## 统计代写|线性回归代写linear regression代考|Variable Selection

Variable selection, also called subset or model selection, is the search for a subset of predictor variables that can be deleted without important loss of information. A model for variable selection in multiple linear regression can be described by
$$Y=\boldsymbol{x}^T \boldsymbol{\beta}+e=\boldsymbol{\beta}^T \boldsymbol{x}+e=\boldsymbol{x}_S^T \boldsymbol{\beta}_S+\boldsymbol{x}_E^T \boldsymbol{\beta}_E+e=\boldsymbol{x}_S^T \boldsymbol{\beta}_S+e$$
where $e$ is an error, $Y$ is the response variable, $\boldsymbol{x}=\left(\boldsymbol{x}_S^T, \boldsymbol{x}_E^T\right)^T$ is a $p \times 1$ vector of predictors, $\boldsymbol{x}_S$ is a $k_S \times 1$ vector, and $\boldsymbol{x}_E$ is a $\left(p-k_S\right) \times 1$ vector. Given that $\boldsymbol{x}_S$ is in the model, $\boldsymbol{\beta}_E=\mathbf{0}$ and $E$ denotes the subset of terms that can be eliminated given that the subset $S$ is in the model.

Since $S$ is unknown, candidate subsets will be examined. Let $\boldsymbol{x}_I$ be the vector of $k$ terms from a candidate subset indexed by $I$, and let $x_O$ be the vector of the remaining predictors (out of the candidate submodel). Then
$$Y=\boldsymbol{x}_I^T \boldsymbol{\beta}_I+\boldsymbol{x}_O^T \boldsymbol{\beta}_O+e .$$
Definition 3.7. The model $Y=\boldsymbol{x}^T \boldsymbol{\beta}+e$ that uses all of the predictors is called the full model. A model $Y=\boldsymbol{x}_I^T \boldsymbol{\beta}_I+e$ that only uses a subset $\boldsymbol{x}_I$ of the predictors is called a submodel. The full model is always a submodel. The sufficient predictor $(\mathrm{SP})$ is the linear combination of the predictor variables used in the model. Hence the full model has $S P=\boldsymbol{x}^T \boldsymbol{\beta}$ and the submodel has $S P=\boldsymbol{x}_I^T \boldsymbol{\beta}_I$.

## 统计代写|线性回归代写linear regression代考|Bootstrapping Variable Selection

The bootstrap will be described and then applied to variable selection. Suppose there is data $\boldsymbol{w}_1, \ldots, \boldsymbol{w}_n$ collected from a distribution with cdf $F$ into an $n \times p$ matrix $\boldsymbol{W}$. The empirical distribution, with cdf $F_n$, gives each observed data case $\boldsymbol{w}_i$ probability $1 / n$. Let the statistic $T_n=t(\boldsymbol{W})=t\left(F_n\right)$ be computed from the data. Suppose the statistic estimates $\boldsymbol{\mu}=t(F)$. Let $t\left(\boldsymbol{W}^\right)=t\left(F_n^\right)=T_n^*$ indicate that $t$ was computed from an iid sample from the empirical distribution $F_n$ : a sample of size $n$ was drawn with replacement from the observed sample $\boldsymbol{w}_1, \ldots, \boldsymbol{w}_n$.

Some notation is needed to give the Olive (2013a) prediction region used to bootstrap a hypothesis test. Suppose $\boldsymbol{w}1, \ldots, \boldsymbol{w}_n$ are iid $p \times 1$ random vectors with mean $\boldsymbol{\mu}$ and nonsingular covariance matrix $\boldsymbol{\Sigma}{\boldsymbol{w}}$. Let a future test observation $\boldsymbol{w}f$ be independent of the $\boldsymbol{w}_i$ but from the same distribution. Let $(\overline{\boldsymbol{w}}, \boldsymbol{S})$ be the sample mean and sample covariance matrix where $$\overline{\boldsymbol{w}}=\frac{1}{n} \sum{i=1}^n \boldsymbol{w}i \text { and } \boldsymbol{S}=\boldsymbol{S} \boldsymbol{w}=\frac{1}{\mathrm{n}-1} \sum{\mathrm{i}=1}^{\mathrm{n}}\left(\boldsymbol{w}{\mathrm{i}}-\overline{\boldsymbol{w}}\right)\left(\boldsymbol{w}{\mathrm{i}}-\overline{\boldsymbol{w}}\right)^{\mathrm{T}}$$
Then the $i$ th squared sample Mahalanobis distance is the scalar
$$D_{\boldsymbol{w}}^2=D_{\boldsymbol{w}}^2(\overline{\boldsymbol{w}}, \boldsymbol{S})=(\boldsymbol{w}-\overline{\boldsymbol{w}})^T \boldsymbol{S}^{-1}(\boldsymbol{w}-\overline{\boldsymbol{w}}) .$$
Let $D_i^2=D_{\boldsymbol{w}i}^2$ for each observation $\boldsymbol{w}_i$. Let $D{(c)}$ be the $c$ th order statistic of $D_1, \ldots, D_n$. Consider the hyperellipsoid
$$\mathcal{A}n=\left{\boldsymbol{w}: D{\tilde{\boldsymbol{w}}}^2(\overline{\boldsymbol{w}}, \boldsymbol{S}) \leq D_{(c)}^2\right}=\left{\boldsymbol{w}: D_{\boldsymbol{w}}(\overline{\boldsymbol{w}}, \boldsymbol{S}) \leq D_{(c)}\right}$$
If $n$ is large, we can use $c=k_n=\lceil n(1-\delta)\rceil$. If $n$ is not large, using $c=$ $U_n$ where $U_n$ decreéses to $k_n$, can improye small sample performance. Olivê (2013a) showed that $(3.10)$ is a large sample $100(1-\delta) \%$ prediction region for a large class of distributions, although regions with smaller volumes may exist.

## 统计代写|线性回归代写linear regression代考|Variable Selection

$$Y=\boldsymbol{x}^T \boldsymbol{\beta}+e=\boldsymbol{\beta}^T \boldsymbol{x}+e=\boldsymbol{x}_S^T \boldsymbol{\beta}_S+\boldsymbol{x}_E^T \boldsymbol{\beta}_E+e=\boldsymbol{x}_S^T \boldsymbol{\beta}_S+e$$

$$Y=\boldsymbol{x}_I^T \boldsymbol{\beta}_I+\boldsymbol{x}_O^T \boldsymbol{\beta}_O+e .$$

## 统计代写|线性回归代写linear regression代考|Bootstrapping Variable Selection

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