In
statistics, the Jarque–Bera test is a
goodness-of-fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
test of whether sample data have the
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal ...
and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
matching a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The test is named after
Carlos Jarque and
Anil K. Bera.
The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution.
The
test statistic
A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifi ...
''JB'' is defined as
:
where ''n'' is the number of observations (or degrees of freedom in general); ''S'' is the sample
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal ...
, ''K'' is the sample
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
:
:
:
where
and
are the estimates of third and fourth
central moment
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random ...
s, respectively,
is the sample
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
, and
is the estimate of the second central moment, the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
.
If the data comes from a normal distribution, the ''JB'' statistic
asymptotically has a
chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
with two
degrees of freedom, so the statistic can be used to
test
Test(s), testing, or TEST may refer to:
* Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities
Arts and entertainment
* ''Test'' (2013 film), an American film
* ''Test'' (2014 film), ...
the hypothesis that the data are from a
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. The
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
is a joint hypothesis of the skewness being zero and the
excess kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosi ...
being zero. Samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0 (which is the same as a kurtosis of 3). As the definition of ''JB'' shows, any deviation from this increases the JB statistic.
For small samples the chi-squared approximation is overly sensitive, often rejecting the null hypothesis when it is true. Furthermore, the distribution of
''p''-values departs from a
uniform distribution and becomes a
right-skewed unimodal distribution
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.
Unimodal probability distribution
In statistics, a unimodal ...
, especially for small ''p''-values. This leads to a large
Type I error
In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
rate. The table below shows some ''p''-values approximated by a chi-squared distribution that differ from their true alpha levels for small samples.
:
(These values have been approximated using
Monte Carlo simulation in
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
)
In
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
's implementation, the chi-squared approximation for the JB statistic's distribution is only used for large sample sizes (> 2000). For smaller samples, it uses a table derived from
Monte Carlo simulations in order to interpolate ''p''-values.
History
The statistic was derived by Carlos M. Jarque and Anil K. Bera while working on their Ph.D. Thesis at the Australian National University.
Jarque–Bera test in regression analysis
According to Robert Hall, David Lilien, et al. (1995) when using this test along with multiple regression analysis the right estimate is:
:
where ''n'' is the number of observations and ''k'' is the number of regressors when examining residuals to an equation.
Implementations
ALGLIBincludes an implementation of the Jarque–Bera test in C++, C#, Delphi, Visual Basic, etc.
*
gretl
gretl is an open-source statistical package, mainly for econometrics. The name is an acronym for ''G''nu ''R''egression, ''E''conometrics and ''T''ime-series ''L''ibrary.
It has both a graphical user interface (GUI) and a command-line inter ...
includes an implementation of the Jarque–Bera test
*
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
includes an implementation of the Jarque-Bera test ''JarqueBeraTest'' in the ''HypothesisTests'' package.
*
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
includes an implementation of the Jarque–Bera test, the function "jbtest".
*
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
statsmodels
Statsmodels is a Python package that allows users to explore data, estimate statistical models, and perform statistical tests. An extensive list of descriptive statistics, statistical tests, plotting functions, and result statistics are available f ...
includes an implementation of the Jarque–Bera test, "statsmodels.stats.stattools.py".
*
R includes implementations of the Jarque–Bera test: ''jarque.bera.test'' in the package ''tseries'', for example, and ''jarque.test'' in the package ''moments''.
*
Wolfram includes a built in function called, JarqueBeraALMTest
and is not limited to testing against a Gaussian distribution.
See also
*
D'Agostino's K-squared test
In statistics, D'Agostino's ''K''2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to gauge the compatibility of given data with the null hypothesis that the data is a realizatio ...
, another test based on kurtosis and skewness.
References
Further reading
*
*
*
*
*
{{DEFAULTSORT:Jarque-Bera test
Normality tests