h = jbtest (x) performs the Jarque-Bera test of the null hypothesis that the sample in the vector x comes from a normal distribution with unknown mean and variance, against the alternative that it does not come from a normal distribution. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise. x must be a vector of real values; NaN values are treated as missing and removed.

The Jarque-Bera test statistic is $$ JB = \frac{n}{6} \left( s^2 + \frac{(k-3)^2}{4} \right), $$ where $n$ is the sample size, $s$ is the sample skewness, and $k$ is the sample kurtosis. Under the null hypothesis it is asymptotically chi-square distributed with two degrees of freedom.

h = jbtest (x, alpha) performs the test at the significance level alpha, a scalar in the range $(0,1)$. The default is $0.05$.

h = jbtest (x, alpha, mctol) computes a Monte-Carlo approximation of the p-value instead of interpolating the embedded table. mctol is the maximum Monte-Carlo standard error allowed for the p-value; the number of simulated samples is chosen accordingly. Use this for small samples, where the chi-square approximation is inaccurate, or for significance levels outside $[0.001,\; 0.5]$.

[h, p] = jbtest (…) also returns the p-value p of the test. p is clamped to the tabulated range $[0.001,\; 0.5]$, as MATLAB clamps it, and a warning is issued when the value lies outside that range. The warning is an addition here: MATLAB clamps silently, so a p-value reported as $0.001$ or $0.5$ there may be a bound rather than an estimate, with nothing to say so.

[h, p, jbstat, critval] = jbtest (…) also returns the test statistic jbstat and the critical value critval at significance level alpha. The null hypothesis is rejected when jbstat > critval.

Note: for $n\; \backslash le\; 2000$ the p-value and critical value are obtained by interpolating an embedded critical-value table (the same approach MATLAB uses); for larger samples the large-sample chi-square approximation with two degrees of freedom is used instead. The embedded table was generated here by Monte-Carlo simulation, so it is itself an estimate of the true null quantiles. MATLAB’s table is likewise a Monte-Carlo estimate but from a different simulation, so the two tables agree only to about two decimal places. As a result the reported p-value and critical value, and (in a narrow band of statistic values around the critical value) the test decision h, can differ slightly from MATLAB in edge cases. These differences are an unavoidable consequence of the Monte-Carlo origin of both tables, not a difference in method. Supply mctol for a direct Monte-Carlo p-value.

See also: kstest, adtest, lillietest

Source Code: jbtest