P-value calculation for various normality tests.

ad_pval(x, stdn = TRUE)

sw_pval(x)

bc_pval(x)

Arguments

x: Vector of observations.
stdn: Logical, whether or not to test against standard normal distribution, or take mean and variance to be unknown.

Value

p-value of the normality test.

Details

The following normality tests are currently implemented:

ad_pval

The Anderson-Darling test, which assumes iid observations. This is merely a wrapper to nortest::ad.test.

sw_pval

A modified Shapiro-Wilk test, which also assumes iid observations. For sample sizes less than 5000, this function is merely a wrapper to stats::shapiro.test(). For sample sizes greater than 5000, the algorithm due to Royston (1992) used in stats::shapiro.test breaks down. The modification proposed here is to (i) randomly divide the sample into K roughly equal parts each less than 5000, (ii) calculate the Shapiro-Wilk p-value for each, and (iii) note that the minimum of K independent p-values has a Beta(1, K) distribution. It is this second-stage p-value whic is returned by sw_pval.

bc_pval

The so-called "Berkowitz' Correlation" test. That is, An autoregressive model of the form


x[n] = rho * x[n-1] + z[n],    z[n] ~iid N(0,1)

is fit to the data, and a likelihood ratio test against H0: rho = 0 is performed.

Note

The tests above only check for normality, not that x is iid N(0,1). Such modifications will need to be made before the package is released.

References

Royston, P. "Approximating the Shapiro-Wilk W-test for non-normality." Statistics and Computing 2:3 (1992): 117-119. https://doi.org/10.1007/BF01891203.

Berkowitz, J. "Testing density forecasts, with applications to risk management." Journal of Business and Economic Statistics 19:4 (2001): 465-474. https://doi.org/10.1198/07350010152596718.