Asymptotic Normality Distribution of Simulated Minimum Hellinger Distance Estimators for Continuous Models

Certain distributions do not have a closed-form density, but it is simple to draw samples from them. For such distributions, simulated minimum Hellinger distance (SMHD) estimation appears to be useful. Since the method is distance-based, it happens to be naturally robust. This paper is a follow-up to a previous paper where the SMHD estimators were only shown to be consistent; this paper establishes their asymptotic normality. For any parametric family of distributions for which all positive integer moments exist, asymptotic properties for the SMHD method indicate that the variance of the SMHD estimators attains the lower bound for simulation-based estimators, which is based on the inverse of the Fisher information matrix, adjusted by a constant that reflects the loss of efficiency due to simulations. All these features suggest that the SMHD method is applicable in many fields such as finance or actuarial science where we often encounter distributions without closed-form density.


Introduction
In actuarial science and finance, we often have to fit data with a distribution that is continuous.In several instances, though the distribution does not have a closed-form density, it is not complicated to simulate from it.Such distribution can be infinitely divisible.Also, new distributions can be created by means of a mixing mechanism.
For those distributions, Luong and Bilodeau [1] have introduced simulated minimum Hellinger distance (SMHD) estimation.They have shown that the SMHD estimators are consistent in general with less regularity conditions needed than the maximum likelihood estimators and that they have the potential to be robust and have high efficiency.
It is conjectured that, asymptotically, the SMHD estimators could attain the lower bound given by the Fisher information matrix adjusted by a factor which is a constant reflecting the loss of efficiency due to simulations from the parametric models.This constant can be expressed as assumed to remain constant, where n is the original sample size of the data and U is the simulated sample size used to estimate the model density function or distribution.This factor also appears in various methods of estimation based on simulations and reflects the loss of efficiency due to the model density or distribution having to be estimated using a simulated sample drawn from the model distribution.Section 2 of the paper further discusses this factor which appears in simulated unweighted minimum chi-square method and simulated quasi-likelihood method.
In this paper, which can be viewed as a follow-up to the previous paper, we shall show that, indeed, under some regularity conditions, the SMHD estimators will follow an asymptotic normal distribution, and the asymptotic covariance matrix is given by the inverse of the Fisher information matrix adjusted by the , as conjectured in Luong and Bilodeau [1].Consequently, the SMHD estimators are fully efficient among the class of simulated estimators, just as the maximum likelihood (ML) estimators are in the classical set-up.We shall closely follow the work of Tamura and Boos [2] and restrict ourselves to the case of the univariate parametric family to establish asymptotic normality of the SMHD estimators.Under those restrictions, Tamura and Boos worked with kernel density estimates with nonrandom bandwidths and the assumption that the parametric family has a closed-form density.Under those assumptions, they can relax the requirement that the parametric family needs a compact support, as given by Beran [3] in his seminal paper.Rather, they have obtained the result that, in general, if parametric families have positive integer moments of all orders, then the minimum Hellinger distance (MHD) estimators in the univariate case will have the same efficiency as the ML estimators and the asymptotic covariance matrix can be based on the Fisher information matrix just as for the ML estimators.
We shall call version D the version based on a parametric family having a closed-form density.We extend the results to a simulated version, version S, where the parametric family requires a density estimate using a random sample drawn from the parametric family as introduced in Luong and Bilodeau [1].The results we obtain in this paper can be summarized as follows: under the same conditions required by Tamura and Boos [2], the SMHD estimators are fully ef-Open Journal of Statistics ficient among the class of simulated estimators just as the MHD estimators are fully efficient when they are based on a parametric family with a closed-form density.It only requires an adjustment by a constant which depends on τ, the ratio between the sample size U drawn from the parametric family and the original sample size n of the data.Since the constant τ can be controlled, the efficiency of SMHD estimators will be close to that of the MHD estimators of the classical version D.
Furthermore, since minimum distance estimators are in general robust, it makes SMHD estimators applicable whenever there is a need for robustness and evidence that data are contaminated.
In actuarial science and finance, there are many useful densities without closed forms with semi-heavy tails which satisfy the requirements needed for asymptotic efficiency of the SMHD estimators.For examples in actuarial science, see Klugman, Panjer and Wilmot [4], or Luong [5].For examples in finance, see Schoutens [6], or Grigoletto and Provasi [7].
To establish asymptotic normality, we shall also make use of Theorem 7.1, given by Newey and McFadden [8], that provides conditions and results for estimators obtained by minimizing or maximizing a nonsmooth objective function.
In addition, we need the concepts of continuity in probability and differentiability in probability, as they will be used in this paper to justify the asymptotic distribution of SMHD estimators.
For count data, Luong, Bilodeau and Blier-Wong [9] used a similar approach to establish asymptotic normality for the SMHD estimators in the discrete case.
The paper is organized as follows.The classical version, version D, is re-examined in section 2. We also extract the relevant results given by Tamura and Boos [2] which are subsequently needed for developing the simulated version, i.e., version S. Version S is studied in section 3.In section 3.1, we define the notions of continuity in probability and differentiability in probability.Those two notions are needed to apply the Theorem 7.1 given by Newey and McFadden [8].Section 3.2 shows that the Theorem 7.1 is applicable to SMHD estimation.Hence, though the objective function to be minimized is nonsmooth, we can establish asymptotic normality for the SMHD estimators and show that the SMHD estimators attain the lower bound within the class of simulated estimators just as, for the parametric model, the MHD or ML estimators attain the lower bound based on the Fisher information matrix when simulations are not needed because the parametric model density has a closed-form expression.

Hellinger Distance Estimation: Classical Results
In this section, we shall review some of the results already established by Tamura and Boos [2] but will focus on the univariate set-up for Hellinger distance estimation.Subsequently, building on their work, we shall establish the asymptotic normality of Hellinger distance estimators for the simulated version, i.e., asymptotic normality of the SMHD estimators, in section 3. SMHD estimators have been introduced and consistency for version S has been established in our previous paper; see Luong and Bilodeau [1].Hence, section 3 will complete the results already obtained.We shall define some notation before restating Theorem 4.1 given by Tamura and Boos [2] as Theorem 1 below.Their theorem is for the multivariate case but, when restricted to the univariate case, some simplifications can be made as the bias term in their theorem will converge to zero in probability, i.e., 0 p n nB  → , and thus can be ignored.Also, we want to use the notation that is directly related to the notions of Fisher information matrix and ML estimation.
We assume we have independent and identically distributed observations where ( ) is a kernel density estimate based on the sample with nonrandom bandwidth n h , and ω is the kernel density used to obtain MHD estimators.This is version D and we shall state the relevant results in this section.
For version S, which we will consider in section 3, the model density ( ) by Tamura and Boos [2] (pages 225-226) as Theorem 1, which is version D as given below.We shall also highlight some of the results in their proof to be used for version S in the next section together with Theorem 7.1 given by Newey and McFadden [8].The proof of Theorem 1 can be found in the proof of Theorem ( ) and θ is first-order as efficient as ˆML θ , where ˆML θ is the vector of classical ML estimators.
Here are the eight conditions to meet: 1) The kernel density ω used to construct the density estimate has a compact support W and the bandwidth n h used satisfies the property ( ) 2) The parameter space Θ is compact and 0 θ is an interior point.
3) The parameterization of the model has no problem of identification, i.e., if as n → ∞ .

5)
The ratio ( ) 7) The Fisher information matrix ( ) I θ exists and we can interchange the order of differentiation and integration so that Equation ( 5) holds.
8) The function ( ) has first partial derivatives vector s  θ and second partial derivatives matrix s  θ , and all the partial derivatives are continuous with respect to θ .Conditions 1, 2 and 3 are standard and easily satisfied.Regarding condition 4, Tamura and Boos [2] commented that it is almost equivalent to It is not difficult to see this equivalence as we often restrict our attention to and the use of Markov's type of inequality allows us to obtain this equivalence.Furthermore, Tamura and Boos [2] also commented that conditions 4 and 5 will be satisfied if X has all positive integer order moments.Despite this restriction, many useful distributions in finance fall into this category, even when the distribution has semi-heavy tails as mentioned earlier.Condition 6, imposed directly on the parametric family, is often met for parametric families encountered in practice.As for the bandwidth n h , we can choose n h such that 0 n h → and n nh → ∞ as n → ∞ , thus meeting the requirements set in con- dition 1 for the bandwidth.
We shall follow these recommendations and will show, in section 3.2, that, for version S, the SHMD estimators given by the vector ˆS θ which minimizes the objective function as given by Equation (3) will have the following asymptotic normality distribution: .

Open Journal of Statistics
The factor  also appears in other simulated methods of inference.It is used to discount the efficiency of the minimum unweighted chi-square method to obtain the efficiency of the simulated version; see Pakes and Pollard [10] (page 1069).Similarly, it is used as a discount factor for obtaining the efficiency of the simulated quasi-likelihood method by discounting the efficiency of the related quasi-likelihood method; see Smith [11] (page S69).This factor can be interpreted as a universal adjusting factor when the true distribution is replaced by an estimate using simulated samples.With this interpretation, the SMHD estimators can be viewed as estimators which attain the lower bounds among the class of estimators based on simulated techniques.Before we proceed, we would like to extract a few results given by Tamura and Boos [2] in their proof of Theorem 4.1.These results will be needed to prove asymptotic normality for version S. For version D, from the proof of their Theorem 4.1 and by taking into account the bias 0 p n nB  → for the univariate case, we have the following results using equality in distribution: ( ) where n F is the commonly used sample distribution function and F θ is the model distribution function.Now, under the commonly used assumption ( ) if interchanging the order of integration and differentiation is permissible, the last equality can be re-expressed as ( ) from which we can see easily that θ is as efficient as ˆML θ .Besides, θ is ro- bust, whereas that may not be the case for ˆML θ .We can also see that, using Equations ( 9)-( 13), we have the following equalities: ( ) Beran [3] had obtained these results earlier but using a compact support assumption for the parametric family and random bandwidths; see Equations (3.7) and (3.12) given by Beran [3] (pages 451-452).

Asymptotic Normality Distribution for SMHD Estimators
The equalities given by Equations ( 9)-( 16) will be used for establishing asymptotic normality for SMHD estimators in section 3.2.A few notions, namely the notions of continuity in probability and differentiability in probability, which extend the related notions in classical real analysis for nonrandom functions, are needed and they will be presented in section 3.1.

Some Preliminary Notions
These notions have been introduced and discussed for SMHD estimation for count data, see Luong, Bilodeau and Blier-Wong [9] (pages 201-203).They are reproduced below to make it easier to follow the results of this paper and make the paper more self-contained.

Definition 1 (Continuity in probability)
A sequence of random functions θ whenever * → θ θ .Equivalently, for any 0 ε > and . This can be viewed as a stochastic version, or an extension, of the classical definition of continuity in real analysis.
It is well known that the supremum of a continuous function on a compact domain is attained at a point in the compact domain; see Davidson and Donsig [12] (page 81) or Rudin [13] (page 89) for this classical result.The equivalent property for a random function which is only continuous in probability is that the supremum of the random function is attained at a point in the compact domain in probability.This property will be given as property 1 given below.
In order to use Theorem 7.1 of Newey and McFadden [8], we need to consider the compact domain of the form and we note that, as n → ∞ , 0 n δ → , and ( ) Property 1 The random function ( ) n g θ , which is continuous in probability and bounded in probability on a compact set Θ , will attain its supremum on a point of Θ in probability.
The justification of this property is similar to the deterministic case, which is a classical result in real analysis.For the random case, again, it suffices to pick a sequence { } j θ in Θ with the property that ( ) ( ) Beside the concept of continuity in probability, we also need the concept of differentiability in probability which is given below.

Definition 2 (Differentiability in probability)
A sequence of random functions  , where ( ) 0, 0, , 0,1, 0, , 0 with 1 appearing only in the j th entry.We also require ( ) ( ) j n v θ be continuous in probability for 1, 2, , j m =  .We can let the derivatives vector be denoted as From definition 2, we can see that differentiability in probability is a notion which parallels the classical notion of differentiability, where each partial derivative of the nonrandom function is required to be continuous.
A similar notion of differentiability in probability has been used in the stochastic processes literature; see Gusak et al. [14].A more stringent differentiability notion, namely differentiability in quadratic means, has been introduced to study the local asymptotic normality property for a parametric family; see Keener [15] (page 326).Also, see Pollard [16] for the notion of stochastic differentiability, which is also more stringent than differentiability in probability.
Below are the assumptions we need to make to establish asymptotic normality for SMHD estimators in section 3.2, and they appear to be satisfied in general.
For the simulated version, we implicitly assume that the sample size U used to draw samples from the parametric family { } ing equality in distribution, we have ( ) If the remainder of the approximation is small, we also have Before defining the remainder term ( ) n R θ , we note that the following ap- proximation Regularity conditions 1 -3 of Theorem 2 can easily be checked.Condition 4 is a consequence of the results already obtained for version D. The most difficult condition to be verified is condition 5.Because it involves technicalities, its verification will be done toward the end of this section.
Here, assuming all conditions can be validated, we apply Theorem 2 for SMHD estimation with ˆS =  θ θ .
Clearly, the objective function Using assumptions 1 -4 and by performing limit operations as for finding derivatives in real analysis, we can conclude

( )
S n nQ θ is differentiable in prob- ability with derivatives vector ( ) where 0 n is a positive integer, by invoking the Dominated Convergence Theorem if necessary.
We have ( ) , and, using Equations ( 24)-(26), we can conclude ( ) ( ) From the results given by Equations ( 27) and ( 28), asymptotic properties suggest that the SMHD estimators will have high efficiency in large samples as the lower bound for simulated estimators is attained.We should also keep 10 { } Hence, the use of Theorem 2 is justified for the SMHD estimators.Moreover, SMHD estimators are robust as they are obtained by minimizing a distance; see Donoho and Liu [17] and Lindsay [18].

Conclusion
Asymptotic properties established in this paper suggest that SMHD estimators are very efficient for large samples for parametric models where all the positive integer moment exists.For the subset of such parametric models that have no closed-form densities, as often are encountered in finance and actuarial science, SMHD estimators appear to be very suitable for large samples based on asymptotic normality results obtained.For any parametric family failing to have finite moments of all positive integer orders, SMHD estimators remain consistent and robust, but large-scale simulation studies seem to be necessary to study the efficiency of the estimators for the specific parametric model being considered.
original sample given by the data.SMHD estimators are obtained by minimizing

4. 1 by
Tamura and Boos [2].Theorem 1 If we can find a sequence of positive numbers { } n α with n α → ∞ as n → ∞ , then, provided the following conditions 1 -8 are met, the MHD estima- tors θ obtained by minimizing Equation (1) have an asymptotic normal dis- tribution and attain the Cramer-Rao lower bound based on the Fisher information matrix, i.e.,

f
θ is proportional to the sample size n, i.e., the loss of efficiency due to simulations and the same seed should be used to generate simulated samples across different values of θ .To assess the performance of the SMHD estimators in finite samples, we need simulation studies which are based on the parametric family being considered as asymptotic theory, despite being quite general, might not be applicable for finite samples, especially with sample size 100 n