# Research Group "Stochastic Algorithms and Nonparametric Statistics"

## Seminar "Modern Methods in Applied Stochastics and Nonparametric
Statistics" Summer semester 2010

last reviewed: Sept, 01, 2010, Christine Schneider

Peter Mathé (WIAS Berlin)

Mathematics of Conjugate Gradient Iteration

Abstract: not available

Nicole Krämer (WIAS Berlin)

Conjugate Gradient Regularization - a Statistical Framework for Partial
Least Squares Regression

Abstract: In this talk, I will show that Partial Least Squares (PLS) regression is
universally consistent.

PLS comprises a class of methods for dimensionality reduction and
prediction. PLS has proven to be successful in a wide range of
applications. However, the derivation of its statistical properties
often remains a challenging task. This is due to the fact that PLS
constructs latent components that depend on the response. While this
typically leads to good performance in practice, it makes the
statistical analysis more involved.

The main goal of my talk is to motivate a statistical framework for PLS
that is based on its equivalence to conjugate gradient regularization.
In this framework, it is possible to study important statistical
properties of PLS in a concise way. In particular, I will derive
universal consistency properties and optimal convergence rates.
Moreover, these results cast PLS into the theory of statistical inverse
problems.

(joint work with Gilles Blanchard)

Jörg Polzehl/Karsten Tabelow (WIAS Berlin)

Some statistical problems in Magnetic Resonance Imaging

Abstract: We present some statistical problems that occur in the context of
functional Magnetic Resonance Imaging (fMRI) and Diffusion-weighted
Imaging (DWI). The talk will introduce aspects of the analysis of both
medical imaging facilities and present open statistical problems.

Le-Minh Ho (HU Berlin)

Saddle point model selection with application to multiple testing

Abstract: not available

Volodja Spokoiny (WIAS/HU Berlin)

Parametric estimation methods I

Abstract: The talk discusses the problem of parameter estimation in a
non-classical situation when

- the parametric assumption is possibly misspecified
- the sample size is fixed and can be even small.

The main results are a local central bound and a global deviation bound.
It will be shown that these results imply some concentration properties, risk bounds and
coverage probability statements.

Gregory Temnov (University College Cork)

Extended stability property for exponential families: a model for financial applications

Abstract:A class of distributions originating from an exponential family and having a property related to the strict stability is described. An explicit form for the characteristic function is given. It appears that
the resulting class coincides with a subclass of Tweedie distributions,
respectively it is related to stable distributions and includes Inverse
Gaussian distribution and Levy distribution as special cases. Due to its
origin, the proposed distribution has a sufficient statistic. Besides,
it combines stability property at lower scales with an exponential decay
of the distribution's tail and has an additional flexibility due to the
convenient parameterization. All of it provides a motivation for using
the proposed class in financial modelling. Apart from the basic model,
we consider a generalization related to geometric stable distributions.
tba

Volodja Spokoiny (WIAS/HU Berlin)

Parametric estimation methods II

Abstract: tba

Volodja Spokoiny (WIAS/HU Berlin)

Parametric estimation methods III

Abstract: tba

Jim Gatheral

tba

Abstract: tba

Valentin Patilea (INSA, Rennes)

Bandwidth-Robust Inference with Conditional Moment Restrictions

Abstract:We propose a new estimation method for models defined by conditional moment restrictions. Our method optimizes a criterion that resembles a statistic based on smoothing techniques used for specification testing. Depending on whether the smoothing parameter is fixed or decreases to zero with the sample size, our approach defines a whole class of estimators. We develop a theory for estimation and inference with focus on robustness with respect to the bandwidth parameter, that is, our results hold uniformly over a wide range of bandwidths that allows for fixed as well as vanishing bandwidths. We show consistency and asymptotic normality of our estimator, allowing for non-smooth moment restrictions, such as quantile restrictions. In the case where the bandiwdth goes to zero, we propose an efficient version of our estimator. We develop inference procedures based on a distance metric statistic to test restrictions on parameters and we study a new bootstrap method to approximate the critical values of these tests.

Robert Liptser (Tel Aviv University)

Modification of Beneš approach. New proof and applications

< p>

Abstract:It is well known the Girsanov exponent
$$
\mathfrak{z}_t=\exp\Big(\int_0^ta(s,B_{[0,s]})dB_s-\frac{1}{2}\int_0^ta^2(s,B_{[0,s]})ds\Big)
$$
being solution of Doleans-Dade's equation
$
\mathfrak{z_t}=1+\int_0^t\mathfrak{z}_sa(s,B_{[0,s]})dB_s
$
generated by Brownian motion $B_t$, is a positive supermartingale
with $\E\mathfrak{z}_t\le 1$. If $\E \mathfrak{z}_T=1$, then the
random process $(\mathfrak{z}_t)_{t\in[0,T]}$ is the martingale.
Sufficient conditions of
$$
\E\mathfrak{z}_T=1 \eqno(1)
$$
to hold represent an important question both in theory and
applications: finance, weak solution of It\^o's equations, absolutely continuity
of diffusion processes distributions, filtering, Bayes's formulas, etc.
The property $\E\mathfrak{z}_T=1$ heavily depends on a functional
$a(t,x_{[0,t]})$ of arguments $\{t, x_{[0,t]}\}$, where
$(x_t)_{t\in[0,T]}$ belongs to a space of continuous function supported on $[0,T]$.
The Bene\^s condition guaranteing (1) reads as:
$$
a^2(t,x_{[0,t]})\le \text{\rm
const.}\Big[1+\sup_{s\in[0,t]}x^2_s\Big]_{t\in[0,T]}
\eqno(2)
$$
and looks like a linear growth condition of past dependent coefficient.
It is incomparably simpler than Novikov, Kazamaki, Krylov conditions and it is compatible
with many $a(t,x_{[0,t]})$ which can not be served by applying different approaches.
The original Bene\^s's proof of (1) uses a discrete time approximation of
$\mathfrak{z}_T$.
As any approximation approach it imposes additional assumptions on $a(t,x_{[0,t]})$ especially
in a vector case or some extended setting as:
\begin{align*}
\mathfrak{z}_t=1+\int_0^t\mathfrak{z}_sa(s,X_{[0,s]})dB_s, \quad
X_t=X_0+\int_0^tb(s,X_{[0,s]})dB_s.
\end{align*}
We give a new proof of Bene\^s approach which is of independent interest.
It is compatible with the above mentioned setting and even if $B_t$ is replaced by purely
discontinuous martingale $M_t$ with independent increments:
$
\mathfrak{z}_t=1+\int_0^t\mathfrak{z}_{s-}a(s,M_{[0,s)})dM_s.
$
In this case $(x_t)_{t\in[0,t]}$ belong to the Skorokhod space and
$$
a^2(t,x_{[0,t]})\le \text{\rm
const.}\Big[1+\sup_{s\in[0,t]}x^2_{s-}\Big]_{t\in[0,T]}.
$$ tba

Oleksandr Valinkevych (HU Berlin)

Local polynomial adaptive regression estimation

Abstract: tba

Ronnie Loeffen (WIAS Berlin)

Title:The Ornstein-Uhlenbeck type risk model: absolute ruin and
spectral representation

Abstract: We consider an insurance risk process modeled by a
mean-fleeing Ornstein-Uhlenbeck type process with a subordinator as the
background driving process.
In this model the company earns interest on positive surplus and pays
interest (at the same rate) when the surplus is negative. It is possible
that the company gets absolutely ruined, which is the event where the
premium income can no longer compensate for the interest payments.
In this talk we provide simple expressions for the Laplace transform in
space of both the finite- and infinite-time absolute ruin probability.
Crucial in the derivation is the fact that our risk process is in some
sense dual to another Ornstein-Uhlenbeck type process, namely the one
that was introduced by Barndorff-Nielsen and Shephard in finance to
model stochastic volatility. For the latter process, we give under some
conditions a spectral representation for its transition density, which
is the analogue of the well-known spectral representation for the
classical Ornstein-Uhlenbeck process. As a first application, this
representation allows us to quickly compute finite-time absolute ruin
probabilities.
This is joint work with Pierre Patie (Université Libre de Bruxelles).