R: P-Value for the Asymptotic Anderson-Darling Test Distribution

ad.pval {kSamples} | R Documentation

## P-Value for the Asymptotic Anderson-Darling Test Distribution

### Description

This function computes upper tail probabilities for the limiting distribution of the standardized Anderson-Darling test statistic.

### Usage

    ad.pval(tx,m,version=1)

### Arguments

`tx` | a vector of desired thresholds ≥ 0
`m` | The degrees of freedom for the asymptotic standardized Anderson-Darling test statistic
`version` | `= 1` (default) if P-value for version 1 of the test statistic is desired, otherwise the version 2 P-value is calculated.

### Details

Extensive simulations (sampling from a common continuous distribution) were used to extend the range of the asymptotic P-value calculation from the original [.01,.25] in Table 1 of the reference paper to 36 quantiles corresponding to P = .00001, .00005, .0001, .0005, .001, .005, .01, .025, .05, .075, .1, .2, .3, .4, .5, .6, .7, .8, .9, .925, .95, .975, .99, .9925, .995, .9975, .999, .99925, .9995, .99975, .9999, .999925, .99995, .999975, .99999. Note that the entries of the original Table 1 were obtained by using the first 4 moments of the asymptotic distribution and a Pearson curve approximation.

Using `ad.test`, 1 million replications of the standardized AD statistics with sample sizes n.i=500, i=1,…,k were run for k=2,3,4,5,7 (k=2 was done twice). These values of k correspond to degrees of freedom m=k-1=1,2,3,4,6 in the asymptotic distribution. The random variable described by this distribution is denoted by T_m. The actual variances (for n_i=500) agreed fairly well with the asymptotic variances.

Using the convolution nature of the asymptotic distribution, the performed simulations were exploited to result in an effective simulation of 2 million cases, except for k=11, i.e., m=k-1=10, for which the asymptotic distribution of T_{10} was approximated by the sum of the AD statistics for k=7 and k=5, for just the 1 million cases run for each k.

The interpolation of tail probabilities P for any desired k is done in two stages. First, a spline in 1/sqrt(m) is fitted to each of the 36 quantiles obtained for m=1,2,3,4,6,8,10,∞ to obtain the corresponding interpolated quantiles for the m in question.

Then a spline is fitted to the log((1-P)/P) as a function of these 36 interpolated quantiles. This latter spline is used to determine the tail probabilities P for the specified threshold `tx`, corresponding to either AD statistic version. The above procedure is based on simulations for either version of the test statistic, appealing to the same limiting distribution.

### Value

a vector of upper tail probabilities corresponding to `tx`

### References

Scholz, F. W. and Stephens, M. A. (1987), K-sample Anderson-Darling Tests, Journal of the American Statistical Association, Vol 82, No. 399, 918–924.

### See Also

`ad.test`, `ad.test.combined`

### Examples

    ad.pval(tx=c(3.124,5.65),m=2,version=1)
    ad.pval(tx=c(3.124,5.65),m=2,version=2)

[Package kSamples version 1.2-9 Index]