The chi-squared test rejects the null hypothesis if the [chi square]-statistic exceeds the critical [chi square] value of a

chi-squared distribution evaluated at degrees-of-freedom DF and a specified significance percentage.

Besides the

chi-squared distribution, kernel density estimation is also utilized to determine the threshold [16].

If p = 1, that is, A is defined as a random variable, then it has a

Chi-squared distribution with 2n degrees of freedom, that is, A ~ [chi square] (2n).

anomala was well approximated by a

chi-squared distribution (Fig.

In this test, the value of the statistic calculated was compared to the tabulated critical value of the

chi-squared distribution (upper 5% quantile), with pxq degrees of freedom at 5% level of significance; when the calculated value is greater than the tabulated value, the null hypothesis is rejected, and it is concluded that the two groups considered in the study are not independent.

and considering [[DELTA].sub.t] = 1 for the sake of simplicity, [[epsilon]'.sub.l] follows a noncentral

Chi-squared distribution with noncentrality parameter [mathematical expression not reproducible].

We determine the value "ChiSquared" of [chi square] statistics (if the null hypothesis is true, then the statistics is compatible with

chi-squared distribution of r-1 or of o r-k-1 degrees of freedom, when there were k parameters of distribution estimated from the sample).

If we replace the ACF with the ACF estimator we obtain test function BP (Box-Pierce) which means that the test statistic has

Chi-squared distribution (Box and Pierce, 1970):

Under the null hypothesis of independence, the statistics has an asymptotic

chi-squared distribution with a single degree of freedom.

For the special case of q = [1, [0'.sub.m-1]' (i.e., the payoff of the first test asset is a gross return and the rest are excess returns), the estimate of the Lagrange multiplier associated with the first test asset, [[??].sub.1], is T-consistent and shares the weighted

chi-squared distribution of [[??].sup.2.sub.T] under the assumption of a correctly specified model.

The parameters set for the data collection were: T-distribution with 3 degrees of freedom, Uniform distribution in the interval of (-[1/2], [1/2]),

chi-squared distribution with 2 degrees of freedom, a mean of one, and the standard deviation set as the coefficients of variation (c.v.).