부분 선형 모형

부분 선형 모델은 모수 및 비모수 요소를 포함하므로 반파라메트릭 모델의 한 형태입니다.알려진 비모수 원소의 가설이 유효한 경우 부분 선형 모형에 최소 제곱 추정기를 적용할 수 있습니다.부분 선형 방정식은 Engle, Granger, Rice 및 Weiss(1986)에 의해 온도와 전기 사용 간의 관계를 분석하는 데 처음 사용되었다.1997년 기업의 생산 수익성의 경우, 미시경제학 분야에서의 부분 선형 모델의 전형적인 적용은 트리파티에 의해 제시되었다.또한 일부 다른 학문 분야에서도 부분 선형 모델이 성공적으로 적용되었습니다.1994년, Zeger와 Diggle은 생체 인식에 부분 선형 모델을 도입했습니다.환경과학에서는 Parda-Sanchez 외.2000년에 수집된 데이터를 분석하기 위해 부분 선형 모델을 사용했습니다.지금까지 부분 선형 모델은 다른 많은 통계 방법에서 최적화되었다.1988년 로빈슨은 최소 제곱 추정기를 만들기 위해 비모수 요소를 테스트하기 위해 Nadaraya-Washton 커널 추정기를 적용하였고, 그 후 1997년 Truong에 의해 국소 선형 방법이 발견되었다.

부분 선형 모형

개요

대수 방정식

부분 선형 모델의 대수식은 다음과 같이 작성된다.

$\displaystyle y_{i}=\display_{T}^{i}\beta +f(T_{i})+\mu _{i}$ ^[1]

방정식 성분 개요

$§$ $Ti \displaystyle$ \ $delta _{T}^$ i $\delta _{T}^{i}$ } $T_{i}$ \displaystyle $T_{$ i $T_{i}$ : $\delta _{T}^{i}$ 변수의 벡터.독립적으로 랜덤 또는 고정 분포 변수.

$\beta$ {\ $displaystyle \beta$ $\beta$ : 측정 파라미터.

$\mu _{i}$ i $\mu _{i}$ \ $displaystyle$ \ $mu$ _ { i $\mu _{i}$ : 통계정보의 랜덤에러(0 평균).

$f(T_{i})$ ( $f(T_{i})$ ) { $displaystyle$ f $(T_{i})$ : 부분 선형 모델에서 부품을 측정합니다.

가정^[1]

볼프강, 화량, 지티 가오는 고정 및 랜덤 설계 조건 하에서 부분 선형 모델의 가정과 발언을 고려한다.

무작위로 분포된 경우,

$L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ j ( $L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ ) $L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ ( $L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ i , $L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ ) { $displaystyle L_{j}(T_{i$ }) = $E$ (\ $delta$ _ {i,j} $T_{i})$ 및 $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ $=$ $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ , $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ - $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ ( $\mu _{i,j}=\delta _{i,j}-E(\delta _{i,j}|T_{i})$ i, $)$ _ $display$ style \ $mu$ {i} _ $L_{j}(T_{i})=E(\delta _{i,j}|T_{i})$ { $i}$

$E(||\delta _{1}||^{3})|T=t)$ ( $E(||\delta _{1}||^{3})|T=t)$ $E(||\delta _{1}||^{3})|T=t)$ ) $E(||\delta _{1}||^{3})|T=t)$ $E(||\delta _{1}||^{3})|T=t)$ ( \ $displaystyle E$ ( \ $displaystyle$ _ { $1}$ ^{3 $}$ T $= t$ ) } t $E(||\delta _{1}||^{3})|T=t)$ 값이 0 ~ 1일 때 양의 무한대보다 작으며, $\delta _{1}-E(\delta _{1}|T_{1})$ 1 - $\delta _{1}-E(\delta _{1}|T_{1})$ ( $\delta _{1}-E(\delta _{1}|T_{1})$ 1 $\delta _{1}-E(\delta _{1}|T_{1})$ ) { $display$ \ style _ 1 - $E$ ( \ $delta _ 1$ } } T } { 1 $\delta _{1}-E(\delta _{1}|T_{1})$ } { 1 } } } } ） e e e e e e e e when when when when when when when when when when when when when when when when when when when when when when when when when when when when랜덤 에러 「」는 $(\delta _{i},T_{i})$ ( 「 $(\delta _{i},T_{i})$ 」 , $)(\displaystyle (\delta$ _ ${i})$ 와는 무관합니다. $T_{i$

$\delta _{i}$ i \ $display$ style \ $display style$ $\delta _{i}$ _ { $i$ } $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ 0 ~1의 L $L_{j}$ $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ _ { $j$ $L_{j}$ } $L_{j}$ , $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ $\delta _{i}$ $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ \ $display$ style _ { i $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ } \ i $=$ $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ j ( $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ ) + $\delta _{ij}=L_{j}(T_{i})+\mu _{ij}$ i \ $display$ style \ $display style$ _ { i j $} =$ $L_{j}(T_{$ i})+\ $mu$ $_$ {ij $}.$ 여기서 인자 i 값은 1 ~n, 인자 $\mu _{ij}$ j 값은 1 ~ $p$ 이며, 에러 $\mu _{ij}$ $}$ 는 $\mu _{ij}$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ n $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ / n $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ 1 $\lim _{n\to \infty }1/n\sum _{i=1}^{n}\mu _{i}\mu _{i}^{T}=\sum$ n $μi$ to \ $ftylim$ $\mu _{ij}$ { $n }$ 을 만족합니다 $.$ $T}=\sum$

최소 제곱(LS) 추정치^[1]

최소 제곱 추정기를 적용하는 전제 조건은 비모수 성분의 존재이며 랜덤 분포 및 고정 분포 사례에서 실행됩니다.

최소 제곱 추정기를 적용하기 전에 Engle, Granger, Rice 및 Weiss의 평활화 모델을 먼저 도입해야 합니다.모델의 대수 함수는 Y $=$ $Y=\delta ^{T}\beta +f(t)$ $Y=\delta ^{T}\beta +f(t)$ β $Y=\delta ^{T}\beta +f(t)$ ( $Y=\delta ^{T}\beta +f(t)$ ) $Y=\delta ^{T}\beta +f(t)$ { $displaystyle$ Y=\ $delta$ ^{ $T}\beta +f(t)}($ 2)로 $Y=\delta ^{T}\beta +f(t)$ 된다.

볼프강, 량, 가오(1983)는 쌍이 $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ 1/ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ i $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ E { $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ - $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ i $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ β - $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ ( $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ 1 / $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ E { $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ - $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ - $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ - $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ / $display$ $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ $style$ 을 $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$ 한다고 가정한다 $.$ $Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{$ $Y_{i}-\delta$ _ ${i}^T}-f(T_{i}\}^2}}($ 3). $1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}\beta -f(T_{i})\}^{2}=min1/n\textstyle \sum _{i=1}^{n}\displaystyle E\{Y_{i}-\delta _{i}^{T}-f(T_{i})\}^{2}$

즉, $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 1 ≤ i $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ + $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 1 ( $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ) $= t$ $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 2 $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ + f $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ( Ti $\delta _{i}^{T}\beta _{1}+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ){ $displaystyle$ _ { $i}^{T}\beta$ _ { $1}+f_{$ 1}( $T_{i$ }=\ $delta _{i$ }^{f} $_{f$ } $_beta}$ _beta } _bet2 .

$f_{1}=f_{2}and\beta _{1}=\beta _{2}$ , $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ a $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ d $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ $=$ $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ 2 ({ $displaystyle$ f_{1}= $f_{$ 2} $and\solid$ _ ${1$ }=\ $solid _$ {2 $f_{1}=f_{2}and\beta _{1}=\beta _{2}$ 입니다.

무작위 분포 사례에서 볼프강, 화량 및 지티 가오는 모든 1µi ≤ n, $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ [ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ( $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ i , $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ) $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 1 + $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 1 ( $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $i$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 2 + $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ( $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ ) \ $display$ E [ \ { $i }$ $]$ 라고 가정한다. $T_{i}]=\delta$ _ ${i}^{T}\beta$ 1 $+f_{1}(T_{i$ })=\ $delta$ _{ $i}^{T}\beta$ _ ${2}+f_{2}(T_{i$ })}( $E[Y_{i}|(\delta _{i},T_{i})]=\delta _{i}^{T}\beta 1+f_{1}(T_{i})=\delta _{i}^{T}\beta _{2}+f_{2}(T_{i})$ 4)

$E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ E $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ { $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ i $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ 1 $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ } $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ { $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ T $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ 2 - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ 2 ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ) $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ + ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ) T $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ E { ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ i $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - E [ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ] $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ I $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ) ( $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ I $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ - $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ ) $E\{Y_{i}-\delta _{i}^{T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{Y_{i}-\delta _{i}^{T}\beta _{2}-f_{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2})^{T}E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}(\beta _{1}-\beta _{2})$ $Y_{i}-\delta _{i}^T}\beta _{1}-f_{1}(T_{i})\}^{2}=E\{}$ $Y_{i}-\delta _{i}^{T}\beta _{2}(T_{i})\}^{2}+(\beta _{1}-\beta _{2}^{T}E\{(\delta _i}_{i})_{delta}_{i}_{delta}_{deli})_{i}$ $T)\}({1$ }-\ $display_{$ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ $})$ $\beta _{1}=\beta _{2}$ 1 $=$ $\beta _{1}=\beta _{2}$ 2(「 $\beta _{1}=\beta _{2}$ $\displaystyle$ _ ${$ $1$ }=\ $display_{$ $2$ }) $\beta _{1}=\beta _{2}$ 、 $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ E $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ [ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ （ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ - $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ ）） $、$ { $displaystyle$ E \ $ti$ 。 $함수$ (1)에서 증명되었듯이 T $}\}$ 은 $E\{(\delta _{i}-E[\delta _{i}|T_{i}])(\delta _{i}-E[\delta _{i}|T_{i}]^{T})\}$ 양수입니다. $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ j ( $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ ) $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ [ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ - E $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ [ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ i $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ $]$ ( \ $display$ style $f$ _ { $j$ （ $T$ _ { $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ ） ]= $E[Y_{i}$ $T_$ $i$ }]-E $[\delta _{i$ }^{T $}\beta _{j} T_{$ i}}는 $f_{j}(T_{i})=E[Y_{i}|T_{i}]-E[\delta _{i}^{T}\beta _{j}|T_{i}]$ f 1 $f_{1}=f_{2}$ = $f_{1}=f_{2}$ $f_{1}=f_{2}$ 1과 $f_{1}=f_{2}$ 됩니다 $.$

고정 분포 케이스에서 스무딩 모델 (2)의 계수를 { $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ 1) , $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ . $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ . $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ . $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ . $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ , $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ f ( $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ ) $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ d $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ r $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ Q ( $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ - x $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ ) { $display$ style \ { $f$ ( $T_{ n$ }^{ }로 파라미터화함 $T}=\opega_{r} 및\opega_{r$ }= $Q($ $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ $\beta)}($ $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ 서 $\{f(T_{1}),.........,f(T_{n})\}^{T}=\omega _{r}and\omega _{r}=Q(Y-x\beta )$ Q $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ ( ( ( T $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ - $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ T { $displaystyle$ Q = \ $obega$ ( \ ^ { $T$ } \ ） ^- $1$ } \ 、、 $obega$ ^{ $T$ } $Q=\omega (\omega ^{T}\omega )^{-1}\omega ^{T}$ ）。

(4)와 동일한 가정 (1)에 따라 $\beta _{1}=\beta _{2}$ $\beta _{1}=\beta _{2}$ $\beta _{1}=\beta _{2}$ $\beta _{1}=\beta _{2}$ 2 ({ $displaystyle \displaystyle _$ ${$ 1} = $f_$ { $2})$ 및 $\beta _{1}=\beta _{2}$ $f_{1}=f_{2}$ 1 $f_{1}=f_{2}$ $f_{1}=f_{2}$ $f_{1}=f_{2}$ ({ $displaystyle f_{1$ } = $f_{$ 2})를 $f_{1}=f_{2}$ 1 $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ / $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ - $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ 1 - $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ 1 - $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ r $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ 1) $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ T ( $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ - $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ )로 가정한다. $1/nE\{(Y-X\beta _{1}-\omega _{r1})^{T}(Y-X\beta _{1}-\omega _{r1})\}=1/nE{(Y-X\beta _{2}-\omega _{r2})^{T}(Y-X\beta _{2}-\omega _{r2})}+1/n(\beta _{1}-\beta _{2})^{T}X^{T}(1-Q)X(\beta _{1}-\beta _{2})$ ${T}(Y-X\beta_{2}-\obega_{r2})+1/n(\beta_{1}-\beta_{2})^$ ${T}X^{T}(1-Q)X(\beta_{1}-\beta_{2$

계수 $\delta _{i},T_{i},Y_{i}$ i $\delta _{i},T_{i},Y_{i}$ $\delta _{i},T_{i},Y_{i}$ 、 $\delta _{i},T_{i},Y_{i}$ \ $displaystyle$ \ $delta$ _ { $\delta _{i},T_{i},Y_{i}$ } 를 전제로 합니다. $T_{i$ }, $Y_{i$ 여기서 i는 양의 정수)는 $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ i $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ i $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ β + $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ ( $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ + $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ i $\psi _{ni}(t)$ { $display y$ _ { i } $=$ \ $display$ _ { $i$ } \ $beta + f$ ( $T$ _ { $i$ } + \ $mu _$ { i } ) $\psi _{ni}(t)$ + $\psi _{ni}(t)$ ( tsi )를 $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ $y_{i}=\delta _{i}^{T}\beta +f(T_{i})+\mu _{i}$ 하고 양의 무게 함수 $\psi _{ni}(t)$ n ( \ $spi )$ 을 설정합니다 $.$ $aystyle \ni$ $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ n ( $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ ; $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ ) $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ i ( $t$ ) （ $Y$ i - $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ $ti$ ） = \ $sum _$ { i $f_{n}(t;\beta )=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta )$ = $1$ }^{ $n}$ \ $psi$ _ { $ni （ Y$ _ i } \ $delta$ } _ { t } ）。 $le \telde _{LS}=\{{\tilde {\tilde$ $T}{\tilde {\delta }\}^{-1}{\tilde {\delta$ $T}{\tilde {Y$ f $)$ { $displaystyle$ f $(n)}$ 의 $f(n)$ 비모수 추정치는 ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ n ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ( ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ) ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ n ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ i ( ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ) ( ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ - ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ S ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ ) { $displaystyle { f _ n }$ ( t ) = ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ 1 $sumi =$ { displaysty } 로 ${\hat {f_{n}}}(t)=\sum _{i=1}^{n}\psi _{ni}(t)(Y_{i}-\delta _{i}^{T}\beta _{LS})$ 된다.ibitted, 분산 $\sigma ^{2}$ ${\$ 2 {\ $displaystyle$ \displaystyle ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ $^{2$ }는 $\sigma ^{2}$ ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ ^ ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ 2 ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ = 1 / ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ 1 n ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ ( ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ ~ - ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ i ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ T ~ $β$ ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ S ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ ) { $displaystyle$ { $hat }$ { $2 =$ 1/ $n sum$ _ ${\hat {\sigma }}_{n}^{2}=1/n\sum _{i=1}^{n}({\tilde {Y_{i}}}-{\tilde {\delta _{i}^{T}}}\beta _{LS})$ i $=$ 1 $n$ } { n } { $n$ } { $tilde$ { $tilde$ $T}}\beta$ _ ${LS$

부분 선형 모델의 역사와 응용

부분 선형 모델의 실제 적용은 1986년 ^[1]Engle, Granger, Rice 및 Weiss의 데이터 분석을 위해 처음 검토되었다.

그들의 관점에서, 평균 소득, 상품 가격, 소비자 구매 능력 및 기타 경제 활동과 같은 교란 요인이 막대하기 때문에 온도와 전기 소비 간의 관련성은 선형 모델로 표현할 수 없다.요인 중 일부는 서로 관련이 있으며 관측 결과에 영향을 미칠 수 있습니다.따라서 모수 인자와 비모수 인자를 모두 포함하는 부분 선형 모형을 도입했습니다.부분 선형 모델은 데이터의 선형 변환을 활성화하고 단순화합니다(Engle, Granger, Rice and Weiss, 1986).또한 연구에 스무딩 스플라인 기법을 적용했습니다.

1994년 Zeger와 Diggle이 생체측정학에서 부분 선형 모델을 적용한 사례가 있다.이 논문의 연구 목표는 HIV(Human Immunity-Deficiency Virus) 혈청변환기에서 CD4 세포량의 진화 주기이다(Zeger and Diggle, 1994).^[2]CD4 세포는 인체 면역 기능에 중요한 역할을 한다.Zeger와 Diggle은 CD4 세포의 변화량을 측정함으로써 질병의 진행을 평가하는 것을 목표로 했다.CD4 세포의 수는 신체 나이와 흡연 행동 등과 관련이 있습니다.실험에서 관측 데이터 그룹을 지우기 위해 Zeger와 Diggle은 작업에 부분 선형 모형을 적용했습니다.부분 선형 모델은 CD4 셀의 평균 손실 시간 추정에 주로 기여하고 데이터 비교 과정을 단순화하기 위해 다른 공변량의 시간 의존성을 조정하며, 부분 선형 모델은 관측된 그룹이 진행 곡선을 추정하기 위한 전형적인 곡선의 편차를 특징짓는다.CD4 셀의 양이 변화하고 있습니다.부분 선형 모델에 의해 부여된 편차는 CD4 세포의 변화량이 느린 관찰 대상을 인식하는 데 잠재적으로 도움이 된다.

1999년 Schmalensee와 Stoker(1999)는 경제학 분야에서 부분 선형 모델을 사용했다.그들의 연구의 독립적인 변수는 미국의 휘발유 수요이다.논문의 주요 연구 대상은 미국의 휘발유 소비와 장기 소득 탄력성의 관계이다. 마찬가지로 교란 변수도 매우 많아 상호 영향을 미칠 수 있다.따라서 Schmalemsee와 Stoker는 부분 선형 ^[3]모델을 적용하여 모수와 비모수 사이의 데이터의 선형 변환 문제를 다루기로 결정했다.

환경과학 분야에서는 2000년(Prada-Sanchez, 2000년)^[4]에는 부분 선형 모델을 사용해 이산화황 오염을 예측했고, 이듬해에는 린과 캐롤이 클러스터 데이터에 부분 선형 모델을 적용했다(Lin and Carroall, 2001년).^[5]

부분 선형 모델 개발

2010년 Liang의 논문(Liang, 2010)에 따르면 스무딩 스플라인 기법은 1986년 Engle, Heckman 및 Rice에 의해 부분 선형 모델로 도입되었다.그 후 Robinson은 1988년 부분 선형 모형에서 비모수 요인에 대해 사용 가능한 LS 추정기를 발견했습니다.같은 해, Specman은 ^[6]프로파일 LS 방식을 추천했습니다.

부분 선형 모델의 기타 계량형 도구

커널 회귀도 부분 선형 모델에 도입되었습니다.Specman이 개발한 국소 상수법이나 1997년 Hamilton과 Truong에 의해 발견되고 1997년 Opsomer와 Ruppert에 의해 수정된 국소 선형 기법은 모두 커널 회귀에 포함된다.Green 등, Opsomer 및 Ruppert는 커널 기반 방법의 중요한 특징 중 하나는 beta의 root-n 추정자를 찾기 위해 언더스무팅이 이루어졌다는 것을 발견했다.그러나 1988년 Specman의 연구와 1994년 Severini와 Staniswalis의 연구는 이러한 제한이 취소될 수 있다는 것을 증명했다.

부분 선형^[6] 모델에서의 대역폭 선택

부분 선형 모델에서의 대역폭 선택은 혼란스러운 문제입니다.Liang은 프로파일 커널 기반 방식과 백핏 방식을 적용하여 이 대역폭 선택에 대한 가능한 해결책을 그의 문헌에서 다루었습니다.또한 백핏 방식을 위한 언더무트의 필요성과 프로파일 커널 기반 방식이 최적의 대역폭 선택을 할 수 있는 이유는 Liang에 의해 정당화되었습니다.일반적인 계산 전략은 비모수 함수를 추정하기 위해 양의 문헌에 적용되었다.또한, 부분 선형 모델에 대한 불이익 스플라인 방법 및 강도 높은 시뮬레이션 실험을 도입하여 불이익 스플라인 방법, 프로파일 및 백핏 방법의 수치적 특징을 발견했다.

커널 기반 프로파일 및 백핏^[6] 방법

$E(Y|T)={E(X|T)}^{T}\beta +g(T)$ ( $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ T ) $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ ( $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ T ) $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ + $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ ( $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ ) { $displaystyle$ E ( Y $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ T ) = {E $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ ( $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ X $E(Y|T)={E(X|T)}^{T}\beta +g(T)$ T ) }^{ $T}$ \ $beta + g$ ( $T$ ) }

$Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ - $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ( $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ T ) $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ( $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ - $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ( $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ) $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ + $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ( \ $display Y$ - E ( Y $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ T ) = $xx$ X - E ( X $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ T ) $Y-E(Y|T)=({X-E(X|T)})^{T}\beta +\epsilon$ ^ ${T}\beta +\epsilon}$

$다음$ 으로 모든 랜덤 벡터 변수 $\xi$ {\({ $displaystyle \xi$ 에 대해 E $E(\xi |T)$ ${\hat {E}}(\xi |T)$ $E(\xi |T)$ ${\hat {E}}(\xi |T)$ T ) ${display {E}(xi$ T $)}$ 가 ${\hat {E}}(\xi |T)$ E( $t$ T $)$ 의 커널 회귀 추정기라고 ${\hat {E}}(\xi |T)$ 합니다. ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ ~ ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ C - ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ E( ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ T ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ $E(\xi |T)$ ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ 。 $E(\xi$ T $),\textstyle \sum$ _ ${X$ T $}\displaystyle$ = $cov{X-E(X$ T ${\tilde {\xi }}=\xi -E(\xi |T),\textstyle \sum _{X|T}\displaystyle =cov{X-E(X|T)}$ 예: ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ ~ ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ $=$ ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ E ( ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ ) ${\tilde {X}}_{i}=X_{i}-E(X_{i}|T_{i})$ { $displaystyle$ { $tilde {X}_{i}-E_{I}.$ $Y_{n}^{$ $T}$ $Y=(Y_{1},...,Y_{n})^{T}$ , g, T는 비슷합니다. $m$ x ( $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ) $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ( X $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ) 、 $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ( $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ) $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ ( X $T =$ t ) $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ = E ( X $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ $T = t$ ) , $m$ _ { $y$ } = $E$ ( Y $T$ = $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ t ) $m_{x}(t)=E(X|T=t),m_{y}(t)=E(Y|T=t)$ E ( $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ T = t )So $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ ） $\psi (m_{x},m_{y},\beta ,Y,X,T)={X-m_{x}(T)}[Y-m_{y}(T)-{X-m_{x}(T)^{T}\beta }]$ ${T}\beta }}$

프로파일-커널 기반 ${\hat {\beta _{p}}}$ ${\hat {\beta _{p}}}$ p ${\hat {\beta _{p}}}$ ^(\ $displaystyle\hat\hat_{p}})$ 는 ${\hat {\beta _{p}}}$ 다음과 같이 해결합니다.

$0=\sum _{i=1}^{n}\psi(\hat {m_{x}}}, {\hat {m_{y}}}, \hat, Y_i}, X_{i})T_{i}$

${\hat {m_{x}}},{\hat {m_{y}}}$ 서 m x ${\hat {m_{x}}},{\hat {m_{y}}}$ , ${\hat {m_{x}}},{\hat {m_{y}}}$ y $^$ { $style {m_{x}},$ {\ $hat {m_{y}}}}$ 은 ${\hat {m_{x}}},{\hat {m_{y}}}$ mx 및 my의 커널 추정치입니다.

불이익 스플라인법^[6]

벌칙 스플라인 방법은 1996년 에일러와 마르크스에 의해 개발되었다.2000년에는 Ruppert와 Carroll, 1999년에는 Brumback, Ruppert 및 Wand가 LME 프레임워크에서 이 방법을 사용했습니다.

$g(t)$ g $g(t)$ $)$ { $displaystyle$ g $(t)}$ 는 $g(t)$ g $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ ) $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ 0 $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ + $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ 1 $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ + . + $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ p t $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ + $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ k $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ k ( $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ - $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ k $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ ) $=$ \ $displaystyle g(t,$ \ $displaystyle$ _ $0$ } + _ _ { 1 $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ } + t로 $g(t,\tau )=\tau _{0}+\tau _{1}t+...+\tau _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k})^{p}$ 할 수 있다고 가정합니다. $+\display _{p}t^{p}+\textstyle \sum _{k=1}^{K}\displaystyle b_{k}(t-\xi _{k}^{p})$

$p\geqslant 1$ 서 p $p\geqslant 1$ 1 { $displaystyle$ p \ $geqslant$ 1 $p\geqslant 1$ an $p\geqslant 1$ 、 $\xi _{1}<...<\xi _{k}$ 1 < . . < ${\$ $k$ \ $displaystyle$ \ $xi$ _ {1} $\xi _{1}<...<\xi _{k}$ <... $<\xi$ _ ${k}$ 는 $\xi _{1}<...<\xi _{k}$ 고정 매듭입니다 $a_{+}=max(a,0).$ + $a_{+}=max(a,0).$ $m$ $a_{+}=max(a,0).$ x $a_{+}=max(a,0).$ ( $a_{+}=max(a,0).$ , $a_{+}=max(a,0).$ ) $a_{+}=max(a,0).$ . { $displaystyle$ a $\tau =(tau_{0},...,\tau _{p})^{T}$ { + } $=$ max ( $a_{+}=max(a,0).$ $\tau =(tau_{0},...,\tau _{p})^{T}$ , $0$ ) $\tau =(tau_{0},...,\tau _{p})^{T}$ $\tau =(tau_{0},...,\tau _{p})^{T}$ { $displaystyle$ \ $displaystyle =$ ( $u$ 0 $\tau =(tau_{0},...,\tau _{p})^{T}$ , $\tau =(tau_{0},...,\tau _{p})^{T}$ . $0$ , \ $displaystyle _$ 0 , \ $p$ ) }^{ $\tau =(tau_{0},...,\tau _{p})^{T}$ } $Y=X^{T}\beta +g(T,\tau )+\epsilon$ $}$ Y $\tau =(tau_{0},...,\tau _{p})^{T}$ $Y=X^{T}\beta +g(T,\tau )+\epsilon$ $Y=X^{T}\beta +g(T,\tau )+\epsilon$ $Y=X^{T}\beta +g(T,\tau )+\epsilon$ β + $Y=X^{T}\beta +g(T,\tau )+\epsilon$ ( $Y=X^{T}\beta +g(T,\tau )+\epsilon$ , $Y=X^{T}\beta +g(T,\tau )+\epsilon$ ) + ${\$ ( \ $displaystyle$ Y = $Y=X^{T}\beta +g(T,\tau )+\epsilon$ X $Y=X^{T}\beta +g(T,\tau )+\epsilon$ ^ { T } \ $beta + g$ ( $T$ , \ $tau$ ) + \ $ipsilon$ $Y=X^{T}\beta +g(T,\tau )+\epsilon$ 。 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ 스플라인 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ ( $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ p $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ T $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ , $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ 、、、、 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ 、 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ ) $T}}},{\hat {ps}^{T}}^{T}/(\beta^{T},\tau^{T})^$ { $T}}$ 은 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}of(\beta ^{T},\tau ^{T})^{T}$ (는) 다음과 같이 정의됩니다 $.$

$\displaystyle \sum _{i=1}^{n}[Y_{i}-X_{i}^{T}\beta _{i}-g(T_{i},\tau)]^{2}+\alpha \sum _{k=1}^{K}b_{k}^2}}$

$\alpha$ 서 $\alpha$ α {\ $displaystyle \alpha}$ 는 $\alpha$ 스무딩 파라미터입니다.

1999년 ^[7]Brumback 등이 언급한 바와 같이 추정기 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ p $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ T $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ ^, $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ s $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ ^) $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ ({\ $hat {beta _{ps}^)$ $T}}},{\hat {{ps}^{T}}^{$ T $}}$ 는 $({\hat {\beta _{ps}^{T}}},{\hat {\tau _{ps}^{T}}})^{T}$ LME 모델에 근거한 $\beta$ β(\ $displaystyle\beta)$ 의 $\beta$ 추정치와 동일합니다.

$=$ ${T}+Zb+\epsilon$

여기서 $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ ( $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ ) $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ 1 $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ . . $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ x $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $\Lambda ={\begin{pmatrix}x_{11}&...&x_{1d}&1&T_{1}&...&T_{1}^{p}\\x_{21}&...&x_{2d}&1&T_{2}&...&T_{2}^{p}\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\.&...&.&.&.&...&.\\x_{n1}&...&x_{nd}&1&T_{n}&...&T_{n}^{p}\end{pmatrix}}$ $displaystyle \Lambda = displays{pmatrix}x_{11}&...$ $x_{1d}&1&$ $T_{1}&...$ $&T_{1}^{p}\x_{212}&...$ $x_{2d}&1&$ $T_{2}&...$ $T_{2}^{p}\.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.&.$ $\\x_{n1}&...$ $&x_{nd}&1&$ $T_{n}&...$ &T_{n}^{p}\end{pmatrix}}}, Z)((T1− ξ 1)p...(티원 − ξ K)+p(T 2− ξ 1)p...(T 2− ξ K)+p...............(Tnξ 1−)p...(Tn− ξ K)+p){\displaystyle Z={\be.진{ $pmatrix}(T_{1}-\xi _{1})^{p}&...$ $&(T_{1}-\xi _{K}_{+}^{p}\\(T_{2}-\xi _{1})^{p}&...$ $&(T_{2}-\xi _{K}_{+}^{p}\.&.&.\.&.\.&.\.&.\(T_{n}-\xi _{1})^{p}&...$ $&(T_{n}-\xi _{K}_{+}^{p}\end{pmatrix}}$

$b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ 서 b $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ( $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ 1 , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ . , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ) $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ( $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ b $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ) 、 $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ( $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ 1 $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ . , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ) $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ T ~ $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ ( $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ , $b=(b_{1},...,b_{k})^{T}\backsim (0,\sigma _{b}^{2}),\epsilon =(\epsilon _{1},...,\epsilon _{n})^{T}\sim (0,\sigma _{\epsilon }^{2})$ 2 ) { $display$ b = ( $b$ _ { 1 , ... $b$ _ { $k$ } }^2 $T}\backsim(0,\b}^{2}),\ipsilon =(\ipsilon _{1}...\ipsilon _{n})^{$ $\alpha =\sigma _{\epsilon }^{2}/\sigma _{b}^{2}$ $}\sim (0,\sim$ _{\ $silon }^{2$ 및 $\alpha =\sigma _{\epsilon }^{2}/\sigma _{b}^{2}$ $=$ $\alpha =\sigma _{\epsilon }^{2}/\sigma _{b}^{2}$ 2 $\alpha =\sigma _{\epsilon }^{2}/\sigma _{b}^{2}$ / $\alpha =\sigma _{\epsilon }^{2}/\sigma _{b}^{2}$ 2 (\ $displaystyle \alpha$ =\ $silon$ _ {\ $silon$ }^{2 $}/\silon$ _ ${b}^2$ 매트릭스는 위 프레임워크에 대해 패널티 처리된 스플라인을 더 부드럽게 보여줍니다.

레퍼런스

^ ^a ^b ^c ^d Hardle, Liang, JiTi, WolfGang, Hua, Gao (2000). Partially linear model. PHYSICA-VERLAG.{{cite book}}: CS1 maint: 여러 이름: 작성자 목록(링크)
^ Zeger, Scott L.; Diggle, Peter J. (1994). "Semiparametric Models for Longitudinal Data with Application to CD4 Cell Numbers in HIV Seroconverters". Biometrics. 50 (3): 689–699. doi:10.2307/2532783. ISSN 0006-341X. JSTOR 2532783. PMID 7981395.
^ Schmalensee, Richard; Stoker, Thomas M. (1999). "Household Gasoline Demand in the United States" (PDF). Econometrica. 67 (3): 645–662. doi:10.1111/1468-0262.00041. hdl:1721.1/50215. ISSN 1468-0262.
^ Prada‐Sánchez, J. M.; Febrero‐Bande, M.; Cotos‐Yáñez, T.; González‐Manteiga, W.; Bermúdez‐Cela, J. L.; Lucas‐Domínguez, T. (2000). "Prediction of SO2 pollution incidents near a power station using partially linear models and an historical matrix of predictor-response vectors". Environmetrics. 11 (2): 209–225. doi:10.1002/(SICI)1099-095X(200003/04)11:2<209::AID-ENV403>3.0.CO;2-Z. ISSN 1099-095X.
^ Carroll, Raymond J.; Lin, Xihong (2001-12-01). "Semiparametric regression for clustered data". Biometrika. 88 (4): 1179–1185. doi:10.1093/biomet/88.4.1179. ISSN 0006-3444.
^ ^a ^b ^c ^d Liang, Hua (2006-02-10). "Estimation in Partially Linear Models and Numerical Comparisons". Computational Statistics & Data Analysis. 50 (3): 675–687. doi:10.1016/j.csda.2004.10.007. ISSN 0167-9473. PMC 2824448. PMID 20174596.
^ Brumback, Babette A.; Ruppert, David; Wand, M. P. (1999). "Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior: Comment". Journal of the American Statistical Association. 94 (447): 794–797. doi:10.2307/2669991. ISSN 0162-1459. JSTOR 2669991.

[:0-1] Hardle, Liang, JiTi, WolfGang, Hua, Gao (2000). Partially linear model. PHYSICA-VERLAG.{{cite book}}: CS1 maint: 여러 이름: 작성자 목록(링크)

[2] Zeger, Scott L.; Diggle, Peter J. (1994). "Semiparametric Models for Longitudinal Data with Application to CD4 Cell Numbers in HIV Seroconverters". Biometrics. 50 (3): 689–699. doi:10.2307/2532783. ISSN 0006-341X. JSTOR 2532783. PMID 7981395.

[3] Schmalensee, Richard; Stoker, Thomas M. (1999). "Household Gasoline Demand in the United States" (PDF). Econometrica. 67 (3): 645–662. doi:10.1111/1468-0262.00041. hdl:1721.1/50215. ISSN 1468-0262.

[4] Prada‐Sánchez, J. M.; Febrero‐Bande, M.; Cotos‐Yáñez, T.; González‐Manteiga, W.; Bermúdez‐Cela, J. L.; Lucas‐Domínguez, T. (2000). "Prediction of SO2 pollution incidents near a power station using partially linear models and an historical matrix of predictor-response vectors". Environmetrics. 11 (2): 209–225. doi:10.1002/(SICI)1099-095X(200003/04)11:2<209::AID-ENV403>3.0.CO;2-Z. ISSN 1099-095X.

[5] Carroll, Raymond J.; Lin, Xihong (2001-12-01). "Semiparametric regression for clustered data". Biometrika. 88 (4): 1179–1185. doi:10.1093/biomet/88.4.1179. ISSN 0006-3444.

[:1-6] Liang, Hua (2006-02-10). "Estimation in Partially Linear Models and Numerical Comparisons". Computational Statistics & Data Analysis. 50 (3): 675–687. doi:10.1016/j.csda.2004.10.007. ISSN 0167-9473. PMC 2824448. PMID 20174596.

[7] Brumback, Babette A.; Ruppert, David; Wand, M. P. (1999). "Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior: Comment". Journal of the American Statistical Association. 94 (447): 794–797. doi:10.2307/2669991. ISSN 0162-1459. JSTOR 2669991.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

Search

부분 선형 모형

네임스페이스

더

목차