/ e ,
otherwise,
which allows explicit approximation, and agrees closely
with numerical solutions to the original problem.
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
Unresolved question:
Conclusion
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
Conclusion
Unresolved question:
WHY and HOW does the local time problem
agree so well with
the original exceedance-probability version?
References
Introduction
Monte Carlo
Brownian band
Kobayashi Maru
Conclusion
References
Kendall, W. S., J. Marin, and C. P. Robert
(2004, November).
Brownian confidence bands on Monte Carlo
output, .
Submitted to Statistics and Computing, also
Warwick CRiSM Working paper No. 05-2.
References