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The Jarque-Bera’s fitting test for normality is a celebrated and powerful one. In this paper, we consider general Jarque-Bera tests for any distribution function (
*df*) having at least 4
*k* finite moments for
*k* ≥ 2. The tests use as many moments as possible whereas the JB classical test is supposed to test only skewness and kurtosis for normal variates. But our results unveil the relations between the coeffients in the JB classical test and the moments, showing that it really depends on the first eight moments. This is a new explanation for the powerfulness of such tests. General Chi-square tests for an arbitrary model, not only normal, are also derived. We make use of the modern functional empirical processes approach that makes it easier to handle statistics based on the high moments and allows the generalization of the JB test both in the number of involved moments and in the underlying distribution. Simulation studies are provided and comparison cases with the Kolmogorov-Smirnov’s tests and the classical JB test are given.

In this paper, we are concerned with generalizations of Jarque-Bera’s (JB) [

Let

These statistics are designed to estimate the theoretical skewness and kurtosis given by

H0: X follows a Gaussian normal law, we have

has an asymptotic chi-square distribution with two degrees of freedom under the null hypothesis of normality. Jarque-Bera’s test consists in rejecting H0 when T_{n} is far from zero. We will find below that the constants 6 and 24 used in (2), actually, are closely related to the first four even moments of a

Our objective here is to generalize JB’s test to a general df G by considering high moments

Actually, JB’s test only checks the third and fourth moments of X while the coefficients of the JB statistic (2) uses the first eight moments of X. Our guess is that we would have better tests if we are able to simultaneously check all the first (2k) moments for some k ≥ 2. To this purpose, we consider the following statistics, that is the normalized centered empirical moments (NCEM),

where

are the

whenever the (4p)^{th} moment exists. Finally we consider C^{1}-class functions

Our general test is based on the following statistics, for k ≥ 2,

which almost-surely

as

From such a general result, we are able to derive a normality test by using it with

We are going to establish a general asymptotic normality of

Our results will show that these tests based on the 2k moments, need, in fact, the eight 4k moments for computing the variance. This unveils that the classical JB’s test is not based only on the kurtosis and the skewness but also on the sixth and the eighth moments. To describe the complete form of the Jarque-Bera method, put

The JB’s test for a

with the particular coefficients

As an illustration of what proceeds, consider a distribution following a double-gamma distribution

centered and has a kurtosis coefficient equal to 3. It is rejected from normality by the JB test. If only the skewned and kurtosis do matter, it would not be the case. Actually, the rejection comes from the parameters

The rest of the paper is organized as follows. In Subsection 2.1 of Section 2, we begin to give a concise of reminder the modern theory of functional empirical processes that is the main theoretical tool we use for finding the asymptotic law of (5). Next in Subsection 2.2, we establish general results of the consistency of (5) and its asymptotic law, consider particular cases in Subsection 2.3, propose chi-square universal tests in Subsection 2.4 and finally state the proofs in Subsection 2.5. We end the paper by Section 3 where simulation results concerning the normal and double-exponential models are given.

We here express that in all the sequel the limits are meant as

Since the empirical functional process is our key tool here, we are going to make a brief reminder on this process associated with

where f is a real measurable function defined on

and

It is known (see van der Vaart [

at least in finite distributions.

This linearity will be useful for our proofs. We are now in position to state our main results.

First introduce this notation for_{i} and g_{i}, ^{1}-functions with values in

and

Here are our main results.

Theorem 1 Let

where

Corollary 1 (Normality test). Let X be a

Then

where

and

Let G be an arbitrary df with a 4k^{th} finite moment for k ≥ 2, this is^{1}-functions f_{i} and g_{i},

value

^{2} is either the exact variance

Our guess is that using a greater value of k makes the test more powerful since the equality in distribution of univariate r.v.’s means equality of all moments when they exist (see page 213 in [_{1} for which the p-value exceeds 5% would suggest it has the same eight moments as G, which is highly improbable. Simulation studies in Section 0 support our findings. Remark that we have as many choices as possible for the functions the

Unfortunately, in the simulation studies reported below, we noticed that the plug-in estimator

Now let us show how to derive chi-square tests from Theorem 1.

Suppose that X is a symmetrical distribution. We have from Theorem 1 that

Since X is symmetrical, that is

and

By reminding that

where

Corollary 2 Let

For a standard normal random variable, we get

Corollary 3 Let G be a Gaussian df. Then

We see that we obtain an infinite number of tests for the normality. For example, for

Consider (15) and put

Corollary 4 Let

converges in law to a

It is now time to prove Theorem 1 before considering the simulation studies.

Since G has at least first 4k moments finite, we are entitled to use the finite-distribution convergence of the empirical function process

where

Now the law of

By the delta-method, we have

and then

and next, by noticing from 17 that

where

This completes the proof of the theorem. The proof of the corollary is a simple consequence of the theorem.

We want to focus on illustrating how performs the general test for usual laws such as Normal and Double Gamma ones. It is clear that the generality of our results that are applicable to arbitrary d.f.’s with some finite k^{th}-moment

In this paper, we want to set a general and workable method to simulate and test two symmetrical models. The normal and the double-exponential one with density

Once these results are achieved, we would be in position to handle a larger scale simulation research following the outlined method. Specially, fitting financial data to the generalized hyperbolic model is one the most interesting applications of our results.

We first choose all the functions f_{i} equal to f_{0} and all the functions g_{i} equal to g_{0}. We fix k = 3, that is we work with the first twelve moments. As a general method, we consider two df’s G_{1} and G_{2}. We fix one of them say G_{1} and compute _{1}. We generate samples of size n from one the df’s (either G_{1} or G_{2}) and compute

and report the mean value t^{*} of the replicated values of

p-value_{1} and low for samples from G_{2}. The results are compared with those given by the Kolmogorov- Smirnov test (KST) and when the data are Gaussian, they are compared with the outcomes from JB’s classical test.

We consider the following cases: G_{1} is a Gaussian r.v_{2} is double-exponential law _{3} is a double-gamma law

sity

The choice

We recall that the variance of our statistic depends on the first 4k moments.

Simulation study.

Testing the model with

and for n = 100,

and for n = 1000,

where JB is the classical Jarque-Berra statistic, pJB is the p-value of the JB test, KS is the Kolmogorov-smirnov statistic and pKS is the related p-value. Our model accepts the normality and this is confirmed by JB’s test and by the Klmogorov-Smirnov test (KST). The simulation results are very stable and constantly suggest acceptance.

Testing the double-exponential versus the normal model.

Recall that the values _{p} coincide and the test is only based on the moments. Indeed, using data from

and for n = 22

Our test rejects the

Testing the double-gamma versus the normal model.

Let use

and for n = 22

We have similar results. Ou test rejects the

Analysing the tables above, we conclude that our test performs better the JB’s test against a double-gamma df with same skewness and kurtosis than a normal df for small sample sizes around ten and this is real advantage for small data sizes. Even for k = 2, our test is performant for the small values n = 11 and n = 12.

Double-exponential model

We point out that the statistic

Here, we do not have the Jarque-Berra test to confirm the results.

Simulation. Testing the model with

The simulation results are very stable and constantly suggest acceptance.

Testing normal data. Using normal data gives

The

We obtain good results for

while the normal model is rejected as illustrated below:

It is important to mention here that the KST is very powerful is rejecting the normal model with double-ex- ponential and double-gamma data with extremely low p-value’s.

We propose a general test for an arbitrary model. The methods are based on functional empirical processes theory that readily provides asymptotic laws from which statistical tests are derived. They depend on an integer k such that the pertaining df has 4k first finite moments. We get two kinds of tests. A general one based on functions f_{i} and g_{i}, _{i}, g_{i}, and k, the test performs well for small samples (n = 20) both for accepting the normal model and rejecting other models. We also show that for suitable choice of f_{i} and g_{i}, the test for the double-exponential model is also successful, but for sizes greater that n = 150. In upcoming papers, we will focus on detailed results on specific models and try to found out, for each case, suitable value of the parameters of the tests ensuring good performances for small data. A paper is also to be devoted to simulation studies for the khi-square tests and their applications to financial data.