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Wald Test
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import numpy as np | |
import matplotlib.pyplot as plt | |
def simulate_bernoulli_trials(p, n): | |
return np.random.binomial(1, p, n) | |
def wald_test_statistic(data, p_null): | |
p_hat = np.mean(data) | |
se = np.sqrt(p_hat * (1 - p_hat) / len(data)) | |
return (p_hat - p_null) / se | |
def rejection_probability(lambda_value, n_trials, p_null=0.5, threshold=1.645): | |
p_values = [p_null - lambda_value / np.sqrt(n) for n in n_trials] | |
print(p_values) | |
rejection_rates = [] | |
wald_stats = [] | |
for n, p in zip(n_trials, p_values): | |
data = simulate_bernoulli_trials(p, n) | |
wald_stat = wald_test_statistic(data, p_null) | |
rejected = 1 if wald_stat > threshold else 0 | |
wald_stats.append(wald_stat) | |
rejection_rates.append(rejected) | |
return rejection_rates, wald_stats | |
plt.figure(figsize=(10, 6)) | |
n_trials = list(range(100, 10000001, 50000)) | |
# Parameters | |
lambda_value = 0.01 | |
rejection_rates, wald_stats = rejection_probability(lambda_value, n_trials, p_null=0.5) | |
print(lambda_value, rejection_rates, wald_stats) | |
# plt.plot(n_trials, rejection_rates, marker='v', color='orange') | |
plt.plot(n_trials, wald_stats, marker='v', color='orange', label=r'$\lambda='+str(lambda_value)+'$') | |
# Parameters | |
lambda_value = 0.1 | |
rejection_rates, wald_stats = rejection_probability(lambda_value, n_trials, p_null=0.5) | |
print(lambda_value, rejection_rates, wald_stats) | |
# plt.plot(n_trials, rejection_rates, marker='o', color='blue') | |
plt.plot(n_trials, wald_stats, marker='o', color='blue', label=r'$\lambda='+str(lambda_value)+'$') | |
# Parameters | |
lambda_value = 1 | |
rejection_rates, wald_stats = rejection_probability(lambda_value, n_trials, p_null=0.5) | |
print(lambda_value, rejection_rates, wald_stats) | |
# plt.plot(n_trials, rejection_rates, marker='x', color='red') | |
plt.plot(n_trials, wald_stats, marker='x', color='red', label=r'$\lambda='+str(lambda_value)+'$') | |
# Parameters | |
lambda_value = 2 | |
rejection_rates, wald_stats = rejection_probability(lambda_value, n_trials, p_null=0.5) | |
print(lambda_value, rejection_rates, wald_stats) | |
# plt.plot(n_trials, rejection_rates, marker='+', color='green') | |
plt.plot(n_trials, wald_stats, marker='+', color='green', label=r'$\lambda='+str(lambda_value)+'$') | |
# Plotting | |
plt.hlines(1.6445, 0, n_trials[-1], label=r'Rejection $Z>1.6445$') | |
# plt.plot(n_trials, rejection_rates, marker='o') | |
plt.xlabel('Sample Size n') | |
plt.ylabel('Wald Test value') | |
plt.title('Rejection Probability of $H_0$ vs. Sample Size') | |
plt.grid(True) | |
plt.legend() | |
plt.show() |
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