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Dataplot Vol 1 Auxillary Chapter

JARQUE BERA TEST

Name:
    JARQUE BERA TEST
Type:
    Analysis Command
Purpose:
    Perform a Jarque-Bera test for normality.
Description:
    Most tests for normality are based either on comparing the empirical cumulative distribution with the theoretical normal cumulative distribution (Kolmogorov-Smirnov, Anderson-Darling, Chi-Square) or empirical quantiles with the theoretical normal quantiles (PPCC, Wilk-Shapiro). In contrast, the Jarque-Bera test is based on the sample skewness and sample kurtosis.

    The Jarque-Bera test statistic is defined as:

      JB = (N/6)*(S**2 + (1/4)*(K - 3)**2)

    with S, K, and N denoting the sample skewness, the sample kurtosis, and the sample size, respectively.

    For sample sizes of 2,000 or larger, this test statistic is compared to a chi-squared distribution with 2 degrees of freedom (normality is rejected if the test statistic is greater than the chi-squared value).

    The chi-square approximation requires fairly large sample sizes to be accurate. For sample sizes less than 2,000, the critical value is determined via simulation. Specifically, one hundred thousand normal samples with the same mean and standard deviation as the original data sample are generated and the Jarque-Bera test statistic computed to generate the reference distribution.

Syntax 1:
    JARQUE BERA TEST <y>             <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
    MULTIPLE JARQUE BERA TEST <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of 1 to 30 response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 3:
    REPLICATED JARQUE BERA TEST <y> <x1> ... <xk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <x1> ... <xk> is a list of 1 to 6 group-id variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    JARQUE BERA TEST Y1
    JARQUE BERA TEST Y1 SUBSET TAG > 2
    MULTIPLE JARQUE BERA TEST Y1 TO Y10
    REPLICATED JARQUE BERA TEST Y X
Note:
    Syntax 1 and Syntax 2 support matrix arguments. Syntax 2 supports the TO syntax.

    For Syntax 3 (the REPLICATED form), the variables must all have the same number of observations.

Note:
    The following statistics are also supported:

      LET A = JARQUE BERA TEST Y
      LET A = JARQUE BERA TEST CDF Y
      LET A = JARQUE BERA TEST PVALUE Y
Default:
    None
Synonyms:
    JARQUE BERA NORMALITY TEST
    JARQUE BERA
Related Commands: Reference:
    Brani Vidakovic (2011), "Statistics for Bioengineering Sciences: with MATLAB and WINBUGS Support," Springer, pp. 521-522.
Applications:
    Distributional Fitting, Assumption Testing
Implementation Date:
    2012/6
Program:
     
    SKIP 25
    READ ZARR13.DAT Y1
    READ ZARR15.DAT Y2
    SET WRITE DECIMALS 4
    MULTIPLE JARQUE BERA TEST Y1 Y2
        
    The following output is generated
                Jarque-Bera Test for Normality
     
    Response Variable: Y1
     
    H0: The Data Are Normally Distributed
    Ha: The Data Are Not Normally Distributed
     
    Summary Statistics:
    Total Number of Observations:                        195
    Sample Mean:                                      9.2614
    Sample Standard Deviation:                        0.0227
    Sample Skewness:                                 -0.0085
    Sample Kurtosis:                                  3.0490
    Sample Minimum:                                   9.1968
    Sample Maximum:                                   9.3279
     
    Test Statistic Value:                             0.0219
    CDF Value:                                        0.0101
    P-Value:                                          0.9898
     
     
    Percent Points of the Reference Distribution
    -----------------------------------
      Percent Point               Value
    -----------------------------------
               25.0    =          0.553
               50.0    =          1.256
               75.0    =          2.403
               80.0    =          2.774
               90.0    =          4.044
               95.0    =          5.679
               97.5    =          8.034
               99.0    =         12.567
     
    Conclusions (Upper 1-Tailed Test)
    ----------------------------------------------
      Alpha    CDF   Critical Value     Conclusion
    ----------------------------------------------
        10%    90%            4.044      Accept H0
         5%    95%            5.679      Accept H0
       2.5%  97.5%            8.034      Accept H0
         1%    99%           12.567      Accept H0
     
     
                Jarque-Bera Test for Normality
     
    Response Variable: Y2
     
    H0: The Data Are Normally Distributed
    Ha: The Data Are Not Normally Distributed
     
    Summary Statistics:
    Total Number of Observations:                        211
    Sample Mean:                                      9.2772
    Sample Standard Deviation:                        0.1117
    Sample Skewness:                                 -0.0067
    Sample Kurtosis:                                  2.4502
    Sample Minimum:                                   9.0274
    Sample Maximum:                                   9.5219
     
    Test Statistic Value:                             2.6585
    CDF Value:                                        0.7840
    P-Value:                                          0.2159
     
     
    Percent Points of the Reference Distribution
    -----------------------------------
      Percent Point               Value
    -----------------------------------
               25.0    =          0.557
               50.0    =          1.266
               75.0    =          2.415
               80.0    =          2.786
               90.0    =          4.084
               95.0    =          5.718
               97.5    =          8.066
               99.0    =         12.467
     
    Conclusions (Upper 1-Tailed Test)
    ----------------------------------------------
      Alpha    CDF   Critical Value     Conclusion
    ----------------------------------------------
        10%    90%            4.084      Accept H0
         5%    95%            5.718      Accept H0
       2.5%  97.5%            8.066      Accept H0
         1%    99%           12.467      Accept H0
        

Date created: 07/05/2012
Last updated: 07/05/2012
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