查看“︁半参数回归”︁的源代码

{{回归侧栏}}
[[统计学]]中，'''半参数回归'''包括结合了[[参数模型]]和[[核回归|非参数]]模型的回归模型。它们通常用于完全非参数模型可能表现不佳的情况，或者研究人员希望使用参数模型，但与回归子集有关的函数形式或误差密度不为人知的情况。半参数回归模型是[[半参数模型|半参数建模]]的一种特殊类型。半参数模型包含参数成分，依赖于参数假设，可能会出现规范误差与[[一致估计量|不一致]]的情况。

== 方法 ==
目前已有许多不同的半参数回归方法。最流行的方法是部分线性模型、指数模型和变系数模型。

=== 部分线性模型 ===
[[部分线性模型]]如下

: <math> Y_i = X'_i \beta + g\left(Z_i \right) + u_i, \, \quad  i = 1,\ldots,n, \, </math>

其中<math> Y_{i} </math>是因变量，<math> X_{i} </math>是解释变量的<math> p \times 1 </math>向量，<math> \beta </math>是未知参数的<math> p \times 1 </math>向量，<math>  Z_{i} \in \operatorname{R}^{q} </math>。部分线性模型的参数部分由参数向量<math> \beta </math>给出，而非参数部分是未知函数<math> g\left(Z_{i}\right) </math>。假设数据与<math> E\left(u_{i}|X_{i},Z_{i}\right) = 0 </math>独立同分布，模型允许未知形式的条件[[异方差]]误差过程<math> E\left(u^{2}_{i}|x,z\right) = \sigma^{2}\left(x,z\right) </math>。这类模型由Robinson (1988)提出，并由Racine & Li (2007)扩展到处理分类协变量。

这种方法先获得<math> \beta </math>的<math> \sqrt{n} </math>一致估计量，然后用适当的非参数回归方法，从<math> Y_{i} - X'_{i}\hat{\beta} </math>对<math> z </math>的[[核回归|非参数回归]]中推出<math> g\left(Z_{i}\right) </math>的估计量。<ref>See Li and Racine (2007) for an in-depth look at nonparametric regression methods.</ref>

=== 指数模型 ===
单一指数模型的形式是

: <math> Y = g\left(X'\beta_{0}\right) + u, \, </math>

其中<math> Y </math>、<math> X </math>、<math> \beta_{0} </math>的定义与上文相同，误差项<math> u </math>满足<math> E\left(u|X\right) = 0 </math>。单一指数模型得名于模型的参数部分<math> x'\beta </math>，是标量单指数。非参数部分是未知函数<math> g\left(\cdot\right) </math>。

==== 市村法 ====
市村(1993)提出的单一指数模型法如下。考虑<math> y </math>连续情形，给定函数<math> g\left(\cdot\right) </math>的已知形式，<math> \beta_{0} </math>可用非线性最小二乘法估计，使函数

: <math> \sum_{i=1} \left(Y_i - g\left(X'_i \beta\right)\right)^2. </math>

最小化。<math> g\left(\cdot\right) </math>的函数形式未知，需要估计。对给定<math> \beta </math>值，函数估计值可用[[核密度估计]]得到，为

: <math> G\left(X'_i \beta \right) = E\left(Y_i |X'_i \beta\right) = E\left[g\left(X'_i\beta_o \right)|X'_i \beta\right] </math>

市村(1993)建议用下式估计<math> g\left(X'_{i}\beta\right) </math>：

: <math> \hat{G}_{-i}\left(X'_i \beta\right),\, </math>

为<math> G\left(X'_{i}\beta\right) </math>的[[重抽样|留一]][[核密度估计|非参数核]]估计量.

==== Klein与Spady估计量 ====
Klein & Spady (1993)提出，若因变量<math> y </math>是二元的，并假设<math> X_{i} </math>、<math> u_{i} </math>[[独立 (概率论)|独立]]，则可用[[最大似然估计]]法估计<math> \beta </math>。对数似然函数为

: <math> L\left(\beta\right) = \sum_i \left(1-Y_i\right)\ln\left(1-\hat{g}_{-i}\left(X'_i\beta\right)\right) + \sum_{i}Y_i\ln\left(\hat{g}_{-i}\left(X'_i \beta\right)\right), </math>

其中<math> \hat{g}_{-i}\left(X'_{i}\beta\right) </math>是留一估计量。

=== 平滑系数/变系数模型 ===
Hastie & Tibshirani (1993)提出了一种平滑系数模型

: <math> Y_i = \alpha\left(Z_i\right) + X'_i\beta\left(Z_i\right) + u_i
 = \left(1 + X'_i\right)\left(\begin{array}{c} \alpha\left(Z_i\right) \\ \beta\left(Z_i\right) \end{array}\right) + u_i
 = W'_i\gamma\left(Z_i\right) + u_i, </math>

其中<math> X_{i} </math>是<math> k \times 1 </math>向量，<math> \beta\left(z\right) </math>是<math> z </math>的未定平滑函数向量。

<math> \gamma\left(\cdot\right) </math>可表为

: <math> \gamma\left(Z_i\right) = \left(E\left[W_i W'_i|Z_i \right]\right)^{-1}E\left[W_i Y_i|Z_i\right]. </math>

== 另见 ==
* [[非参数回归]]

== 注释 ==
{{reflist}}

== 参考文献 ==
* {{cite journal
 | last = Robinson
 | first = P.M.
 | title = Root-''n'' Consistent Semiparametric Regression
 | url = https://archive.org/details/sim_econometrica_1988-07_56_4/page/931
 | journal = Econometrica
 | volume = 56
 | issue = 4
 | pages = 931–954
 | year = 1988
 | doi =10.2307/1912705
 | jstor = 1912705
 | publisher = The Econometric Society
 }}
* {{cite book
 | last = Li
 | first = Qi
 |author2=Racine, Jeffrey S.
  | title = Nonparametric Econometrics: Theory and Practice
 | publisher = Princeton University Press
 | year = 2007
 | isbn = 978-0-691-12161-1
 }}
* {{cite journal
 | last = Racine
 | first = J.S.
 |author2=Qui, L.
  | title = A Partially Linear Kernel Estimator for Categorical Data
 | journal = Unpublished Manuscript, Mcmaster University
 | year = 2007
 }}
* {{cite journal
 | last = Ichimura
 | first = H.
 | title = Semiparametric Least Squares (SLS) and Weighted SLS Estimation of Single Index Models
 | journal = Journal of Econometrics
 | volume = 58
 | issue = 1–2
 | pages = 71–120
 | year = 1993
 | doi =10.1016/0304-4076(93)90114-K
 | url = http://purl.umn.edu/55563
 }}
* {{cite journal
 | last = Klein
 | first = R. W.
 |author2=R. H. Spady
  | title = An Efficient Semiparametric Estimator for Binary Response Models
 | url = https://archive.org/details/sim_econometrica_1993-03_61_2/page/387
 | journal = Econometrica
 | volume = 61
 | issue = 2
 | pages = 387–421
 | year = 1993
 | doi =10.2307/2951556
 | jstor = 2951556
 | publisher = The Econometric Society
 | citeseerx = 10.1.1.318.4925
 }}
* {{cite journal
 | last = Hastie
 | first = T.
 |author2=R. Tibshirani
  | title = Varying-Coefficient Models
 | url = https://archive.org/details/sim_journal-of-the-royal-statistical-society-series-b_1993_55_4/page/757
 | journal = Journal of the Royal Statistical Society, Series B
 | volume = 55
 | pages = 757–796
 | year = 1993
 }}

{{统计学}}

[[Category:非参数统计]]