随机向量 x服从 p 元正态分布 ,回归系数b , 给定 的条件下,y是0,1,y等于1的概率是标准正态分布到bx的积分(iv)用信赖域算法和局部线性近似 编程实现b的最大似然估计 从上述模型中产生独立同分布观测样本 .的 python代码(不使用minize函数和optimize包并且产生结果)
时间: 2024-01-22 20:17:34 浏览: 17
以下是一个可能的实现,其中使用了信赖域算法和局部线性近似来进行最大似然估计:
```python
import numpy as np
from scipy.stats import norm
# 生成随机观测样本
p = 3
n = 100
x = np.random.normal(size=(n, p))
b = np.array([1, 2, 3])
eps = np.random.normal(size=n)
y = (norm.cdf(np.dot(x, b)) + eps) > 0.5
# 定义似然函数和其导数
def log_likelihood(b):
log_l = 0
for i in range(n):
xb = np.dot(x[i], b)
if y[i] == 1:
log_l += np.log(norm.cdf(xb))
else:
log_l += np.log(1 - norm.cdf(xb))
return log_l
def log_likelihood_grad(b):
grad = np.zeros(p)
for i in range(n):
xb = np.dot(x[i], b)
cdf = norm.cdf(xb)
pdf = norm.pdf(xb)
if y[i] == 1:
grad += x[i] * pdf / cdf
else:
grad -= x[i] * pdf / (1 - cdf)
return grad
# 定义信赖域算法
delta_0 = 1
eta_1 = 0.25
eta_2 = 0.75
eta_3 = 2
tol = 1e-6
max_iter = 100
b_hat = np.zeros(p)
delta = delta_0
for k in range(max_iter):
# 计算近似模型
f_k = lambda b: log_likelihood(b_hat) + np.dot(b - b_hat, log_likelihood_grad(b_hat)) + 0.5 * np.dot(b - b_hat, np.dot(delta * np.eye(p), b - b_hat))
f_k_grad = lambda b: log_likelihood_grad(b_hat) + delta * (b - b_hat)
# 求解近似模型的最小值
p_k = -f_k_grad(b_hat) / delta
b_new = b_hat + p_k
# 计算实际下降量和预测下降量
actual_decrease = log_likelihood(b_new) - log_likelihood(b_hat)
predicted_decrease = -np.dot(f_k_grad(b_hat), p_k) - 0.5 * np.dot(p_k, np.dot(delta * np.eye(p), p_k))
rho_k = actual_decrease / predicted_decrease
if rho_k < eta_1:
delta = delta * 0.25
elif rho_k >= eta_2:
delta = min(eta_3 * delta, delta_0)
# 更新估计值
if rho_k >= eta_1:
b_hat = b_new
# 判断收敛
if np.linalg.norm(p_k) < tol:
break
print("最大似然估计值为:", b_hat)
```