Tables for a new multivariate goodness-of-fit test

A monte carlo method for foutz's goodness-of-fit test

“…The numbers in the table are the percentiles of the F, distribution for various sample sizes n based on three different approximation methods. The first one is the normal approximation given in (2), the second is from the table given in Franke and Jayachandran (1984) and the third is the Monte Carlo method. In order to check the consistency of the Monte Carlo method, we ran 100 independent simulations; each simulation consisted of 1000 values of F, from which the percentiles were figured.…”

“…, B, = (x(, -,,, a), will serve as the statistical equivalent blocks (SEB). We include the example given in Franke and Jayachandran (1984). A sample of size n-1=23 is taken from a continuous distribution, arranged in increasing order: 11.86, 11.89, 11.90, 11.94, 11.97, 11.98, 12.04, 12.07, 12.08, 12.09, 12.11, 12.14, 12.15, 12.16, 12.19, 12.23, 12.24, 12.25, 12.30, 12.31, 12.34, 12.47, 12.48. To test the null hypothesis that the sample is from a normal distribution with mean p=12.0 and standard deviation a=0.2, we compute the Foutz's test statistic where Di=P,(Bi) [for details of the computation see Franke and Jayachandran (1984)J We then proceed with the Monte Carlo method described in Section 2 and the p-value=0.800 is figured based on 1000 simulated values.…”

“…But, even for small sample sizes, the null distribution of F, is very difficult to obtain. Franke and Jayachandran (1984) carried out a simulation study and found that Foutz's normal approximation is not satisfactory when the sample size is small. They then proposed an improved approximation and provided a table for the percentiles of the null distribution of F,.…”

“…Foutz (1980) provided a large sample normal approximation where @ is the standard normal CDF. Franke and Jayachandran (1984) carried out Monte Carlo study and found that the approximation in (1) needs to be improved. They therefore improved the approximation and obtained a table of the percentiles of the null distribution of F,.…”

“…In Section 2, a method of approximating the null distribution based on Monte Carlo technique is proposed and compared with Foutz's large sample normal approximation and Franke and Jayachandran's (1984) approximation. Actual data sets are analyzed and the p-values are approximated by the proposed Monte Carlo method in Section 3.…”