# 统计理论|MATH2801 Theory of Statistics代写

To see that this solution has minimum norm, first factor $Z$, as
$$X=Q R U^{\mathrm{T}} \text {, }$$
and form the Moore-Penrose inverse as
$$X^{+}=U\left[\begin{array}{cc} R_{1}^{-1} & 0 \ 0 & 0 \end{array}\right] Q^{\mathrm{T}} \text {. }$$
Now let
$$\widehat{\beta}=X^{+} Y .$$

This is a least squares solution (that is, we have chosen a specific least squares solution).
Now, let
$$Q^{\mathrm{T}} Y=\left(\begin{array}{l} c_{1} \ c_{2} \end{array}\right),$$
where $c_{1}$ has exactly $r$ elements and $c_{2}$ has $n-r$ elements, and let
$$U^{\mathrm{T}} b=\left(\begin{array}{l} t_{1} \ t_{2} \end{array}\right),$$
where $b$ is the variable in the norm $|Y-X b|_{2}$ that we seek to minimize, and where $t_{1}$ has $r$ elements.

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