1
$\begingroup$

Suppose one would like to test that a sample of observations comes from Uniform(0,1) distribution. Instead of applying the Kolmogorov-Smirnov test on the sample, one may first apply the inverse CDF (quantile function) of $N(0,1)$ random variable and then apply a likelihood ratio (LR) test for $H_0\colon \mu=0,\sigma^2=1$ on these transformed observations. In finance, this idea dates back to Berkowitz "Testing Density Forecasts, With Applications to Risk Management" (2001) who refers back to Rosenblatt "Remarks on a Multivariate Transformation" (1952). Berkowitz suggests this trick allows us to gain power as LR test is uniformly most powerful (UMP).

I wonder where this power gain is coming from and at what expense (unless this is free lunch). And if this is free lunch, should we never use one-sample Kolmogorov-Smirnov test but always apply the transformation?

$\endgroup$

1 Answer 1

1
$\begingroup$

There is no free lunch. Berkowitz (2001) himself explains that the LR test is UMP w.r.t. a one-sided alternative such as $\mu<0,\sigma^2>1$. The power comes from going from a nonparametric to a parametric test. This entails assuming the shape of a normal curve for the transformed observations and then only examining the two moments, mean and variance. Thus it assumes away a number of violations of normality of the transformed data (uniformity of original data).

Consequently, it is not true that we should never use the Kolmogorov-Smirnov test on the original sample but always apply the transformation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.