ajb.norm.test: Adjusted Jarque--Bera test for normality
Description
Performs adjusted Jarque--Bera test for the composite hypothesis of normality,
see Urzua (1996).
Usage
ajb.norm.test(x, nrepl=2000)
Arguments
x
a numeric vector of data values.
nrepl
the number of replications in Monte Carlo simulation.
Value
A list with class "htest" containing the following components:
statisticthe value of the adjusted Jarque--Bera statistic.
p.valuethe p-value for the test.
methodthe character string "Adjusted Jarque-Bera test for normality".
data.namea character string giving the name(s) of the data.
Details
The adjusted Jarque--Bera test for normality is based on the following statistic:
$$AJB = \frac{(\sqrt{b_1})^2}{\mathrm{Var}\left(\sqrt{b_1}\right)} + \frac{(b_2 - \mathrm{E}\left(b_2\right))^2}{\mathrm{Var}\left(b_2\right)},$$
where
$$\sqrt{b_1} = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^3}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^{3/2}},
\quad
b_2 = \frac{\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^4}{\left(\frac{1}{n}\sum_{i=1}^n(X_i - \overline{X})^2\right)^2},$$
$$\mathrm{Var}\left(\sqrt{b_1}\right) = \frac{6(n-2)}{(n+1)(n+3)},
\quad
E\left(b_2\right) = \frac{3(n-1)}{n+1},
\quad
\mathrm{Var}\left(b_2\right) = \frac{24n(n-2)(n-3)}{(n+1)^2(n+3)(n+5)}.$$
The p-value is computed by Monte Carlo simulation.
References
Urzua, C. M. (1996): On the correct use of omnibus tests for normality. --- Economics Letters, vol. 53, pp. 247--251.