Simulated Minimum Cramér-Von Mises Distance Estimation for Some Actuarial and Financial Models ()
1. Introduction
In actuarial science or finance we often model losses or log-returns with distribution functions where neither the distribution function nor its corresponding density function has a closed form expression yet it is not complicated to draw random samples from these distributions. It is clear that likelihood methods are complicated in such a situation.
For statistical inferences using models with these features, we shall assume to have independent and identically distributed (iid) observations
which have a common distribution as
with model distribution and density given respectively by
and
. Neither
nor
has a closed form expression but often its moment generating function (mgf)
has a closed form expression. The vector of parameters of interest is
The compound Poisson distribution used in actuarial sciences and jump diffusion distribution in finance are typical examples for these types of models. Furthermore, in many circumstances distributions derived from the increments of Lévy processes also display these characteristics and it is of interest to make inferences for the vector of parameters. We shall illustrate the situation with example 1 and example 2 below.
Example 1
In this example, we shall consider the compound Poisson gamma distribution which is commonly used in actuarial science and it arises from the compound Poisson processes which also belong to the class of Lévy processes.
The compound gamma distribution is the distribution of a random variable
representable as a random sum, i.e.,
with the
’s being iid with a common gamma distribution with the density function given by
and the moment generating function is given by
. The random variable
follows a Poisson distribution with parameter
and it is assumed that the
’s and
are independent.
Note that the moment generating function of
is
(1)
and from the mgf
the first there cumulants can be found and they are given by
(2)
The vector of parameters is
.
It is not difficult to simulate from the distribution of
but the density function of
has no closed form, see Klugman et al. [1] (p. 143) for the series representation of this density and strictly speaking this is not a continuous model but a hybrid model where there is a probability mass assigned to the origin. Furthermore, continuous distributions created using a mixing mechanism also leads to continuous mixture distributions without closed form density but simulated samples often can be drawn from such distributions. These distributions are commonly used in actuarial science and they are given by Klugman et al. [1] (pp. 62-65); see Luong [2] for other distributions with similar features used in actuarial science.
Lévy processes are also used in finance and they can be used as alternative models to the classical Brownian motion. The distributions of the increments of these processes can be more flexible than the normal distribution, they can be asymmetric and have fatter tail than the normal distribution. Consequently, they are more suitable to model log-returns of assets in finance. The following double exponential jump diffusion distribution is an illustration of an alternative distribution to the normal distribution which is the distribution of the increments of a Brownian motion.
Example 2
The double exponential jump diffusion model is a special case of a larger class of jump diffusion models where the distribution for the jumps follows an asymmetric Laplace distribution instead of the classical normal distribution as in the classical jump-diffusion model introduced by Merton [3] . This distribution has six parameters and it has been studied by Kou [4] , Kou and Wang [5] . A sub- model which is the double exponential jump diffusion model with only five parameters has been found very useful for modeling log-returns of stocks, see Tsay [1] (pp. 311-319). We shall call this model, the KWT model. Exact pricing for European call option for this model is also possible with the use of some special functions. The distribution can be represented as the distribution of
with
The
’s are iid with a common distribution and mgf given respectively by
(3)
. (4)
The distribution function of the double exponential distribution is
for
and
for
. (5)
Since this distribution has an explicit expression, simulated samples drawn from the double exponential distribution can be based on the inverse method.
Tsay [6] (p. 312) also gives additional properties of the distribution of
, i.e., the mean and variance are given respectively by
(6)
It is assumed that the
’s,
and
are independent,
follows a Poisson distribution with parameter
and
has a normal distribution
.
It is easy to see that the mgf of
is given by
. (7)
From
, the first five cumulants can be found and they are given by
(8)
and
(9)
The vector of parameters is
.
For models introduced by these examples if we use methods of moments (MM) to estimate the parameters, the MM estimators will lack of robustness properties and they might not even be efficient as for models with more than two parameters, MM estimators will depend on polynomials of degree higher or equal to three hence will be unstable in the presence of outliers. Estimators based on empirical characteristic functions procedures such as the KL procedures of Feurverger and McDunnough [7] involves an arbitrariness of choice of points to match the empirical characteristic function with its model counterpart which motivates us in this paper to extend Cramer-von Mises estimation to a simulated version (version S). The classical Minimum Cramer Von-Mises estimators (version D) are given by the vector
which minimize the objective function
or equivalently (10)
(11)
as given by Duchesne et al. [8] with
and
are respectively the sample and model distribution. The MCVM estimators are known to be robust,
is the commonly used sample distribution and
is
the indicator function. Note that if it is easy to draw samples from
, we can construct the simulated sample distribution function
using
observations drawn from
similarly and minimize instead the following objective function
(12)
to obtain estimators. We shall call these estimators simulated MCVM (SMCVM) estimators and denoted them by the vector
and we shall call this version, version S. The method is numerically relatively simple to implement using simplex direct search methods which are derivative free. Packages like R already have built in function to minimize a function using the Nelder-Mead simplex method. The SMCVM method does not require a proxy model like other simulated methods such as methods based on indirect inference, see Garcia et al. [9] , also see Smith [10] . Therefore, the method appears to be useful for actuarial science and finance where there are needs to analyze data using these types of distributions. It can also be viewed as a natural extension of the classical MCVM methods proposed by Hogg and Klugman [11] (p. 83) where the asymptotic properties of the estimators have been established by Duchesne at al. [8] . Like Simulated minimum Hellinger (SMHD) method proposed by Luong and Bilodeau [12] , the new method is robust and it is even easier to implement than SMHD method as it makes use of sample distribution functions instead of density estimates. Furthermore, it can handle models like the compound Poisson model which displays a probability mass at the origin where SMHD method might not be suitable but comparing to SMHD estimators, the SCVM estimators might not be as efficient as the SMHD estimators for continuous models.
The paper is organized as follows. Following the approach in section 3 by Pakes and Pollard [13] (pp. 1037-1043) who make use of the Euclidean space and Euclidean norm to establish asymptotic properties of estimators, the Hilbert space
is used in this paper with a natural norm extending respectively the Euclidean space and the commonly used Euclidean norm. Asymptotic properties for both the CVM estimators and the SCVM estimators can be established using a unified approach by considering minimizing the norm of a random function to obtain estimators and they are given in section 2. This approach also facilitates the use of the available results of their Theorems given in section 3 by Pakes and Pollard [13] as most of the results of their Theorems continue to hold in
. The SMCVM estimators are shown to be consistent and have an asymptotic normal distribution. Their asymptotic covariance can be estimated using the influence function approach which were used by Duchesne et al. [8] . An estimate for the covariance matrix is also given in section 2 and by having such an estimate, it will make hypotheses testing for parameters easier to handle. Section 3 displays results of a limited simulation study using the compound Poisson gamma model and the double exponential jump distribution where we compare the SMCVM estimators with methods of moment (MM) estimators. For both models, it appears that the SCVM estimators are much more efficient than MM estimators using the overall relative efficiency criterion.
2. Asymptotic Properties of the SCVM Estimators
2.1. The Space l2 and Its Norm
We can make use elegant results of Theorem 3.1 and 3.2 in section 3 of the paper by Pakes and Pollard [13] (pp. 1037-1043) to investigate asymptotic properties of CVM estimators and simulated CVM estimators (SCVM). For asymptotic results of estimators using simulations in their seminal works, Pakes and Pollard [13] consider estimators obtained by minimizing the Euclidean norm of a vector of random functions. The vector of random functions in their set up belong to the Euclidean space. If we used their results to investigate CVM estimation it is more convenient to consider the Hilbert space
with infinite dimension which generalize the Euclidean space and the following norm
defined below which generalizes the Euclidean norm.
For an element
which belongs to
, define
assumed to be finite. Clearly,
is a norm for
and it
generalizes naturally the Euclidean norm. Also, a vector
is of finite dimension p, hence belongs to the Euclidean space then it belongs to
with
. The space
and the norm
have been studied in functional analysis or real analysis, see Davidson and Donsig [14] (pp. 137-141) for example.
For a matrix
in
, define
. With the space
, most of the results of their Theorems in section 3 are valid and only some minor changes are needed.
For estimation, we assume that we have a random sample which consist of n iid observations
from a continuous parametric family with distribution
. We also assumed that
has no closed form expression but simulated samples can be drawn from
.The commonly used sample distribution function is denoted by
. The vector of parameters is denoted by
.
Define the following vectors of random functions
(13)
for version D and it is easy to see that
. (14)
Equivalently,
, (15)
if
has support on the real line and
, if
is the distribution of a nonnegative random variable. Using the set up given by section 3 in Pollard and Pakes [13] , the classical MCVM estimators can be viewed as the vector of values which minimize
or
as defined by expression (15).
For the simulated version of MCVM estimation, i.e., version S, define
(16)
with
being the sample distribution function based on the simulated sample of observations of size S drawn out
. Then the SCVM estimators given by the vector
is obtained by minimizing
. (17)
Clearly, both versions of MCVM estimation can be treated in a unified way using this set up, we also have
in probability. For both versions, let
,
and we have
,
, with
being an expression which converges to 0 in probability.
We shall restate Theorem 3.1 given by Pakes and Pollard [13] (p. 1038) assuming the space
and its norm as defined earlier are used so that it is more suitable for CVM estimation. The condition ii) which requires
in their Theorem 3.1 can be replaced by
as only this condition is used in their proof. Note that the set up for their Theorem is very general, we only need to verify their conditions for estimators obtained by minimizing the objective function of the form
(18)
Theorem 1: Under the following conditions, the estimators given by the vector
converges in probability to
, the vector of the true parameters, i.e.,
.
1)
,
is the parameter space assumed to be compact.
2)
.
3)
for each
,
is an expression bounded in probability.
Clearly for the SMCVM estimators given by the vector
which minimizes
will satisfy condition 1) and 2) of Theorem 1 as
only at
if the parametric family is well parameterized
which is the case in general. Note that the integrand of the integral defined by expression (10) is nonnegative and smaller or equal to one. Therefore, in probability,
for
The condition 3) is satisfied in general which implies consistency for the SCVM estimators, we then have
. Note that since
is always bounded it is not surprising that it generates robust estimators. For more on robustness in the sense of bounded influence functions for the SMCVM estimators see section (2.2.2). Also, observe that
remains consistent even the parametric models are only hybrid, i.e., with some discontinuity points such as in the case of the compound Poisson models. Now we turn our attention to the question of asymptotic normality for
and discuss informally the arguments used to establish asymptotic normality for
first and the formal arguments will follow subsequently from the proofs of Theorem 3.3 by Pakes and Pollard [13] (pp. 1040-1043). A version of their Theorem 3.3 is restated as Theorem 2 below.
Since
is not differentiable, the traditional Taylor expansion argument cannot be used to establish asymptotic normality of estimators obtained by minimizing
. Here, we assume
is differentiable with derivative matrix
, it means Fréchet differentiable with respect to the norm
for
; see Luenberger [15] for the notion of Fréchet differentiability and see chapter 3 of the book by Luenberger [15] for the notion of Hilbert space.
If the property of differentiability holds then we can define the random function
to approximate
with
(19)
Let
and
be the vectors which minimize
and
respectively. The ideas behind the proofs for asymptotic normality of Theorem (3.3) of Pakes and Pollard are if the approximation of the original objective function
which is not differentiable by a differentiable one namely
is of the right order then the vector
which minimizes
and
, the vector which minimizes
are asymptotically equivalent, i.e., we have:
1)
or using equality in distribution,
and it is easy to see that
can be expressed explicitly as
since
is an affine transformation.
2)
,
is an expression converging to 0 in probability at a faster rate than
.
Note that the matrix
is of rank m with m columns but infinite number of rows given by
with
and
An estimate of this matrix
is
and is defined by expression (33) in section (2.2), consequently we can estimate
by its estimate
using the corresponding elements
extracted from
,
(20)
Under these conditions, it suffices to work with
and
to derive asymptotic distribution for of
. A regularity condition for the approximation is of the right order given by their Theorem 3.3 which is the most difficult to check is given as
by Pakes and Pollard [13] (p. 1040).
A slightly more stringent condition which obviously implies the above regularity condition is
. (21)
For simulated methods for this condition to hold, in general independent samples for each
cannot be used, see Pakes and Pollard [13] (p. 1048). Otherwise, only consistency can be guaranteed for estimators using version S, see section 2.2.2 for the same seed issue. For version S, the simulated samples are assumed to have size
and the same seed is used across different values of
to draw samples of size
. These two assumptions are quite standard for simulated methods of inferences, see section 9.6 for method of simulated moments (MSM) given by Davidson and McKinnon [16] (p. 384), also see Smith [10] (p. S66) for this assumption for his simulated quasi-likelihood estimators. For numerical optimization to find the minimum of
, we rely on direct search simplex methods which are derivative free. Chong and Zak [17] (pp. 273-278) provides a good overview of derivative free simplex algorithm.
2.2. Asymptotic Normality
In this section, we shall state Theorem 2 which is essentially Theorem (3.3) given by Pakes and Pollard [13] and comment on the conditions needed to verify asymptotic normality for the MCVM estimators for version D and S.
Theorem 2
Let
be a vector of consistent estimators for
, the unique vector which satisfies
.
Under the following conditions:
1) The parameter space
is compact.
2)
3)
is differentiable at
with a derivative matrix
of full rank
4)
for every sequence
of positive numbers which converge to zero.
5)
,
is an expression bounded in probability.
6)
is an interior point of the parameter space
, assumed to be compact.
Then, we have the following representation which will give the asymptotic distribution of
in Corollary 1, i.e.,
, (22)
or equivalently, using equality in distribution,
. (23)
The proofs of these results follow from the results used to prove Theorem 3.3 given by Pakes and Pollard [13] . For expression (22) or expression (23) to hold only condition 5) of Theorem 2 is needed and used in their proofs of Theorem 3.3 and there is no need to assume that
has an asymptotic distribution. Clearly MCVM estimators or SCVM estimators are obtained by minimizing
hence they will satisfy the condition 2) of Theorem 2 with
as defined by expression (14) or expression (17) depending it is version D or version S being considered.
Therefore, for version D,
,
as defined by expression (13)
And for version S,
,
as defined by expression (16).
From the result of the Theorem, it is easy to see that we can obtain the main result of the following corollary which gives the asymptotic covariance matrix of the estimators.
Corollary 1.
Let
, if
and
,
is full rank and symmetric then
with
(24)
The matrices
and
depend on
, and we adopt the notations
.
These results are proved by Pakes and Pollard [13] , see the proofs of their Theorem (3.3). We just need to verify these conditions are met for SMCVM estimation. Before verifying these conditions for both versions of MCVM estimation, the following assumptions are needed to verify the condition 4 of Theorem 2 which is the most difficult condition to verify. We need to define the following sequence of functions,
as it will be used later,
Assumption 1
1) As
and
, for version S of CVM estimation
, (25)
Is the conditional expectation on
of the expression inside the bracket.
2) The sequence of functions
is differentiable with continuous partial derivatives,
, the expectation is under
and using the usual conditioning argument, it can also be expressed
(26)
For the condition 1) of Assumption 1 to hold we cannot use independent samples for different values of
to draw simulated samples for version S of CVM estimation, otherwise
and
cannot converge to 0 in probability. This justifies the same seed must be used to generate random samples for different values of
.
We shall proceed to check the regularity conditions for both versions of MCVM estimation and note that
is the derivative of
in
means that
is the Fréchet derivative at
with the property
As for the Euclidean space, the sufficient condition for differentiability here only requires the partial derivatives
being continuous with respect to
. For the notion of derivative in Hilbert space, see the notion of Fréchet derivative in Luenberger [15] (pp. 171-177) which generalizes the notion of derivative of Euclidean space. The conditions (1-3) of Theorem 2 can be verified easily. The condition (4) of Theorem 2 will be met in general if Assumption 1 holds, see Appendix for details and justifications.
We proceed to find the asymptotic distribution for
. Using expression (22) and expression (23), we shall obtain the asymptotic covariance matrix for the MCVM estimators for both versions. For version D, the asymptotic covariance matrix has been obtained by Duchesne at al. [8] (p. 407), using the influence function approach with the statistical functional
being defined as
and consider the vector of influence function
(27)
is the degenerate distribution at the point
,
.The influence function
is bounded provided that
as a vector of functions of u
is bounded which implies the MCVM estimators are robust for version D. We shall assume this property of bounded influence functions holds implicitly; we shall see this also makes version S robust. Furthermore, based on standard results of robust estimation theory, the representations given by expressions (28) and (31) using influence functions are valid for the statistical functionals being considered. Now since
,
(28)
This is the influence function representation of
for version D and we have
with
for version D,
is the covariance matrix of
,
is given by expression (2.15) in Duchesne et al. [8] (p. 407), with
(29)
, since
.
Replacing
by
and
by
in the above expression leads to approximate the vector
by
with its elements given by
An estimate for the covariance matrix
can be defined as
(30)
Using
, an estimate for the asymptotic covariance matrix of
can be constructed, see expression (2.15) and expression (2.13) given by Duchesne et al. [8] (pp. 406-407). Clearly, the results for version D as given by Duchesne et al. [8] can be reobtained using this unified approach.
Note that the property of asymptotic normality continues to hold even the parametric model fails to be continuous and is only hybrid as in the compound Poisson gamma case. Using the arguments of the next paragraph to establish asymptotic normality, the same conclusion can be reached for version S. The derivation of the asymptotic covariance matrix
for the SCVM estimators is similar. We shall make use of the notion of bivariate statistical functional introduced by expression (1.6) given by Reid [18] (pp. 80-81). This leads to define the bivariate statistical functional
,
We have a representation which is similar to the representation given by expression (28) but using both
and
with
,
is as defined by expression (27) and
is similarly defined with
,
is the degenerate distribution at
and
. Note that
as given by expression (29) can also be reobtained using the bivariate statistical functional with
.
Based on the expression defining
, we have
and
is identical for version D and S. Therefore, for version S, we have the representation
. (31)
Note that the size of the random sample drawn from the model distribution is
and the
’s are iid and have the same distribution as the
’s but the
’s are independent of the
’s as the simulated sample is independent from the original sample represented by the data. Therefore,
. (32)
It is also clear that the elements of
are given by
which converge in probability to the corresponding elements
of the matrix
with
, i.e. (33)
2.3. An Estimate for the Covariance Matrix for SCVM Estimators
The asymptotic covariance matrix of
can be estimated if we can estimate
. Using a result given by Pakes and Pollard (p. 1043), an estimate for
is the matrix
(34)
with 1 occuring at the ith entry of the vector
and
,
and in general we can let
. Note that the columns of
estimate the corresponding columns of
with elements depend on
as mentioned in section (2.2).
Therefore, using results of Corollary 1 we have the asymptotic for version S
with
. (35)
The factor
represents the loss of overall efficiency due to simulations
and can be controlled if we let
. This factor is identical to the one for simulated unweighted minimum chi-square method or the one for simulated quasi-likelihood method, see Pakes and Pollard [13] (p. 1049), also see Smith [10] (p. S69). It suffices to estimate
then we can have an estimate for the asymptotic covariance matrix of the SCVM estimators as clearly
can be estimated by
.
Define
with its elements given by
are as given by expression (20).
An estimate for
for version S can then be defined as
. (36)
Consequently, an estimate
for
can be defined as
(37)
Clearly with
available, it will facilitate hypothesis testing for the parameters of the model.
3. Numerical Study
3.1. MM Estimation for the Compound Poisson Gamma Model
The MM method consists of matching the empirical cumulants with its model counterpart to form estimating equations and solutions will give the moment estimators. For the compound gamma model of example 1 this leads to the system of equations given by
.
The sample mean and variance are given respectively by
and
, the moment estimators can be obtained explicitly. Note from these equations let
and
which implies
and from the last equation, we can solve for
which gives
the MM estimator for
with
. Since the parameter
, we might want to define the moment estimator as
. It is not difficult to obtain
and
the corresponding MM estimators for
and
and when we also consider the constraints imposed on
and
, this leads to define
and
.
3.2. MM Estimation for the KWT Model
For the KWT model, there are five parameters so beside the first three empirical cumulants as defined above we also need the fourth and fifth empirical cumulants with
,
and matching
will give the moment estimators as in the previous example. It might be easier to let
and from these estimating equations, it is not difficult to see that the following two equations
and
depend only on
and
and can be solved numerically to obtain the MM estimators for
and
which are given respectively by
and
. Also, using the first three equations we obtain
.
We might want to redefine these MM estimators by imposing
.
In the limited simulation study, we draw
samples of size n=1000 for each sample and use
.
For the overall asymptotic relative efficiency (ARE) for the compound gamma model we use
, the mean square errors (MSE) are estimated using random samples and displayed in Table 1. The mean square error of an estimator
for
is defined as
.
The range of the parameters being considered is given by
.
We find that the SCVM method is more efficient than MM method, the order of ARE gained by using SCVM method is illustrated with results displayed in Table 1. We also test for various parameters outside the range and we also have
Table 1. Compound Poisson gamma model with
asymptotic overall relative efficiency between SCVM estimators and MM estimators.
The overall efficiency used for comparisons used is
Table 2. Model KWT (
) asymptotic overall relative efficiency between SCVM estimators and MM estimators.
The overall efficiency used for comparisons used is
similar findings.
For the KWT model we use the corresponding asymptotic relative efficiency (ARE) and it is defined as
The mean square errors (MSE) are similarly defined as in the case of the compound gamma model and again estimated using simulated samples. The ARE is a ratio with the total of mean square errors for the SCVM estimators appearing in the numerator and the total of mean square errors of MM estimators appearing in the denominator.
The key findings are illustrated using Table 2 and again SCVM method seems to perform much better than MM method for the common range of parameters used for modeling daily returns of stocks with
0 ≤ λ ≤ 0.010, 0.005 ≤ ω ≤ 0.010 and 0 ≤ μ ≤ 0.001, 0 ≤ σ ≤ 0.008. With the results displayed in Table 2 which give an idea of the order of the overall efficiency gained by using SCVM method, we can see that overall SCVM method is at least 100 time better than MM method for the range of parameters being considered. Clearly, more numerical studies are needed but we do not have the computer resources to conduct larger scale of study being in a small department equipped with only laptop personal computers. Despite the limited nature of the study it does point to better efficiency when using SCVM methods for models having at least three parameters, in general.
4. Conclusion
It appears that SCVM method has the potential to generate more efficient estimators than MM method especially for models with more than two parameters. Like SMHD method, it is also robust and easier to implement than SMHD method as it is based on sample distribution function instead of density estimates. It can handle continuous models with a few discontinuity points with probability masses attached to them where the SMHD method might not be suitable but it might be less efficient than SMHD method for continuous model, in general.
Acknowledgements
The helpful comments of an anonymous referee and the support of the staffs of OJS which lead to an improvement of the presentation of the paper are gratefully acknowledged.
Appendix
In this technical appendix, we shall prove that with the conditions of Assumption 1, the condition 4 of Theorem 2 will hold, i.e.,
i.e.,
uniformly as
and
Now define the sequence of functions
, it suffices to show
uniformly as
and
.
Using Markov’s type inequality, for any
, we have the following inequality
with
as given by expression (26).
Consequently, it suffices to have
uniformly as
and
. Clearly under Assumption 1 we have
pointwise but we need to strengthen it to uniform convergence for
. Therefore, it suffices to have equicontinuity for the sequence
as the domain of the sequence of functions is compact, see Rudin [19] (1974, p. 168). A sufficient condition for this property is the Lipschitz property which is related to the property of differentiability of the sequence of functions, see Davidson and Donsig [14] (2009, p. 88). Since the parameter space is compact and if the sequence
is differentiable hence Lipchitz then with Assumption 1, these properties implies equicontinuity for the sequences of functions
.
For the notion of stochastic equicontinuity a stochastic version of equicontinuity, see Newey and McFadden [20] (1994, pp. 2136-2138) equicontinuity which extend the notion of equicontinuity of deterministic functions of real analysis and section 7 in Newey and McFadden [20] (1994, pp. 2184-2193) on asymptotic normality with non-smooth objective function.