Download On the use of conditional expectation estimators

New Developments in Pure and Applied Mathematics
On the use of conditional expectation estimators
Sergio Ortobelli, Tommaso Lando
difference between the two methods, that are actually aimed at
vs. the
different estimates (i.e. the mathematical function
) and therefore are not comparable. In
random variable
order to compare these two different methodologies, we
based
propose a method to estimate the distribution of
on the kernel non-parametric formula proposed by [1] and [2].
(which, for
Then, if we know the real distribution of
instance, can be easily computed in case of normality), then
we can perform a simulation analysis, drawing a bivariate
, and finally investigate which
random sample from
estimated distribution better fits to the true one.
The paper is organized as follows: in Section II we present
the different methodologies and their properties; in Section III
we examine a method to compare the two estimators, with
assumption of normality; in Section IV we briefly illustrate the
financial interpretation and possible application of the
conditional expected value.
Abstract— This paper discusses two different methods to
estimate the conditional expectation. The kernel nonparametric regression method allows to estimate the regression
function, which is a realization of the conditional expectation
. A recent alternative approach consists in estimating
the conditional expectation (intended as a random variable),
based on an appropriate approximation of the σ-algebra
generated by X. In this paper, we propose a new procedure to
estimate the distribution of the conditional expectation based
on the kernel method, so that it is possible to compare the two
approaches by verifying which one better estimates the true
. In particular, if we assume that the
distribution of
is normally distributed, then
two-dimensional variable
can be computed quite easily,
the true distribution of
and the comparison can be performed in terms of goodness-offit tests.
Keywords—Conditional Expectation, Kernel, Non Parametric,
II. METHODS
Regression.
In this section we describe two different procedures to
evaluate the conditional expected value between two random
and
be integrable random
variables. Let
. Let
variables in the probability space
be a random sample of
independent observations from the bi-dimensional variable
. The first procedure is aimed at estimating the
given
, which is a
conditional expectation of
mathematical function of ; the second method yields an
unbiased and consistent estimator of the random variable
.
The kernel non-parametric regression
It is well known that, if we know the form of the function
(e.g. polynomial, exponential, etc.), then
with several
we can estimate the unknown parameters of
methods (e.g. least squares). In particular, if we do not know
, except that it is a continuous and
the general form of
smooth function, then we can approximate it with a nonparametric method, as proposed by [1] and [2]. Thus,
can be estimated by:
I. INTRODUCTION
W
ithin a bivariate probabilistic framework, this paper
discusses different methods to estimate the conditional
expected value. On the one hand, several well known
methods are aimed at estimating the regression function
, which represents a realization of the
. In particular, the kernel nonrandom variable
parametric regression (see [1] and [2]) allows to estimate
as a locally weighted average, based on the
choice of an appropriate kernel function: the method yields
consistent estimators, provided that the kernel functions and
the random variable satisfy some conditions, described in
Section II. On the other hand, an alternative methodology was
recently introduced by [3] for estimating the random variable
: this method has been proved to be consistent without
requiring any regularity assumption. In this paper we stress the
This paper has been supported by the Italian funds ex MURST 60% 2014,
2015 and MIUR PRIN MISURA Project, 2013–2015, and ITALY project
(Italian Talented Young researchers). The research was also supported
through the Czech Science Foundation (GACR) under project 13-13142S and
through SP2013/3, an SGS research project of VSB-TU Ostrava, and
furthermore by the European Regional Development Fund in the
IT4Innovations Centre of Excellence, including the access to the
supercomputing capacity, and the European Social Fund in the framework of
CZ.1.07/2.3.00/20.0296 (to S.O.) and CZ.1.07/2.3.00/30.0016 (to T.L.).
S.O. Author is with University of Bergamo, via dei Caniana, 2, Bergamo,
Italy; and VŠB -TU Ostrava, Sokolská třída 33, Ostrava, Czech republic; email: [email protected].
T.L. Author is with University of Bergamo, via dei Caniana, 2, Bergamo,
Italy; and VŠB -TU Ostrava, Sokolská třída 33, Ostrava, Czech republic; email: [email protected].
ISBN: 978-1-61804-287-3
,
(1)
where
is a density function such that i)
;
; iii)
when
. The
ii)
is denoted by kernel, observe that kernel
function
functions are generally used for estimating probability
densities non-parametrically (see [4]). It was proved in [1] that
is quadratically integrable then
is a consistent
if
. In particular, observe that, if we denote by
estimator for
244
New Developments in Pure and Applied Mathematics
the joint density of
, the denominator of (1)
, while the
converges to the marginal density of
numerator converges to the function
(note that, if
The method, as defined by (3), requires only that is an
integrable random variable. From a practical point of view,
given n i.i.d. observations of , if we know the probability
corresponding to the i-th outcome , we obtain:
.
(5)
Otherwise, we can give uniform weight to each observation,
:
which yields the following consistent estimator of
is continuous, the
function
intended as a regular conditional probability).
has to be
,
The OLP method
We now describe an alternative non-parametric approach
[3] for approximating the conditional expectation, the method
is denoted by “OLP”, which is an acronym of the authors’
the σ-algebra generated by X (that is,
names. Define by
, where
is the
Borel σ-algebra on ). Observe that the regression function is
,
just a “pointwise” realization of the random variable
. The following
which can equivalently be denoted by
rather than
.
methodology is aimed at estimating
can be approximated by a σ-algebra generated by a
, we
suitable partition of . In particular, for any
of
consider the partition
where
is the number of elements of . Therefore, we are
, which in turn is a consistent
always able to estimate
.
estimator of the conditional expected value
III. COMPARISON IN CASE OF NORMALITY
If we assume that and
i.e.
of the random variable
that
are jointly normally distributed,
, we can obtain the distribution
quite easily. Indeed, we know
,
(7)
in
therefore, as
subsets, where b is an integer number greater than 1 and:
•
(6)
, we obtain that
,
.
(8)
•
•
Of course, if we simulate data from
and approximate
with the estimator
defined in (3), we can
finally compare the true and the theoretical (estimated)
distribution by performing a goodness-of-fit test. Differently,
the kernel non-parametric regression method does not allow to
, but only yields a consistent estimator of
estimate
. However, assume that the random variable
is
and moreover
(that is,
independent from
and
): in this case we can estimate
with
.
Starting with the trivial sigma algebra
, we can
obtain a sequence of sigma algebras generated by these
partitions, for different values of k (k=1,…,m,…). For
is the sigma algebra
instance,
,
generated by
s=1,...,b-1 and
. Generally:
,
(2)
and thereby we can also estimate the distribution of
,
. Obviously, the estimate depends
because
on the choice of the kernel function . It is proved that
converges
almost
surely
to
). Moreover, note that also
(
satisfies a weaker convergence property (convergence in
distribution). Indeed, we have that
,
(10)
thus we obtain that
.
Finally, it is possible to compare the two methods by
verifying which one better estimates the distribution of
, future studies will be focused on this issue.
Hence, it is possible to estimate the random variable
by
,
where
(3)
. Indeed, by definition of the
conditional expectation, we can easily verify that
the unique -measurable function such that, for any set
, (that can be seen as a union of disjoint sets, in
we obtain the equality
particular
is
IV. CONDITIONAL EXPECTATION AND FINANCIAL
APPLICATIONS
(4)
It is proved in [3] that
random variable
a.s.
The conditional expectation of a random variable given
another can be especially useful for financial applications. In
particular, we can use conditional expectation estimators
is a consistent estimator of the
, that is,
ISBN: 978-1-61804-287-3
(9)
245
New Developments in Pure and Applied Mathematics
either for ordering the investors choices as suggested by [3] or
to evaluate and exercise those arbitrage opportunities when
applies in the market.
We recall the classical definitions of first and second-orders
stochastic dominance.
First order stochastic dominance: X FSD Y if and only if
(9) yields a consistent estimator of the distribution function of
. In future work it will be possible to compare these
two methodologies by verifying which one better estimates the
, based on simulation analysis and
distribution of
goodness-of-fit tests. Moreover, we recall that these estimators
may have several financial applications such as ordering
investors’ opportunities or identifying arbitrage opportunities.
or, equivalently X FSD Y if and only if
for any increasing function .
Second order stochastic dominance (increasing concave
REFERENCES
[1]
order): X SSD Y if and only if
[2]
or, equivalently X SSD Y if and only if
for any increasing and concave function .
Obviously X FSD Y implies also X SSD Y.
If we assume that X and Y are, for instance, two different
gambles or investments, the financial interpretation of
stochastic orders follows straightforward. Indeed, in this case
X FSD Y means that X is stochastically “larger” than Y, while
X SSD Y indicates a larger expectation of gain and generally
an inferior “risk”. Those investors who prefer X to Y ,
provided that X SSD Y, are generally defined non-satiable risk
averse investors. The following property characterizes the
second order stochastic dominance in terms of conditional
expectation.
[3]
[4]
[5]
[6]
[7]
Super-martingale property. X SSD Y if and only if there exist
two random variables X', Y' defined on the same probability
space that have the same distribution of X and Y such that:
The proof of this property arises from the analysis proposed
by Strassen [5] and is a well-known result of ordering theory
(see also [6]-[7] and the references therein).
Thus, using the empirical evaluation of the conditional
expected value, we can attempt to order the investors choices
as suggested by [3]. On the other hand, using the fundamental
theorem of arbitrage, we know that there exist no arbitrage
opportunities in the market if there exists a risk neutral
martingale measure under which the discounted price process
results a martingale. So, when we assume that the filtration
is the one generated by the price process {Xt}t>0 (assumed
)=
).
to be a Markov process) then we get that
Therefore, this property the contidional expected value
estimator and the fundamental theorem of arbitrage can be
used to estimate the risk neutral measure and the presence of
arbitrage opportunities in the market.
V. CONCLUSION
In this paper, we deal with two methodologies for
estimating the conditional expectation, studying their
properties and analyzing their differences. We also propose a
based on the
procedure to estimate the distribution of
kernel method. We observe that the OLP method proposed by
[3] yields a consistent estimator of the random variable
, while the generalized kernel method, proposed in eq.
ISBN: 978-1-61804-287-3
246
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