The random coefficients model $Y_i={\beta_0}_i+{\beta_1}_i {X_1}_i+{\beta_2}_i {X_2}_i+\ldots+{\beta_d}_i {X_d}_i$, with $\mathbf{X}_i$, $Y_i$, $\mathbf{\beta}_i$ i.i.d, and $\mathbf{\beta}_i$ independent of $X_i$ is often used to capture unobserved heterogeneity in a population. We propose a quasi-maximum likelihood method to estimate the joint density distribution of the random coefficient model. This method implicitly involves the inversion of the Radon transformation in order to reconstruct the joint distribution, and hence is an inverse problem. Nonparametric estimation for the joint density of $\mathbf{\beta}_i=({\beta_0}_i,\ldots, {\beta_d}_i)$ based on kernel methods or Fourier inversion have been proposed in recent years. Most of these methods assume a heavy tailed design density $f_\mathbf{X}$. To add stability to the solution, we apply regularization methods. We analyze the convergence of the method without assuming heavy tails for $f_\mathbf{X}$ and illustrate performance by applying the method on simulated and real data. To add stability to the solution, we apply a Tikhonov-type regularization method.
翻译:随机系数模型 $Y_i ⁇ _0 ⁇ _0 ⁇ ⁇ ⁇ _1 ⁇ i {X_1 ⁇ ⁇ ⁇ ⁇ ⁇ _2 ⁇ i{X_2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ i{X_d ⁇ i$, $Y_i}, $mathbf_beta ⁇ i$ i.d, $mathbf_beta# ⁇ i=$X_i}, 通常用来捕捉人群中未观测到的异性。 我们提议了一种准最大可能性的方法来估计随机系数模型的联合密度分布。 这个方法隐含着对Radon变换以重建联合分布, $y_i, $_i, $mathb ⁇ i i, 和 $xxx 的双数估计, Tieta_d ⁇ i} 以 yeta_d ⁇ _i yleglegle =xxxx 常规方法, 假设了这些稳定度 的正态方法。