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Bandwidth and Stability: more explanation of poles and zeros

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In [[statistics]], the '''Jarque–Bera test''' is a [[goodness-of-fit]] test of whether sample data have the [[skewness]] and [[kurtosis]] matching a [[normal distribution]]. The test is named after [[Carlos Jarque]] and [[Anil K. Bera]]. The [[test statistic]] ''JB'' is defined as

: <math>
\mathit{JB} = \frac{n}{6} \left( S^2 + \frac14 (K-3)^2 \right)
</math>

where ''n'' is the number of observations (or degrees of freedom in general); ''S'' is the sample [[skewness]], and ''K'' is the sample [[kurtosis]]:

: <math>\begin{align}
& S = \frac{ \hat{\mu}_3 }{ \hat{\sigma}^3 }
= \frac{\frac1n \sum_{i=1}^n (x_i-\bar{x})^3} {\left(\frac1n \sum_{i=1}^n (x_i-\bar{x})^2 \right)^{3/2}} \\
& K = \frac{ \hat{\mu}_4 }{ \hat{\sigma}^4 }
= \frac{\frac1n \sum_{i=1}^n (x_i-\bar{x})^4} {\left(\frac1n \sum_{i=1}^n (x_i-\bar{x})^2 \right)^2} ,
\end{align}</math>

where <math>\hat{\mu}_3</math> and <math>\hat{\mu}_4</math> are the estimates of third and fourth [[central moment]]s, respectively, <math>\bar{x}</math> is the sample [[mean]], and
<math>\hat{\sigma}^2</math> is the estimate of the second central moment, the [[variance]].

If the data comes from a normal distribution, the ''JB'' statistic [[asymptotic analysis|asymptotically]] has a [[chi-squared distribution]] with two [[degrees of freedom (statistics)|degrees of freedom]], so the statistic can be used to [[statistical hypothesis testing|test]] the hypothesis that the data are from a [[normal distribution]]. The [[null hypothesis]] is a joint hypothesis of the skewness being zero and the [[excess kurtosis]] 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 in fact true. Furthermore, the distribution of p-values departs from a uniform distribution and becomes a right-skewed uni-modal distribution, especially for small p-values. This leads to a large [[Type I error]] rate. The table below shows some p-values approximated by a chi-squared distribution that differ from their true alpha levels for small samples.

:{| class="wikitable"
|+Calculated p-value equivalents to true alpha levels at given sample sizes
! True α level !! 20 !! 30 !! 50 !! 70 !! 100
|-
! 0.1
| 0.307 || 0.252 || 0.201 || 0.183 || 0.1560
|-
! 0.05
| 0.1461 || 0.109 || 0.079 || 0.067 || 0.062
|-
! 0.025
| 0.051 || 0.0303 || 0.020 || 0.016 || 0.0168
|-
! 0.01
| 0.0064 || 0.0033 || 0.0015 || 0.0012 || 0.002
|}
(These values have been approximated by using [[Monte Carlo simulation]] in [[Matlab]])

In [[MATLAB]]'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.<ref name="MathWorks">{{cite web|url=http://www.mathworks.com/access/helpdesk/help/toolbox/stats/jbtest.html|title=Analysis of the JB-Test in MATLAB|publisher=MathWorks|accessdate=May 24, 2009}}</ref>

==History==
Considering normal sampling, and √''β''1 and ''β''2 contours, {{harvtxt|Bowman|Shenton|1975}} noticed that the statistic ''JB'' will be asymptotically ''χ''2(2)-distributed; however they also noted that “large sample sizes would doubtless be required for the ''χ''2 approximation to hold”. Bowman and Shelton did not study the properties any further, preferring [[D’Agostino’s K-squared test]].

Around 1979, Anil Bera and [[Carlos Jarque]] while working on their dissertations on regression analysis, have applied the [[Lagrange multiplier principle]] to the [[Pearson distribution|Pearson family of distributions]] to test the normality of unobserved regression residuals and found that the ''JB'' test was asymptotically optimal (although the sample size needed to “reach” the asymptotic level was quite large). In 1980 the authors published a paper ({{harvnb|Jarque|Bera|1980}}), which treated a more advanced case of simultaneously testing the normality, [[homoscedasticity]] and absence of [[autocorrelation]] in the residuals from the [[linear regression model]]. The ''JB'' test was mentioned there as a simpler case. A complete paper about the JB Test was published in the ''International Statistical Review'' in 1987 dealing with both testing the normality of observations and the normality of unobserved regression residuals, and giving finite sample significance points.

==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:

: <math>
\mathit{JB} = \frac{n-k}{6} \left( S^2 + \frac14 (K-3)^2 \right)
</math>

where ''n'' is the number of observations and ''k'' is the number of regressors when examining residuals to an equation.

==References==
{{reflist}}
<HR>
{{refbegin}}
* {{cite journal
| first1 = K.O. | last1 = Bowman
| first2 = L.R. | last2 = Shenton
| title = Omnibus contours for departures from normality based on √''b''1 and ''b''2
| year = 1975
| journal = Biometrika
| volume = 62 | issue = 2
| pages = 243–250
| jstor = 2335355
| ref = CITEREFBowmanShenton1975
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = Efficient tests for normality, homoscedasticity and serial independence of regression residuals
| year = 1980
| journal = Economics Letters
| volume = 6 | issue = 3
| pages = 255–259
| doi = 10.1016/0165-1765(80)90024-5
| ref = CITEREFJarqueBera1980
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence
| year = 1981
| journal = Economics Letters
| volume = 7 | issue = 4
| pages = 313–318
| doi = 10.1016/0165-1765(81)90035-5
| ref = CITEREFJarqueBera1981
}}
* {{cite journal
| first1 = Carlos M. | last1 = Jarque | authorlink1 = Carlos Jarque
| first2 = Anil K. | last2 = Bera
| title = A test for normality of observations and regression residuals
| year = 1987
| journal = International Statistical Review
| volume = 55 | issue = 2
| pages = 163–172
| jstor = 1403192
| ref = CITEREFJarqueBera1987
}}
* {{cite book
| first = | last = Judge
| coauthors = et al.
| title = Introduction and the theory and practice of econometrics
| year = 1988
| edition = 3rd
| pages = 890–892
}}
* {{cite book
| first1 = Robert E. | last1 = Hall
| first2 = David M. | last2 = Lilien
| coauthors = et al.
| title = EViews User Guide
| year = 1995
| pages = 141
}}
{{refend}}

== Implementations ==
* [http://www.alglib.net/statistics/hypothesistesting/jarqueberatest.php ALGLIB] includes implementation of the Jarque–Bera test in C++, C#, Delphi, Visual Basic, etc.
* [[gretl]] includes an implementation of the Jarque–Bera test
* [[R (programming language)|R]] includes implementations of the Jarque–Bera test: ''jarque.bera.test'' in package ''tseries'', for example, and ''jarque.test'' in package ''moments''.
* [[Matlab|MATLAB]] includes implementation of the Jarque–Bera test, the function "jbtest".
* [[Python (programming language)|Python]] statsmodels includes implementation of the Jarque–Bera test, "statsmodels.stats.stattools.py".

{{Statistics}}

{{DEFAULTSORT:Jarque-Bera test}}
[[Category:Normality tests]]

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en>Zen-in: /* Bandwidth and Stability */ more explanation of poles and zeros