Simulated Minimum Cramér-Von Mises Distance Estimation for Some Actuarial and Financial Models

Minimum Cramér-Von Mises distance estimation is extended to a simulated version. The simulated version consists of replacing the model distribution function with a sample distribution constructed using a simulated sample drawn from it. The method does not require an explicit form of the model density functions and can be applied to fitting many useful infinitely divisible distributions or mixture distributions without closed form density functions often encountered in actuarial science and finance. For these models likelihood estimation is difficult to implement and simulated Minimum CramérVon Mises (SMCVM) distance estimation can be used. Asymptotic properties of the SCVM estimators are established. The new method appears to be more robust and efficient than methods of moments (MM) for the models being considered which have more than two parameters. The method can be used as an alternative to simulated Hellinger distance (SMHD) estimation with a special feature: it can handle models with a discontinuity point at the origin with probability mass assigned to it such as in the case of the compound Poisson distribution where SMHD method might not be suitable. As the method is based on sample distributions instead of density estimates it is also easier to implement than SMHD method but it might not be as efficient as SMHD methods for continuous models.


Introduction
In actuarial science or finance we often model losses or log-returns with distri-complicated in such a situation.
For statistical inferences using models with these features, we shall assume to have independent and identically distributed (iid) observations 1 , , n X X  which have a common distribution as X with model distribution and density given respectively by ( ) 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 X representable as a random sum, i.e., The vector of parameters is ( ) , , α β λ ′ = θ . It is not difficult to simulate from the distribution of X but the density function of X 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 submodel 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 X with The i Y 's are iid with a common distribution and mgf given respectively by The distribution function of the double exponential distribution is 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 Y , i.e., the mean and variance are given respectively by It is easy to see that the mgf of X is given by ( ) 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 em- as given by Duchesne 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 2 l 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 2 l . 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.

The Space l 2 and Its Norm
We can make use elegant results of , , x assumed to be finite. Clearly, . is a norm for 2 l and it generalizes naturally the Euclidean norm. Also, a vector ( ) For a matrix For estimation, we assume that we have a random sample which consist of n iid observations 1 , , n X X  from a continuous parametric family with distribu- We also assumed that F θ has no closed form expression but simulated samples can be drawn from Define the following vectors of random functions (13) for version D and it is easy to see that Equivalently, has support on the real line and Clearly, both versions of MCVM estimation can be treated in a unified way using this set up, we also have ( ) Theorem 1: Under the following conditions, the estimators given by the vector  θ converges in probability to 0 θ , the vector of the true parameters, i.e., , Ω is the parameter space assumed to be compact.
2)   [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 ( ) a n Q θ to approximate Let ˆS θ and * θ be the vectors which minimize

Asymptotic Normality
In this section, we shall state Theorem 2 which is essentially Theorem 1) The parameter space Ω is compact.
or equivalently, using equality in distribution, The proofs of these results follow from the results used to prove Theorem 3.3 given by Pakes and Pollard [13]. Therefore, for version D, ( ) n G θ as defined by expression (13) And for version S, ( ) n G θ 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.
The matrices D and V depend on 0 θ , and we adopt the notations These results are proved by Pakes and Pollard [13], see the proofs of their 2) The sequence of functions As for the Euclidean space, the sufficient condition for differentiability here only requires the partial derivatives and consider the vector of influence function x δ is the degenerate distribution at the point , is given by expression (2.15) in Duchesne et al. [8] (p. 407), with with its elements given by An estimate for the covariance matrix 1 V can be defined as Using  1 V , an estimate for the asymptotic covariance matrix of θ 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 2 D 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 It is also clear that the elements of ′ Γ Γ are given by

An Estimate for the Covariance Matrix for SCVM Estimators
The asymptotic covariance matrix of ˆS The factor 1 1 τ + represents the loss of overall efficiency due to simulations and can be controlled if we let 10 τ ≥ . 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] are as given by expression (20).
An estimate for 1 V for version S can then be defined as Consequently, an estimate  Clearly with  2 D available, it will facilitate hypothesis testing for the parameters of the model.

MM Estimation for the Compound Poisson Gamma Model
The The sample mean and variance are given respectively by X and 2 s , the moment estimators can be obtained explicitly. Note from these equations let ( ) For the overall asymptotic relative efficiency (ARE) for the compound gamma model we use The range of the parameters being considered is given by 2 10,1 10,1 10 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  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

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.