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 Tn 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 C1-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
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 4kth finite moment for k ≥ 2, this is
value
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 [
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 kth-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 fi equal to f0 and all the functions gi equal to g0. We fix k = 3, that is we work with the first twelve moments. As a general method, we consider two df’s G1 and G2. We fix one of them say G1 and compute
and report the mean value t* of the replicated values of
p-value
We consider the following cases: G1 is a Gaussian r.v
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
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 fi and gi,