The fault detection rate of using kernel density estimation to calculate the threshold is slightly higher than the

chi-squared distribution. Most of the faults can be detected well and they have high FDRS and low FARS.

[[gamma].sub.1] has a

chi-squared distribution with 2([N.sub.T] - N - L + 1) degrees of freedom.

anomala, step lengths appeared to have a finite distribution well approximated by a

chi-squared distribution (Fig.

Studying various tests for covariance matrices in samples with P=2, 3 and 5 variables in different matrix structures (modification in the variance, in the correlation, and in both), including the LRT, to test the hypothesis that the covariance matrix of a process ([summation]) is equal to a particular matrix ([[summation].sub.0]), PINTO (2009) reported that the LRT approximated by the

chi-squared distribution does not control type I errors for samples of size n=10 in bivariate situations and for samples of sizes n=10 and 25 in situations with P=3 and 5 variables.

(3) Furthermore, this result helps us to reconcile the form of the asymptotic approximation proposed in Theorem 1 with the weighted

chi-squared distribution that arises in some special cases as in Example 2 above.

=4 k RMAPE1 RMAPE3 RMAPE5 RMAPE7 RMAPE9 U 3 RMAPE1 -- 0.000 0.000 0.000 0.000 RMAPE3 -- -- 0.821 0.117 0.003 RMAPE5 -- -- -- 0.008 0.005 RMAPE7 -- -- -- -- 0.041 5 RMAPE1 -- 0.000 0.000 0.000 0.000 RMAPE3 -- -- 0.082 0.016 0.000 RMAPE5 -- -- -- 0.255 0.001 RMAPE7 -- -- -- -- 0.051 7 RMAPE1 -- 0.000 0.000 0.000 0.000 RMAPE3 -- -- 0.066 0.067 0.003 RMAPE5 -- -- -- 0.803 0.141 RMAPE7 -- -- -- -- 0.038 9 RMAPE1 -- 0.000 0.000 0.000 0.000 RMAPE3 -- -- 0.022 0.012 0.000 RMAPE5 -- -- -- 0.579 0.084 RMAPE7 -- -- -- -- 0.048 Table 12 P-values from Pair-wise T-tests on Means of MAD's from MA(1), MA(3), MA(5), MA(7), MA (9) for the

Chi-squared Distribution with c.v.

From the obtained empirical distributions for the ML QTL positions, [LRT.sub.p] had a

chi-squared distribution with 2 degrees of freedom corrected (Bonferroni correction) for 4.5 to 7 independent traits when testing the seven traits.

A further indication of the importance of using a variable counting time scheme can be seen from the analysis of the cumulative

chi-squared distribution for the refinement of the structure of famotidine (Figure 4).

This deflation factor is then applied to the observed G statistic, which is then compared with the

chi-squared distribution to calculate the final P value.

Denoting the ratio of the restricted value to the unrestricted value as [lambda], the test relies on the fact that -2ln[lambda] follows a

chi-squared distribution.

where [[chi square].sub.1] is a

chi-squared distribution with 1 degree of freedom.