The Laplace transform power mixture operator (LTPM) has been introduced in the literature for creating univariate non-negative distribution and can be viewed as a continuous type of compounding operator and it makes use of LT of a distribution and the LT of a mixing random variable to create a new distribution specified by its LT. It is due to Abate and Whitt [4] (pp. 92-93). They use the acronym PM but to emphasize the explicitly use of LTs with this operator we shall use the acronym LTPM and to distinguish it with a similar power mixture operator introduced by Marshall and Olkin [6] (pp. 834-836) which make explicitly use of distributions instead of LTs. Abate and Whitt [4] also point out that a new ID can be obtained if we start with an ID distribution and the mixing distribution is non-negative continuous and ID. They also give an elegant and insightful interpretation when instead of ID distribution we can consider the corresponding Lévy process behind. The new ID distribution is related to a new Lévy process constructed from a Lévy process subject to a change time process induced by the mixing random variable and since it is ID, the change time process is also a Lévy process. To summarize, we can say that starting with a Lévy process and subject this process to another non-negative Lévy process used as a change of time process then a new Lévy process can be created or equivalently a new ID non-negative distribution is created which is the distribution of the increments of the newly created Lévy process. Observe that the LTPM operator can be used to generate new distributions in general and with further conditions imposed on the procedures, the new distribution can be ID. Consequently, it appears to be natural to generalize this operator to a bivariate version to create bivariate distributions using two univariate LTs and the LT of a mixing random variable, this is the bivariate power mixture operator (BLTPM). Clearly, it is natural extension of the LTPM operator which is used to create univariate distribution. This operator is similar to another power mixture operator which makes use of distributions or survival functions instead of LTs given by Marshall and Olkin [6] (p. 835), we shall use the acronym BDSPM for this operator.

For illustrations, some examples of new bivariate distribution specified by BLT and created using the BLTPM operator will be given for illustrations in section 3.2. The BLTPM operator can also be used to create new distribution with one marginal distribution which is discrete and the other marginal being continuous. These types of bivariate distributions appear to be useful for actuarial science and possibly for other fields as well.

Often, we need to know whether a function can be considered as LT of some univariate random variable, the notion of completely monotonicity of a function is useful for characterizing a function to be a proper LT of a random variable and can be found in Feller [11] (pp. 439-440) but it is reproduced here for completeness as Definition and Theorem below.

Definition

A function ф ( s ) with the domain given by s ≥ 0 is completely monotone (CM) if it possesses derivatives ф ( n ) of all order and ( − 1 ) n ф ( n ) ( s ) ≥ 0 .

Theorem

A function ф ( s ) with the domain given by s ≥ 0 is the LT of a random variable if and only if it is CM.

Now if we assume that the LT of a random variable is f ^ ( s ) so that f ^ α ( s ) is a proper LT, also suppose that the LT of another random variable Y is g ^ ( t ) so that g ^ α ( t ) is a proper LT and α is a non-negative random variable with distribution function H and LT h ^ ( u ) . Furthermore, we assume that conditioning on a realized value α, the LT f ^ α ( s ) which is the LT of a random variable denoted by X ( α ) and the LT g ^ α ( t ) which is the LT of a random variable denoted by Y ( α ) are independent but when integrate out with respect to the distribution of α will give a new bivariate LT for a vector X = ( Y , Z ) ′ ,

L X ( s , t ) = ∫ 0 ∞ f ^ α ( s ) ⋅ g ^ α ( t ) d H ( α )     . (1)

Using the LT of α, it can be re-expressed as

L X ( s , t ) = h ^ ( − log ( f ^ ( s ) ) − log ( g ^ ( t ) ) ) . (2)

Expression (2) gives a definition of the BLTPM operator and it generalizes expression (6.1) given by Abate and Whitt [4] (p. 92) for the univariate LTPM operator. Similarly, if MGFs are used instead and they are given respectively by f M ^ ( s ) , g M ^ ( t ) and h M ^ ( u ) , a new bivariate distribution with BMGF can be created similarly.

M X ( s , t ) = h M ^ ( log ( f M ^ ( s ) ) + log ( g M ^ ( t ) ) ) . (3)

Observe that if X ( α ) , Y ( α ) are ID and α ≥ 0 is continuous and ID then the new bivariate distribution created is also ID. We can interpret this property using Lévy processes, as two independent Lévy processes subject to the same time change Lévy process is a bivariate Lévy process. This generalizes the argument used for the univariate case given by Abate and Whitt [4] (p. 92).

A form of bivariate PM operator, the BSDPM has been used in the literature but with the use of distribution functions or survival functions, see the seminal works of Marshall and Olkin [6] and a class of survival or distribution copula functions are also obtained. This class is defined using a generator which is based on the LT of some nonnegative random variable which forms a subclass of the Archimedean class and will be further discussed in section 3.2. Due to the analogy between the BLTPM operator and BDSPM operator, these distribution or survival copula functions subject to some conditions are also LT copulas. For the Archimedean class, see Klugman et al. [17] (pp. 187-213).

In section 3.2, we shall see that additional requirements are needed when working with LT’s instead of distribution functions or survival functions for the use of these copulas traditionally used to create new bivariate distribution function using prescribed marginal distributions.

This is due to the conditions e − α ф − 1 ( L 1 ( s ) ) and e − α ф − 1 ( L 2 ( t ) ) , α > 0 are proper LT are more difficult to verify than the corresponding conditions, e − α ф − 1 ( F 1 ( x ) ) and e − α ф − 1 ( F 2 ( y ) ) are proper distribution functions with F 1 ( x ) and F 2 ( y ) being distribution functions when the BDSPM operator is used, see Marshall and Olkin [6] (p. 834).

We shall give a few examples below to illustrate the use of the BLTPM operator to create bivariate ID distribution using the LT of a gamma mixing random variable. The same procedure can be used with other mixing random variables such as the inverse Gaussian random variable or the exponential mixture inverse Gaussian (EMIG) random variable. The EMIG distribution as studied by Abate and Whitt [4] (pp. 95-96) is also ID, continuous and nonnegative. This distribution has not received much attention in the statistical literature despite it is used frequently in queueing theory.

Example 3

For modeling two type of claims in insurance for one period of time, we might want to construct a joint continuous model using the BLTPM operator with the mixing random variable following a gamma distribution with LT h ^ ( u ) = ( 1 + u ) − τ , τ > 0 , f ^ ( s ) and g ^ ( t ) are LT of inverse Gaussian distributions with LTs given respectively by

f ^ ( s ) = exp ( θ 1 μ 1 ( 1 − 1 + 2 μ 1 2 s θ 1 ) ) and g ^ ( t ) = exp ( θ 2 μ 2 ( 1 − 1 + 2 μ 2 2 t θ 2 ) ) ,

θ i , μ i > 0 , i = 1 , 2 see Klugman et al. [18] (p. 472) for LT of an inverse Gaussian distribution.

Apply the BLTBPM operator using these LTs, this will give a new bivariate continuous distribution for a random vector X = ( Y , Z ) ′ which is ID and nonnegative with its BLT given by

L X ( s , t ) = ( 1 − θ 1 μ 1 ( 1 − 1 + 2 μ 1 2 s θ 1 ) − θ 2 μ 2 ( 1 − 1 + 2 μ 2 2 t θ 2 ) ) − τ .

From L X ( s , t ) , the Pearson product moment correlation coefficient can be obtained and the coefficient can be used to study the dependence between Y and Z. Other depence measures can also be used to study associativity and dependence, see Chapter 4 of the book by Balakrishnan and Lai [19] (pp. 141-173) and for total positivity dependence, see Barlow and Proschan [20] (pp. 142-150) but we do not go into details into these dependence studies for distributions created using the BLTPM operator as the main focus of the paper are on statistical inference methods using BLTs or BMGFs.

Example 4

Bivariate mixed models with one marginal for counts and the other one for claim amounts appear to be useful in actuarial science. These models can also be constructed using the BLTPM operator. For example, let the random vector X = ( Y , Z ) ′ with Y being discrete and Z being continuous and nonnegative. We might specify h ^ ( u ) = ( 1 + u ) − τ , τ > 0 , f ^ ( s ) = exp ( λ ( e − s − 1 ) ) which is the LT of a Poisson random variable with λ > 0 , let and

g ^ ( t ) = exp ( θ μ ( 1 − 1 + 2 μ 2 t θ ) ) , θ and μ > 0 which is the LT of an inverse Gaussian distribution.

The BLT of the random vector X = ( Y , Z ) ′ created using the BLTPM operator is

L X ( s , t ) = ( 1 − λ ( e − s − 1 ) − θ μ ( 1 − 1 + 2 μ 2 t θ ) ) − τ .

Setting t = 0 we obtain the marginal LT of Y which is given by

L Y ( s ) = ( 1 − λ ( e − s − 1 ) ) − τ . One can recognize that this is a LT of a negative binomial distribution and for this distribtion, the corresponding probability

generating function (PGF) P Y ( s ) = ( 1 − λ ( s − 1 ) ) − τ , see Klugman et al. [17] (p. 83) for the PGF of a negative binomial distribution.

The marginal L Z ( t ) = ( 1 − θ μ ( 1 − 1 + 2 μ 2 t θ ) ) − τ is the LT of a continuous random variable.

Example 5

In this example we shall obtain a bivariate negative binomial distribution using

the BLTPM operator. Let h ^ ( u ) = ( 1 + u ) − τ , τ > 0 , f ^ ( s ) = exp ( λ 1 ( e − s − 1 ) ) and

g ^ ( t ) = exp ( λ 2 ( e − s − 1 ) ) , λ 1 and λ 2 > 0 . Apply the BLTPM operator this gives a

joint distribution for X = ( Y , Z ) ′ with BLT given by

L X ( s , t ) = ( 1 − λ 1 ( e − s − 1 ) − λ 2 ( e − s − 1 ) ) − τ .

This is a BLT of a bivariate negative binomial distribution which is also ID. In the literature, this distribution has been constructed using various methods and by many authors, see Mardia [21] (p. 84). We can see that the BLTPM operator is very useful for constructing bivariate distribution with the property being ID. This property is useful in finance and actuarial sciences as the corresponding bivariate process is a bivariate Lévy process with stationary and independent increments. This property often leads to nice results in risk theory and pricing formulas in finance. Simulation of sample paths from these processes are also simplified using this property.

We briefly mention on how to simulate from bivariate distribution with a specified L X ( s , t ) created by the BLTPM operator. The procedures are similar to the procedures described in section 5 and given by Marshall and Olkin [6] (p. 840) but the inverse transform method might not be necessary explicitly.

For the inverse transform method, see Ross [22] (pp. 69-70).

The main steps are:

1) Generate an observation for the mixing random variable α from the distribution H which has LT h ^ ( u ) .

2) Use the observed value α, generate an observation X ( α ) from a distribution with LT f ^ α ( s ) . Often from the expression of f ^ α ( s ) , we have a procedure to simulate from this distribution and the inverse method might not be needed explicitly at this step.

3) Use the same observed value α, simulate another observation Y ( α ) which is independent of X ( α ) obtained from 2) from a distribution with LT g ^ α ( t ) .

The pair of observations ( X ( α ) , Y ( α ) ) ′ obtained will follow a bivariate distribution with BLT as specified by L X ( s , t ) .

Observe that the BLTPM operator given by expression (2) is similar to the BDSPM operator given by expression 2.2 in Marshall and Olkin [6] (p834) where a class of distribution or survival copulas can be obtained, it is considered below. Without loss of generality we consider distribution copulas which are related to the BDSPM operator. By using the specified marginal distributions F 1 and F 2 , a new bivariate distribution F ( x , y ) can be constructed with these two specified marginals which is given by

F ( x , y ) = ф ( ф − 1 ( F 1 ( x ) ) + ф − 1 ( F 2 ( y ) ) ) , (4)

ф is a univariate LT and its inverse given by ф − 1 .

If we let ф = ( 1 + s ) τ which is a LT of a gamma distribution, its inverse is given by ф − 1 = s − 1 τ − 1 , then we will have the classical distribution or survival

distribution Clayton copula as shown in example 4.2 given by Marshall and Olkin [6] (p. 837). For other copulas of the form given by expression (4), see Shih and Louis [5] .

For new bivariate distribution constructed using the BLTPM, we restrict the attention to nonegative distributions. Note that if the marginal distributions are specified respectively using LTs and given by L 1 ( s ) and L 2 ( t ) and subject to the two functions e − α ф − 1 ( L 1 ( s ) ) and e − α ф − 1 ( L 2 ( t ) ) are proper LT, a BLT of a new distribution with marginal LTs given respectively by L 1 ( s ) and L 2 ( t ) can also be obtained as

L X ( s , t ) = ф ( ф − 1 ( L 1 ( s ) ) + ф − 1 ( L 2 ( t ) ) ) , (5)

which is similar to the procedure as given by expression (4) using distribution or survival functions. This also means that some of the distribution or survival copulas of the class given by expression (4) can also be used as LT copulas. But the drawback of this approach using LT copula here is even by specifying the marginals L 1 ( s ) and L 2 ( t ) being LT of ID distributions, L X ( s , t ) constructed using expression (5) might not necessary be a BLT of a bivariate ID distribution.

It is also more difficult to simulate from a BLT L X ( s , t ) obtained using the LT copula as given by expression (5). Despite the algorithm used to generate a pair of observations from a specified BLT procedure is already given as above but when apply the procedures here we encounter the difficulty on how to simulate from distributions with LTs given by K 1 ( s ) = e − α ф − 1 ( L 1 ( s ) ) and K 2 ( t ) = e − α ф − 1 ( L 2 ( t ) ) as we might not be able to recognize these distributions.The inverse transform method can be applied by using the approximate quantile function based on moment generating functions M 1 ( s ) = e − α ф − 1 ( L 1 ( − s ) ) and M 2 ( s ) = e − α ф − 1 ( L 2 ( − t ) ) to obtain a simulated sample with some accuracy. The approximate quantile functions based on moment generating functions can be obtained explicitly using the saddle point technique and it is given by Arevelillo [23] . If these univariate MGFs do not exist, these LTs can be converted to characteristic functions and the numerical methods based on the inverse method and the approximate quantile functions as described by Glasserman and Liu [24] (pp. 1613-1615) can also be used to generate a pair of observations with some degree of accuracies.

There is a vast literature on survival or distribution copula, we just mention a few here. For a good general review on distribution or survival copula models with emphasis on goodness-of-fit test statistics, see Genest et al. [25] and for a good review of inference methods based on survival copulas used in actuarial science, see Klugman et al. [17] (pp. 187-213), Frees and Valdez [26] .

As mentioned earlier, asymptotic theory for Maximum entropy empirical likelihood (MEEL) and empirical likelihood (EL) for models where we have X 1 , ⋯ , X n a sample of multivariate observations independent and identically distributed (iid) as X which follows a d-variate multivariate distribution F β , β = ( β 1 , ⋯ , β p ) ′ are well established but assuming a set of moment conditions or constraints has been chosen, see Qin and Lawless [16] , Schennach [27] , Owen [28] , Mittelhammer et al. [29] ; also see discussions in section 3.2 by Luong [7] (pp. 471-472).

For applications, the question on how to choose moment conditions or equivalently bases so that MEEL methods have high efficiencies is a relevant one and we would like to address mainly this issue here for models with closed form BLTs or BMGFs as introduced in previous sections. The moment conditions are viewed as constraints and can be identifed with elements of a chosen basis used to project the true score functions as in the univariate case, see Luong [7] (pp. 461-467). Beside estimation, empirical likelihood type methods also give a chi-square test statistics for model testing which is useful for practical applications. We shall emphasize moment conditions and focus on how to choose constraints for models with closed form BLTs or BMGFs in the next section as asymptotic theory is similar to the univariate case and already discussed in section 3.2 as given by Luong [7] .

In this section, we shall recommend a base when using a model BLT to extract moment conditions which can be used as a guideline for forming other bases if needed for applying bivariate MEEL methods.

We shall define the first five g_i’s as follows which make use of the first two moments of X, i.e.,

g 1 ( x ; β ) = y − E β ( z ) ,     g 2 ( x ; β ) = z − E β ( z ) g 3 ( x ; β ) = ( y − E β ( y ) ) 2 − V β ( z ) ,     g 4 ( x ; β ) = ( z − E β ( z ) ) 2 − V β ( z ) , g 5 ( x ; β ) = ( y − E β ( y ) ) ( z − E β ( z ) ) − C o v β ( y , z ) . (6)

The variances of y and z are given respectively by V β ( y ) and V β ( z ) , the covariance between y and z is denoted by C o v β ( y , z ) . Subsequently we will pick the g_i’s directly using the model BLT L β ( s , t ) . We shall choose first four points of the BLT L β ( s , t ) , ( s 1 , t 1 ) , ( s 2 , t 2 ) , ( s 3 , t 3 ) , ( s 4 , t 4 ) which belong to a circle of radius r 1 = 0.01 in the nonnegative quadrant and centered at the origin, i.e., let

( s 1 , t 1 ) = r 1 ( cos ( 0 ) , sin ( 0 ) ) ,     ( s 2 , t 2 ) = r 1 ( cos ( π 6 ) , sin ( π 6 ) )

( s 3 , t 3 ) = r 1 ( cos ( 2π 6 ) , sin ( 2π 6 ) ) ,     ( s 4 , t 4 ) = r 1 ( cos ( 3π 6 ) , sin ( 3π 6 ) )

which lead to define

g 6 ( x ; β ) = e s 1 y + t 1 z − L β ( s 1 , t 1 ) ,     g 7 ( x ; β ) = e s 2 y + t 2 z − L β ( s 2 , t 2 ) , g 8 ( x ; β ) = e s 3 y + t 3 z − L β ( s 3 , t 3 ) ,     g 9 ( x ; β ) = e s 4 y + t 4 z − L β ( s 4 , t 4 ) .

We might want to add another 4 points on the part of the circle of radius r 2 = 0.02 of the nonnegative quadrant and centered at the origin by letting

( s 5 , t 5 ) = r 2 ( cos ( 0 ) , sin ( 0 ) ) ,     ( s 6 , t 6 ) = r 2 ( cos ( π 6 ) , sin ( π 6 ) )

( s 7 , t 7 ) = r 1 ( cos ( 2π 6 ) , sin ( 2π 6 ) ) ,     ( s 8 , t 8 ) = r 2 ( cos ( 3π 6 ) , sin ( 3π 6 ) )

which leads to define

g 10 ( x ; β ) = e s 5 y + t 5 z − L β ( s 1 , t 1 ) ,     g 11 ( x ; β ) = e s 6 y + t 6 z − L β ( s 6 , t 6 ) , g 12 ( x ; β ) = e s 7 y + t 7 z − L β ( s 7 , t 7 ) ,     g 13 ( x ; β ) = e s 8 y + t 8 z − L β ( s 8 , t 8 ) .

If needed, pick another 4 points on the circle of radius radius r 3 = 0.03 of the nonnegative quadrant centered at the origin by letting

( s 9 , t 9 ) = r 3 ( cos ( 0 ) , sin ( 0 ) ) ,     ( s 10 , t 10 ) = r 3 ( cos ( π 6 ) , sin ( π 6 ) )

( s 11 , t 11 ) = r 3 ( cos ( 2π 6 ) , sin ( 2π 6 ) ) ,     ( s 12 , t 12 ) = r 3 ( cos ( 3π 6 ) , sin ( 3π 6 ) )

which leads to define

g 14 ( x ; β ) = e s 9 y + t 9 z − L β ( s 9 , t 9 ) ,     g 15 ( x ; β ) = e s 10 y + t 10 z − L β ( s 10 , t 10 ) , g 16 ( x ; β ) = e s 11 y + t 11 z − L β ( s 11 , t 11 ) ,     g 17 ( x ; β ) = e s 12 y + t 12 z − L β ( s 12 , t 12 ) .

With all these g_i’s, the basis can be formed and given by

B L T = { g 1 ( x ; β ) , ⋯ , g k ( x ; β ) } , k = 17. (7)

Therefore, if the number of parameters m in the model, m < k , bivariate MEEL methods can be used for estimation and model testing using this basis to generate constraints. Note that the choice of g i ( x ; β ) , i = 1 , ⋯ , 17 do not put restrictions on the parameter space as the model BLT is well defined on the nonnegative quadrant for all values of the vector s = ( s , t ) ′ . This might not be the case with the use of the BMGF M β ( s , t ) where the domain of s might depend on β . We have to be ensured that if restrictions are imposed, the restricted parameter space is all we need for applications. The choice of points are based on the intuitve ground that the BLT contains more information at points near the origin and obviously there is some arbitrairiness on the choice of points or equivalently moment conditions for using MEEL methods.

If we use moment conditions based on the BMGF such as estimation for the jump-diffusion model as given by example 2, in general we might want to keep the first five g i ( x ; β ) ’s as given by expression (6) for BLT and add 12 additional g_i’s with chosen points on the circle of radius r = 0.01 centered at the origin without restricting on the first quadrant and define

( s j , t j ) = r ( cos ( j π 6 ) , sin ( j π 6 ) ) ,   j = 0 , 1 , 2 , ⋯ , 11 which leads to define

g 6 ( x ; β ) = e s 0 z 1 + t 0 z 2 − M β ( s 0 , t 0 ) , g 7 ( x ; β ) = e s 1 z 1 + t 1 z 2 − M β ( s 1 , t 1 ) ,                             ⋮ g 17 ( x ; β ) = e s 11 z 1 + t 11 z 2 − M β ( s 11 , t 11 ) .

Consequently, the base

B M G F = { g 1 ( x ; β ) , ⋯ , g 5 ( x ; β ) , g 6 ( x ; β ) , ⋯ , g 17 ( x ; β ) } (8)

which makes use of the model BMGF M β ( s , t ) can be used for generating constraints.

Note that the BMGF for the jump-diffusion model as given by example 2 is well defined for all points s = ( s , t ) ′ . Note that with a basis of k = 17 elements, it will allow the number of parameters p < 17 .It appears that such a base gives a good balance between efficiency and numerically tractability for the MEEL methods and can be used as a guideline to form other bases if necessary in practice.

Asymptotic properties for Multivariate MEEL methods and for EL methods for models with independent and identically distributed of vector of observations are well established, see Qin and Lawless, Mittelhammer et al. [29] (p. 323), Schennach [27] . Now if we use the constraints as defined using a basis B = { g 1 , ⋯ , g k } , MEEL estimators are obtained as the vector β ^ which maximizes the entropy or minimizes

∑ i = 1 n π i * ( β ) ln π i * ( β ) (9)

with

π i * = exp ( − ∑ j = 1 k λ j ( β ) ( ∑ i = 1 n g j ( x i ; β ) ) ) ∑ i = 1 n exp ( − ∑ j = 1 k λ j ( β ) ( ∑ i = 1 n g j ( x i ; β ) ) ) ,   i = 1 , ⋯ , n (10)

subject to

∑ i = 1 n π i * ( λ , β ) ( g j ( x i ; β ) ) = 0 ,   j = 1 , ⋯ , k . (11)

The λ_j's depend on β and are only implicitly defined using expression (11).

Using standard regularity conditions as indicated in the references just listed, also see section 3.2 given by Luong [7] (pp. 471-472), we then have consistency and asymptotic normality for the MEEL estimators given by β ^ , i.e., β ^ → p β 0 , β 0 is the vector of the true parameters,

n ( β ^ − β 0 ) → p N ( 0 , Ω ) ,

Ω = [ E [ ∂ g ( x , β ) ∂ β | β = β 0 ] ( E [ g ( x , β ) g ( x , β ) ′ ] | β = β 0 ) − 1 E [ ∂ g ( x , β ) ∂ β ′ | β = β 0 ] ] − 1 , (12)

g ( x , β ) = ( g 1 ( x , β ) , ⋯ , g k ( x , β ) ) ′ ,   Σ ( β 0 ) = E [ g ( x , β ) g ( x , β ) ′ ] | β = β 0 .

An estimator Ω ^ for Ω is given below,

Ω ^ = [ [ ∑ i = 1 n π ^ i ∂ g ( x i , β ) ∂ β | β = β ^ ] ( ∑ i = 1 n π ^ i g ( x i , β ) g ( x i , β ) ′ | β = β ^ ) − 1     ⋅ [ ∑ i = 1 n π ^ i ∂ g ( x i , β ) ∂ β | β = β ^ ] ] − 1

π ^ i = π i * ( β ^ ) , i = 1 , ⋯ , n . If we let π ^ i = 1 n , i = 1 , ⋯ , n , we have another consistent

estimator for Ω . The asymptotic results are very similar to the ones for the univariate case as given by Luong [7] but for completeness they are reproduced to make the paper easier to follow.

For model testing, it is viewed as testing the validity of the moment conditions, i.e., tesing the null hypothesis H 0 : E β ( g j ( x ) ) , j = 1 , ⋯ , k just as in the univariate case, the expectations are under the true parametric model.

The following test statistics given below is a chi-square test statistics which has an asymptotic chi-square distribution with r = k − p degree of freedom, i.e.,

2 n K L ( π * ( β ^ ) , p n ) = 2 n ( ∑ i = 1 n π i * ( β ^ ) [ log ( π i * ( β ^ ) ) − log ( 1 n ) ] ) → L χ 2 ( k − p ) . (13)

This test might be useful for testing bivariate normality in empirical finance and it is based on the MEEL estimators β ^ and since the basis used spans the true score functions of the bivariate normal model which makes β ^ as efficient as the maximum likelihood estimator β M L ^ , as the projected score functions are identical to the score functions. Furthermore, the asymptotic chi-square distributions are practical for the use of these test statistics and we do not need intensive simulations to implement model testing procedures as in other methods. Also, MEEL methods can be applied for model with a singular part in the domain provided that the model BLT has a closed form expression.

As for the univariare case, a penalty approach can be used which means that we can perform unconstrained minimization using the following objective function with respect to λ = ( λ 1 , ⋯ , λ k ) ′ and β 1 , ⋯ , β p ,

∑ i = 1 n π i * ( λ , β ) ln π i * ( λ , β ) + K 2 [ ( ∑ i = 1 n π i * ( λ , β ) [ g 1 ( x i , β ) ] ) 2 + ⋯ + ( ∑ i = 1 n π i * ( λ , β ) [ g k ( x i , β ) ] ) 2 ] (14)

The penalty constant K is a large positive value, direct search agorithms which are derivative free are recommended and these direct search algorithms are also more stable.

It is important to note that only a local minimizer is found each time using these algorithms, some strategies are needed to identify the global minimizer. Often, we might need a starting vector being close to the vector of the estimators to initialize the algorithm, this is important when working with real data and bivariate models might have many parameters. We might want to adopt the following strategy to look for a good starting vector. Using the two submodels based on the two marginal distributions of the model separately we can fit using univariate MEEL methods using { y i , i = 1 , ⋯ , n } and { z i , i = 1 , ⋯ , n } as described in Luong [7] to obtain respectively β 1 ( 0 ) ^ the vector which estimates the vector of parameters β 1 of the first marginal model and similarly obtain β 2 ( 0 ) ^ which estimates the vector of parameters β 2 of the second marginal model.The vector of parameters of the bivariate model is β and β can be partitioned into three components,i.e.,

β = ( β 1 , β 2 , β 3 ) ′ . Now if we apply MEEL methods with the bivariate model but fixing

β 1 = β 1 ( 0 ) ^ , β 2 = β 2 ( 0 ) ^ and only estimate β 3 , the vector which estimates β 3 is denoted by β 3 ( 0 ) ^ . We can construct the following starting vector

β ^ ( 0 ) = ( β 1 ( 0 ) ^ , β 2 ( 0 ) ^ , β 3 ( 0 ) ^ ) ′ , λ ^ ( 0 ) = ( 0 , ⋯ , 0 ) and fit the bivariate model using penalty method by minimizing jointly with respect to the vector λ and the vector β the objective function as given by expression (14).