Lilliefors Test of Normality

Test for class of normal distributions.

We can write the class or set of normal distributions as: $\left\{N\left(\mu ,{\sigma }^2\right)\mid \mu \in \mathbb{R},{\sigma }^2>0\right\}$

Our Null Hypothesis thus is $H_0:F\in \left\{N\left(\mu ,{\sigma }^2\right)\mid \mu \in \mathbb{R},{\sigma }^2>0\right\}$ or in words, “is the CDF of our data sequence in the class of normal distributions?”

We define our Test Statistic $L_n$ as the supremum of the absolute distances between the ECDF and the CDF of a normal distribution with estimated Arithmetic Mean and Variance.

The distribution of $L_n$ is independent of the Variance and Expectation of the Normal Distribution and may thus be simulated.

To make up for the increasing $n$ we use the square root of it to scale the distribution.

We accept $H_0$:

• if the test statistic is $\le \frac{L_{1-\alpha }}{\sqrt{n}}$
• if $\alpha <p(l)$ with $p(l)=1-L(\sqrt{n}l)$ the p-Value of the test.

Usage in R

todo