Chi-Squared GOF Test for finitely many values

Null Hypothesis

H_{0}:F= G

Test Statistic

\sum_{j=1}^{s}{\frac{(N_{j}-np_{j})^2}{np_{j}}}

data sequence size $n$
frequencies $n_s$ of Item Expression $\xi_s$ (in Random Variable $N_j$ )
probability of Item Expression $p_j$
Some data sequence from unknown CDF $F$ .
A known discrete CDF $G$

Given $H_0$ the Test Statistic is small. Thats why the p-Value is given by

p(t)=\mathbb{P}(T_{n}\geq t)

We can simulate the test statistic by n-fold drawing with replacement from balls with probability $p_{j}$ .

Or use the fact that for $n\rightarrow\infty$ the CDFs of the test statistic will converge to the Chi-Squared Distribution with $s-1$ Degrees of Freedom. This approximation is only good if the Class Condition is met. Otherwise use simulation above.

Example Code

Estimate parameters with Plug-In-Method.

N = 6
n = 100

# Data with unknown probability
x = rbinom(n, 5, 0.8)

# Estimate probability
pt = mean(x) / N

# Use estimates proability to get "real" probabilities for all N outcomes
p = dbinom(0:N, N, pt)

# Calculuate the empirical relative frequencies (probabilities) for our data
tab = c()
for (i in 0:N){
  tab[i+1] = sum(x == i)
}

# Calculate Test statistic (more or less: difference between real and empirical probabilities)
T = sum((tab - n*p)**2 / (n*p))
T

# Use CDF of chi squared to determine p-Value of test statistic
pval = 1 - pchisq(T, df=N-1)
pval