Chi-Squared GOF Test for finitely many values

Null Hypothesis $H_0:F=G$

Test Statistic ${\sum }_{j=1}^s\frac{(N_j-n{p}_j{)}^2}{n{p}_j}$

• 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 $p(t)=\mathbb{P}(T_n\ge t)$

We can simulate the test statistic by n-fold drawing with replacement from balls with probability $p_j$.

Or use the fact that for $n\to \infty$ 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