Suppose that X t is an infinitely divisible nonnegative discrete random variable such that the Laplace transform can be expressed as ( κ ( s ) ) t , t ≥ 0 , where κ ( s ) is the Laplace transform of X, which is nonnegative and infinitely divisible as well. The power mixture (PM) with mixing distribution function H ( y ) and Laplace transform κ H ( s ) of a nonnegative random variable Y is defined as the Laplace transform

η ( s ) = PM ( κ , H ) = ∫ 0 ∞ ( κ ( s ) ) t d H ( t ) = κ H ( − log ( κ ( s ) ) ) .

Furthermore, if H ( y ) is infinitely divisible, then the distribution with Laplace transform η ( s ) is also infinitely divisible. The random variable Y ≥ 0 with distribution H ( y ) can be discrete or continuous but needs to be ID. This is the PM method for creating new parametric families, i.e., using the PM operator. The PM method can be viewed as a form of continuous compounding method. The ID property can be dropped but as a result the new distribution created using the PM operator needs not be ID. For the traditional compounding methods, see Klugman et al. [4] (p141-148). Abate and Whitt [2] also mentioned other methods.

Example 2 (Generalized negative binomial) The generalized negative binomial (GNB) distribution introduced by Gerber [5] can be viewed as a power variance function distribution mixture of a Poisson distribution. The power variance function distribution introduced by Hougaard [6] is obtained by tilting the positive stable distribution using a parameter θ . It is a three-parameter continuous nonnegative distribution with Laplace transform given by

κ H ( s ) = exp { − λ ( θ + s ) α − θ α } ,   λ , θ > 0 ,   0 < α < 1 .

Gerber [5] used a different parameterization and named this distribution generalized gamma. It is also called positive tempered stable distribution in finance.

Let κ ( s ) = e ( e − s − 1 ) be the Laplace transform of a Poisson distribution with rate μ = 1 . The Laplace transform of the GNB distribution can be represented as

η ( s ) = exp ( − λ ( θ − e − s + 1 ) α − θ α ) .

The corresponding pgf can be expressed as

P ( s ) = exp ( − λ ( θ − s + 1 ) α − θ α ) .

The pgf is given by expression (21) in the paper by Gerber [5] . The GNB distribution is infinitely divisible. If stochastic processes are used instead of distributions, the distribution can also be derived from a stochastic process point of view by considering a Poisson process subordinated to a generalized gamma process and obtain the new distribution as the distribution of increments of the new process created. See section 6 of Abate and Whitt [2] (p92-93). See Zhu and Joe [7] for other distributions which are related to the GNB distribution.

Note that, if H ( y ) is discrete, η ( s ) is the Laplace transform of a random variable expressible as a random sum. A random sum is also called stopped sum in the literature, see chapter 9 by Johnson et al. [8] (p343-403). The Neymann-Type A distribution given below is an example of a distribution of a random sum.

Example 3 Let X = ∑ i = 1 Y U i ,the U i ’s conditioning on Y are independent and identically distributed and follows a Poisson distribution with rate ф and Y is distributed with a Poisson distribution with rate λ . Using the Power mixture operator we conclude that the LT for X is

η ( s ) = exp ( λ ( e ф ( e − s − 1 ) − 1 ) ) ,

and the pgf is

P ( s ) = exp ( λ ( e ф ( s − 1 ) − 1 ) ) .

Properties and applications of the Neymann type A distribution have been studied by Johnson et al. [8] (p368-378). The mean and variance of X are given respectively by E ( X ) = λ ф and V ( X ) = λ ф ( 1 + ф ) . From these expressions, moment estimators (MM) have closed form expressions, see section (4.1) for comparisons between MM estimators and SMHD estimators in a numerical study. For applications often the parameter λ is smaller than the parameter ф .

By tilting the density function using the Esscher transform, the Esscher transform operator can be defined and, provided the tilting parameter τ introduced is identifiable, new distributions can be created from existing ones.

Let X be the original random variable with Laplace transform κ ( s ) . The Esscher transform operator which can be viewed as a tilting operator is defined as

η ( s ) = Esscher ( κ , τ ) = κ ( s + τ ) κ ( τ ) .

Let κ ( s ) be the Laplace transform of a positive continuous random variable X. The Laplace transform of Y = X − τ , Y ≥ τ ≥ 0 is given by e τ κ ( s ) . So, we can define the shift operator as

η ( s ) = Shift ( κ , τ ) = e τ κ ( s ) .

In some cases, even the pmf of Y has a closed form but the maximum likelihood (ML) estimators might be attained at the boundaries, the ML estimators might not have the regular optimum properties.

Note that parallel to the closed form pgf expressions for these new discrete distributions, it is often simple to simulate from the new distributions if we can simulate from the original distribution before the operators are applied. For example, let us consider the new distribution obtained by using the Esscher operator. It suffices to simulate from the distribution before applying the operator and apply the acceptance-rejection method to obtain a sample from the Esscher transformed distribution. The situation is similar for new distributions created by the PM operator. If we can simulate one observation from the mixing distribution of Y which gives a realized value t and if it is not difficult to draw one observation from the distribution with LT κ ( s ) t then combining these two steps, we would be able to obtain one observation from the new distribution created by the PM operator. Consequently, simulated methods of inferences offer alternative methods to inferences methods based on matching selected points of the empirical pgf with its model counterpart or other related methods, see Doray et al. [9] for regression methods using selected points of the pgfs. For these methods there is some arbitrariness on the choice of points which make it difficult to apply. The techniques of using a continuum number of points to match are more involved numerically, see Carrasco and Florens [10] . The new methods also avoid the arbitrariness of the choice of points which is needed for the regression methods and the k-L procedures as proposed by Feurverger and McDunnough [11] if characteristic functions are used instead of probability generating functions and they are more robust than methods based on matching moments (MM) in general. We can reach the same conclusions for another class of distributions namely mixture distributions created by other mixing mechanisms, see Klugman et al. [4] , Nadarajah and Kotz [12] , Nadarajah and Kotz [13] . These distributions might not display closed form pmf or the pmf are only expressible only using special functions such as the confluent hypergeometric functions. For these models, likelihood methods might also be difficult to implement.

This leads us to look for alternative methods such as the simulated minimum Hellinger distance (SMHD) methods for count data. We shall consider grouped count data and ungrouped count data. With grouped data, it leads to simulated chi-square type statistics which can be used for model testing for discrete or continuous models. These statistics are similar to the traditional Pearson statistics. For model testing with continuous distributions, continuous observations when grouped into intervals are reduced to count data and we do not need to integrate the model density functions on intervals using SMHD methods, it suffices to simulate from the continuous model and construct sample distribution functions to obtain estimate interval probabilities. Therefore, the scopes of applications of simulated methods are widened due to these features.

We briefly describe the classical minimum Hellinger distance methods introduced by Simpson [14] , Simpson [15] for estimation for count data in the next section and we shall develop inference methods based on a simulated version of this HD distance following Pakes and Pollard [16] (1989), who have developed an elegant asymptotic theory for estimators obtained by minimizing a simulated objective function expressible as the Euclidean norm of a random vector of functions. As an example, they have shown that the simulated minimum chi-square estimators without weight satisfy the regularity conditions for being consistent and have an asymptotic normal distribution, see Pakes and Pollard [16] (p1048). They work with properties of some special classes of sets to check the regularity conditions of their Theorem 3.3. Meanwhile, Newey and McFadden [17] (p2187) work with properties of random functions and introduce a stochastic version of the classical equicontinuity property of real analysis. In this paper, we shall also extend the notion of continuity of real analysis to a version which only holds in probability for random functions which we call continuity in probability for a sequence of random functions which is similar to the notion of continuity with probability one as discussed by Newey and McFadden [17] (p2132) in their Theorem 2.6. We also use the property of the compact domains under considerations shrink as the sample size n → ∞ to verify conditions of Theorem 3.3 given by Pakes and Pollard [16] (1989) for SMHD methods using grouped data and conditions of Theorem 7.1 of Newey and McFadden [17] (p2185) for ungrouped data. This approach appears to be new and simpler that other approaches which have been used in the literature to establish asymptotic normality for estimators using simulations; previous approaches are very general but they are also more complicated to apply. A similar notion of continuity in probability has been introduced in the literature of stochastic processes.

It is worth to mention that simulated methods of inferences are relatively recent. In advanced econometrics textbook such as the book by Davidson and McKinnon [18] , only section 9.6 is devoted to simulated methods of inferences where the authors mention simulated methods of moments (MSM). The simulated version for HD methods will be referred to as version S and the original version which is deterministic will be referred to as version D in this paper. We briefly review the Hellinger distance and chi-square distance below and subsequently develop simulated inference methods for grouped and ungrouped count data using HD distance.

We define MHD estimators as given by the vector for version D and for version S but emphasize version S as version D has been studied by Simpson [14] . Both versions can be treated in a unified way using the following Theorem 1 for consistency which is essentially Theorem 3.1 of Pakes and Pollard [16] (p1038) and the proof has been given by the authors.

Theorem 1 (Consistency)

Under the following conditions converges in probability to:

a), the parameter space Ω is compact

b),

c) for each.

Theorem 3.1 states condition b) as but in the proof the authors just use so we state condition b) as as it is easier to use this condition when there is a need to extend to the infinite dimensional case with the space.

An expression is if it converges to 0 in probability and if it is bounded in probability. In version D and version S for Hellinger distance we have occurs at the values of the vector values of the HD estimators, so the conditions a) and b) are satisfied for both versions and compactness of the parameter space Ω is assumed. Also, for both versions only at and otherwise, this implies that there exist real numbers u and v with such that

Therefore, for both versions of whether deterministic or simulated, the minimum Hellinger distance estimators (MHD) are consistent. Theorem 3.1 of Pakes and Pollard [16] is an elegant theorem, its proof is also concise using the norm concept of functional analysis and it allows many results to be unified. Essentially, the same theorem remains valid with the use of the Hilbert space and its norm instead of the Euclidean space and the Euclidean norm. By using and its norm the consistency for the ungrouped SMHD estimators can also be established but further asymptotic results for the ungrouped SMHD estimators will be postponed and given in Section 3.

Asymptotic normality is more complicated in general. For the grouped case, Theorem 3.3 given by Pakes and Pollard [16] (p1040) can be used to establish asymptotic normality for both versions of Hellinger distance estimators. We shall rearrange results of Theorem 3.3 under Theorem 2 and Corollary 1 given in the next section to make it easier to apply for HD estimation using both versions.

Since the proofs have been given by the authors, we only discuss here the ideas of their proofs to make it easier to follow the results of Theorem 2 and Corollary 1 in Section (2.2.2).

For both versions, but is

not differentiable for version S, the traditional Taylor expansion argument cannot be used to establish asymptotic normality of estimators obtained by minimizing. If we assume is differentiable with derivative matrix, then we can define the random function to approximate with

is based on expression (8) for version D and it is based on expressions (9-10) for version S. Note that is differentiable for both versions.

Let and be the vectors which minimize and respectively. If the approximation is of the right order then and are asymptotically equivalent. This set up will allow a unified approach for establishing asymptotic for MHD estimation for both versions. For version D, it suffices to let and for version S, let.

Under these circumstances, it suffices to work with and for asymptotic properties of and. A regularity condition for the approximation is of the right order which implies the condition (iii) given by their Theorem 3.3, which is the most difficult to check is given as

This condition is used to formulate Theorem 2 below and is slightly more stringent than the condition iii) of their Theorem 3.3 but it is less technical and sufifcient for SMHD estimation. Clearly, for SMHD estimation is as given by expression (9) or expression (10). For simulated unweighted simulated minimum chi-square estimation for this condition to hold, independent samples for each cannot be used, see Pakes and Pollard [16] (p1048). Otherwise, only consistency can be guaranteed for estimators using version S. 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 U. We implicitly make these assumptions for SMHD methods. These two assumption are standard for simulated methods of inferences, see section 9.6 for method of simulated moments(MSM) given by Davidson and McKinnon [19] (p383-394). For numerical optimization to find the minimum of the objective function, we rely on direct search simplex methods which are derivative free and the R package already has prewritten functions to implement direct search methods.

In this section, we shall state a Theorem namely Theorem 2 which is essentially Theorem 3.3 by Pakes and Pollard [16] (p1040-1043) with the condition (4) of Theorem 2 given by expression (9) replacing their condition (iii) in their Theorem 3.3, the condition (4) implies the condition (iii) by being more stringent. We also comment on the conditions needed to verify asymptotic normality for the HD estimators based on Theorem 2.

Theorem 2

Let be a vector of consistent estimators for, the unique vector which satisfies.

Under the following conditions:

1) The parameter space Ω is compact, is an interior point of Ω.

2).

3) is differentiable at with a derivative matrix of full rank.

4) for every sequence of positive numbers which converge to zero.

5).

6) is an interior point of the parameter space Ω.

Then, we have the following representation which will give the asymptotic distribution of in Corollary 1, i.e.,

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 [16] (p1040-1043). For expression (13) or expression (14) to hold, in general only condition 5) of Theorem 2 is needed and there is no need to assume that has an asymptotic distribution. From the results of Theorem 2, it is easy to see that we can obtain the main result of the following Corollary 1 which gives the asymptotic covariance matrix for the HD estimators for both versions.

Corollary 1.

Let, if then with

,

The matrices and depend on we also adopt the notations,.

We observe that condition 4) of Theorem 2 when applies to Hellinger distance or in general involve technicalities. The condition 4) holds for version D, we only need to verify for version S. Note that to verify the condition 4, it is equivalent to verify

and for version S of Hellinger distance estimation, let

and for the grouped case, it is given by

We need to verify that we have the sequence of functions converge uniformly to 0 in probability as and or equivalently, as and.

Note that

We shall outline the approach by first defining the notion of continuity in probability and let which is a compact set. The compactness of this set simplifies proofs and does not appear to be used in previous approaches in the literature. Observe that, it is easy to see that

as. Subsequently we establish being continuous in probability for and using the property that is continuous in probability is attained at a point which belongs to the compact set in probability. This is similar to the property of nonrandom continuous function in real analysis.

Now as which implies and by continuity in probability. Therefore, which means that converges uniformly in probability as. The technical

details of these arguments are given in technical appendices TA1.1 and TA1.2 at the end of the paper, in the section of Appendices.

The notion of continuity in probability has been used in a similar context in the literature of stochastic processes, see Gusalk et al. [25] and will be introduced in the next paragraph and we also make a few assumptions which are summarized by Assumption 1 and Assumption 2 given below along with the notion of continuity in probability. A related continuity notion namely the notion of continuity with probability one has been mentioned by Newey and McFadden [18] in their Theorem 2.6 as mentioned earlier. They also commented that this notion can be used for establishing asymptotic properties of simulated estimators introduced by Pakes [26] . Pakes [26] also has used pseudo random numbers to estimate probability frequencies for some models. For SMHD estimation, we extend a standard result of analysis which states that a continuous function attains its supremum on a compact set to a version which holds in probability. This approach seems to be new and simpler than the use of the more general stochastic equicontinuity condition given by section 2.2 in Newey and McFadden [18] (p2136-2138) to establish uniform convergence of a sequence of random functions in probability. Our approach uses the fact that as the set shrinks to, a property which did not seem to have been

used previously by other approaches to establish as

and. Subsequently, we define the notion of continuity in probability which is similar to the one used in stochastic processes, see Gusak et al. [25] (p33) for a related notion of continuity in probability for stochastic processes.

Definition 1 (Continuity in probability)

A sequence of random functions is continuous in probability at if whenever. Equivalently, for any, there exists a and such that for, whenever. This can be viewed as an extension of the classical

result of continuity in real analysis. It is also well known that the supremum of a continuous function on a compact domain is attained at a point of the compact domain, see Davidson and Donsig [27] (p81) or Rudin [28] (p89) for this classical result. The equivalent property for a random function which is only continuous in probability is the supremum of the random function is attained at a point of the compact domain in probability. The compact domain we study here

is given by and as,. It

might be more precise to use the term sequence of random functions rather than just random function here for the notion of continuity in probability as the random function will depend on n.

Below are the assumptions we need to make to establish asymptotic normality for SMHD estimators and they appear to be reasonable.

Assumption 1

1) The pmf of the parametric model has the continuity property with whenever.

2) The simulated counterpart has the continuity in probability property with whenever. Convergence in probability is denote by.

3) is differentiable with respect to.

In general, the condition 2) will be satisfied if the condition 1) holds and implicitly we assume the same seed is used for obtaining the simulated samples across different values of. For ungrouped data, we also need the notion of differentiability in probability to facilitate the application of Theorem 7.1 given by Newey and McFadden (1994, p2185-2186). Before stating their Theorem 7.1, Newey and McFadden has mentioned the notion of approximate derivative for the use of their Theorem, the definition given below will make it clearer.

Definition 2 (Differentiability in probability)

The sequence of random functions is differentiable with respect to at in probability if, exists and with 1 occurring at the ith entry. Furthermore, the vector is continuous and bounded in probability for all for some. This concept is similar to the notion of

differentiability in real analysis for nonrandom function.

A similar notion of differentiability in probability has been used in stochastic processes literature, see Gusak et al. [25] (p33-34), a more stringent differentiability notion namely differentiability in quadratic mean has also been used to study local asymptotic normality (LAN) property for a parametric family, see Keener [29] (p326). The notion of differentiability in probability will be used in section 3 with Theorem 7.1 of Newey and McFadden [17] to establish asymptotic normality for the SMHD estimators for the ungrouped case. We make the

following assumption for where can be viewed as a proxy model for,

Assumption 2

with the same seed being used across different values of is differentiable in probability with the same derivative vector as where the derivative vector for is

.

This assumption appears to be reasonable, this can be checked by using limit operations as in real analysis with and is continuous in probability.

Since regularity conditions for Theorem 2 and its corollary can be met and they are justified in TA1.1 and TA1.2 in the Appendices, we proceed here to find the asymptotic covariance matrix.

Since for version D is based on expression (8) and for version S is based on expressions (9-10), the asymptotic covariance matrix of version S is just the asymptotic covariance matrix of of version D multiplied by as the simulated sample from is independent

from the sample given by the data, so we can focus on version D and make the adjustment for version S. We need the asymptotic covariance matrix of the

vector first then we can find the matrix and we let for version D and for version S, we shall let.

Recall that form properties of the multinomial distribution, the covariances of and are

and

The covariance matrix of using matrix notations can be expressed as

is a diagonal matrix with diagonal elements and the vector is the transpose of and is the identity matrix of dimension with. Using the delta method the asymptotic covariance matrix of of version D is simply the asymptotic covariance matrix of which is given by

and the asymptotic covariance matrix of, version S is

We then have the vector of HD estimators version D and S given respectively by and with asymptotic distributions given by

,

is the model Fisher information matrix using grouped data as due to using. Let,

,

so

with

.

Therefore for version S,

the simulated sample size is.

Note that for version D, the HD estimators are as efficient as the minimum chi-square estimators or ML estimators based on grouped data. The overall asymptotic relative efficiency (ARE) between version D and S for HD estimation is

simply ARE = and we recommend to set to minimize the loss of efficiency due to simulations.

An estimate for the covariance matrix

The asymptotic covariance matrix of can be estimated if we can estimate. Using a result given by Pakes and Pollard (1989, p1043), an estimate for is the matrix

with 1 occurring at the ith entry of the vector and, and in general we can let. Note that the columns of estimate the corresponding partial derivatives given by the columns of .

For ungrouped data and for version D, it is equivalent to choose with unit length and let. If we choose and let and note that and the is Fisher information matrix for ungrouped data with elements given by

and. We can foresee that the HD estimators are as efficient as ML estimators for version D, a result which is already obtained by Simpson [14] . We postpone till section (3) for a more rigorous approach to justify the related result for version S using Theorem 7.1 given by Newey and McFadden [17] . The SMHD estimators given by for ungrouped data will be shown to have the property

.

Section 3 may be skipped for practitioners if their main interests are only on applications of the results.

In this section, the Hellinger distance is used to construct goodness of fit test statistics for the simple hypothesis

H₀: data comes from a specified distribution with distribution, can be the distribution of a discrete or continuous distribution. The chi-square test statistics and their asymptotic distributions are given below with

The version S is of interest since it allows testing goodness of fit for discrete or continuous distribution without closed form pmfs or density functions, all we need is to be able to simulate from the specified distribution. We shall justify the asymptotic chi-square distributions given by expression (23) and expression (24) below.

Note that

and

for version D. For version S,

.

Using standard results for distribution of quadratic forms and the property of the operator trace of a matrix with , see Luong and Thompson [30] (p247); we have the asymptotic chi-square distributions as given by expression (23) and expression (24). On how to choose the intervals, the problem is rather complex as it depends on the type of alternatives we would like to detect. We can also follow the recommendations of the Pearson’s statistics, see Greenwood and Nikulin [31] ; also see Lehmann [32] (p341) for more discussions and references on this issue.

Just as the chi-square distance, the Hellinger distance can also be used for construction of the test statistics for the composite hypothesis,

H₀: data comes from a parametric model, can be a discrete or continuous parametric model. The chi-square test statistics are given by

for version D and for version S,

where and are the vector of HD estimators which minimize version D and version S respectively and assuming. To justify these asymptotic chi-square distributions, note that we have for version D,

. It suffices to consider the asymptotic distribution of as we have the following equalities in distribution,

, as given by expression (11). Also, using expression (11) and expression (13),

which can be reexpressed as

or

With

,

is based on expression (8) for version D. Consequently,

by noting

,

using and the matrix is of rank with the rank of the matrix is also equal to its trace. The argument used is very similar to the one used for the Pearson’s statistics, see Luong and Thompson [30] (p249).

For version S,

and

with

,

is based on expressions (9-10) for version S. This justifies the asymptotic chi-square distribution for version S as given by expression (25) and expression (26). This version is useful for model testing for nonnegative continuous models without closed form expression densities, see Luong [33] for some positive infinitely divisible distributions without closed form densities used in actuarial sciences. It is also suitable for testing models with support on the real line used in finance such as the jump diffusion model as given by Tsay [34] (p311-319), for example. All we need is to be able to simulate from the model.

As an example for illustration we choose the Neymann Type A distribution with the method of moments (MM) estimators for and which have been given

by Johnson et al. [8] . The MM estimators are given by and

with the sample mean and variance given respectively by and. The MM estimators are classical moment estimators. We perform a limited simulation study to compare the performance of the SMHD estimator which is given by

vs the MM estimators given by.

For the range of parameter values, we let,

are used in the study. For applications often the parameter for the mixing distribution much smaller than the parameter. The SMHD estimators seem to perform much better than the MM estimators, in general. The results are displayed in Table A. The criterion for overall relative

efficiency used is the ratio with denotes

the mean square error of the estimator inside the parenthesis. The mean square error of an estimator for is defined as

.

The ratio ARE can be estimated using simulated data and they are displayed in Table A. Due to limited computing facilities, we only draw samples of size and the simulated sample is fixed at, and the results are summarized using Table A. It takes around one minute using a laptop computer for obtaining the SMHD estimators for one simulated sample. The MM estimators appear to perform reasonably well for some samples but display erratic results for some other samples which account for the loss of efficiency of the MM estimators. Also, the parameter is not well estimated by the moment method but it gives reasonably good estimates for the parameter in general. The MM estimators are based on the sample mean and variance and these statistics are known to be nonrobust. If outliers are present, the MM estimators again might become erratic. The mean square errors (MSE) for estimating the parameters and the corresponding ratios ARE are estimated using the simulated samples and the AREs are displayed in Table A.

For the Poisson model with parameter we compare the performance of the MLE for which is the sample mean vs the SMHD estimator

using the ratio for. For the Poisson

model, the information matrix exists and we can check the efficiency and robustness of the SHD estimator and compare it with the ML estimator which is the sample mean. Since there is only on parameter estimate we are able to fix

Asymptotic relative efficiency between MLE and SHD, for the Poisson model with parameter.

Asymptotic relative efficiency between MLE and SHD for the Poisson model when 10% of data coming from the discrete positive distribution with parameter and, i.e.,.

U = 10000 for the simulated sample size from the Poisson model without slowing down the computations. It appears overall the SHD estimators performs very well for the range of parameters often encountered in actuarial studies, here we observe that the asymptotic efficiencies range from 0.7 to 1.1. We also study a contaminated Poisson model () with observations coming from the Poisson model () and of observations coming from a discrete positive stable (DPS) distribution with the parameter for and has the same value of the Poisson model. We compare the performance of the sample mean for which is the ML estimator vs the SMHD estimator using the contaminated model Poisson model as described and estimate the

for to compare the robustness of

the SMHD estimator vs ML estimator in presence of contamination. The sample mean looses its efficiency and becomes very biased. The results are given at the bottom of Table A which shows that the performs much better than the sample mean which is the ML estimator. For drawing simulated samples from the DPS distribution, the algorithm given by Devroye [37] is used.