R Under development (unstable) (2025-12-21 r89216 ucrt) -- "Unsuffered Consequences"
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> # library(numDeriv)
> #
> # # 1. 数据准备
> # set.seed(123)
> # n <- 1000
> # input_x <- runif(n, 0, 10) # 建议变量名也避开 x
> # obs_y <- 2 + 5 * input_x + rnorm(n, 0, 1.5)
> #
> # # 2. 定义函数：将参数名改为完全不冲突的名称（如 my_x, my_y）
> # nll_func_numeric <- function(theta_val, my_y, my_x) {
> #   b0 <- theta_val[1]
> #   b1 <- theta_val[2]
> #   sig <- theta_val[3]
> #
> #   # 逻辑不变
> #   y_hat <- b0 + b1 * my_x
> #   res <- my_y - y_hat
> #   n <- length(my_y)
> #
> #   nll <- (n/2) * log(2 * pi) + n * log(sig) + sum(res^2) / (2 * sig^2)
> #   return(nll)
> # }
> #
> # # 3. 估计值
> # theta_est <- c(intercept = 2.01, slope = 4.99, sigma = 1.49)
> #
> # # 4. 计算 Hessian
> # # 现在我们可以放心地按位置传递参数了
> # hess_matrix <- hessian(
> #   func = nll_func_numeric,
> #   x = theta_est,    # hessian 要求的求导中心点
> #   my_y = obs_y,     # 显式指定传给 nll_func_numeric 的 my_y
> #   my_x = input_x    # 显式指定传给 nll_func_numeric 的 my_x
> # )
> #
> # # 5. 计算结果
> # print("--- 观测 Hessian 矩阵 ---")
> # print(hess_matrix)
> #
> # # 协方差矩阵与标准误
> # cov_matrix <- solve(hess_matrix)
> # se <- sqrt(diag(cov_matrix))
> # names(se) <- names(theta_est)
> #
> # cat("\n--- 参数估计的标准误 ---\n")
> # print(se)
> 
> proc.time()
   user  system elapsed 
   0.14    0.01    0.14