### 商科代写|计量经济学代写Econometrics代考|ECON 2504

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• Statistical Inference 统计推断
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 商科代写|计量经济学代写Econometrics代考|Some Examples

Each of the loss functions that we discuss in this subsection corresponds to a machine learning algorithm, as thoroughly explained in Bühlmann and Hothorn (2007), Sect. 3. We refer to this article for more properties of these losses and for issues regarding their practical implementation.

• A first canonical example, in the regression setting, is to let $\psi(x, y)=(y-x)^{2}$ (squared error loss), which is 2 -strongly convex in its first argument (Assumption $\mathbf{A}{\mathbf{2}}$ ) and satisfies Assumption $\mathbf{A}{\mathbf{1}}$ as soon as $\mathbb{E} Y^{2}<\infty$. It also satisfies $\mathbf{A}{\mathbf{3}}^{\prime}$, with $\partial{x} \psi(x, y)=2(x-y)$ and $L=2$.
• Another example in regression is the loss $\psi(x, y)=|y-x|$ (absolute error loss), which is convex but not strongly convex in its first argument. Whenever strong convexity of the loss is required, a possible strategy is to regularize the objective via an $L^{2}$-type penalty, and take
$$\psi(x, y)=|y-x|+\gamma x^{2},$$
where $\gamma$ is a positive parameter (possibly function of the sample size $n$ in the empirical setting). This loss is (2 ) -strongly convex in $x$ and satisfies $\mathbf{A}{\mathbf{1}}$ and $\mathbf{A}{\mathbf{2}}$ whenever $\mathbb{E}|Y|<\infty$, with $\xi(x, y)=\operatorname{sgn}(x-y)+2 \gamma x$ (with $\operatorname{sgn}(u)=2 \mathbb{1}{[u>0]}-1$ for $u \neq 0$ and $\operatorname{sgn}(0)=0$ ). On the other hand, the function $\psi(\cdot, y)$ is not differentiable at $y$, so that the smoothness Assumption $\mathbf{A}{3}^{\prime}$ is not satisfied. However,
\begin{aligned} \mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)=& \int\left(\operatorname{sgn}\left(x_{1}-y\right)-\operatorname{sgn}\left(x_{2}-y\right)\right) \mu_{Y \mid X}(\mathrm{~d} y)+2 \gamma\left(x_{1}-x_{2}\right) \ =& \mu_{Y \mid X}\left(\left(-\infty, x_{1}\right)\right)-\mu_{Y \mid X}\left(\left(-\infty, x_{2}\right)\right)+2 \gamma\left(x_{1}-x_{2}\right) \ &-\mu_{Y \mid X}\left(\left(x_{1}, \infty\right)\right)+\mu_{Y \mid X}\left(\left(x_{2}, \infty\right)\right) . \end{aligned}
Thus, if we assume for example that $\mu_{Y \mid X}$ has a density (with respect to the Lebesgue measure) bounded by $B$, then
$$\left|\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)\right| \leq 2(B+\gamma)\left|x_{1}-x_{2}\right|,$$
and Assumption $\mathbf{A}{3}$ is therefore satisfied. Of course, in the empirical setting, assuming that $\mu{Y \mid X}$ has a density precludes the use of the empirical measure $\mu_{n}$ for $\mu_{X, Y}$. A safe and simple alternative is to consider a smoothed version $\tilde{\mu}{n}$ of $\mu{n}$ (based,
• for example, on a kernel estimate; see Devroye and Györfi 1985), and to minimize the functional
• $$• C_{n}(F)=\int|y-F(x)| \tilde{\mu}{n}(\mathrm{~d} x, \mathrm{~d} y)+\gamma \int F(x)^{2} \tilde{\mu}{n}(\mathrm{~d} x) •$$
• over the linear combinations of functions in $\mathscr{F}$.

## 商科代写|计量经济学代写Econometrics代考|Two Algorithms

Let $\operatorname{lin}(\mathscr{F})$ be the set of all linear combinations of functions in $\mathscr{F}$, our collection of base predictors in $L^{2}\left(\mu_{X}\right)$. So, each $F \in \operatorname{lin}(\mathscr{F})$ has the form $F=\sum_{j=1}^{J} \beta_{j} f_{j}$. where $\left(\beta_{1}, \ldots, \beta_{J}\right) \in \mathbb{R}^{J}$ and $f_{1}, \ldots, f_{J}$ are elements of $\mathscr{F}$. Finding the infimum of the functional $C$ over lin $(\mathscr{F})$ is a challenging infinite-dimensional optimization problem, which requires an algorithm. The core idea of the gradient boosting approach is to greedily locate the infimum by producing a combination of base predictors via a gradient-descent-type algorithm in $L^{2}\left(\mu_{X}\right)$. Focusing on the basics, this can be achieved by two related yet different strategies, which we examine in greater

mathematical details below. Algorithm 1 appears in Mason et al. (2000), whereas Algorithm 2 is essentially due to Friedman (2001).

It is implicitly assumed throughout this paragraph that Assumption $\mathbf{A}{\mathbf{1}}$ is satisfied. We recall that under this assumption, the convex functional $C$ is locally bounded and therefore continuous. Thus, in particular, $$\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F),$$ where $\varlimsup \overline{\operatorname{lin}(\mathscr{F})}$ is the closure of lin( $\mathscr{F})$ in $L^{2}\left(\mu{X}\right)$. Loosely speaking, looking for the infimum of $C$ over $\overline{\operatorname{lin}(\mathscr{F})}$ is the same as looking for the infimum of $C$ over all (finite) linear combinations of base functions in $\mathscr{F}$. We note in addition that if Assumption $\mathbf{A}{2}$ is satisfied, then there exists a unique function $\bar{F} \in \overline{\operatorname{lin}(\mathscr{F})}$ (which we call the boosting predictor) such that $$C(\bar{F})=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)$$
Algorithm 1. In this approach, we consider a class $\mathscr{F}$ of functions $f: \mathscr{X} \rightarrow \mathbb{R}$ such that $0 \in \mathscr{F}, f \in \mathscr{F} \Leftrightarrow-f \in \mathscr{F}$, and $|f|_{\mu_{X}}=1$ for $f \neq 0$. An example is the collection $\mathscr{F}$ of all $\pm$-binary trees in $\mathbb{R}^{d}$ using axis parallel cuts with $k$ terminal nodes (plus zero). Each nonzero $f \in \mathscr{F}$ takes the form $f=\sum_{j=1}^{k} \beta_{j} \mathbb{1}{A{j}}$, where $\left|\beta_{j}\right|=1$ and $A_{1}, \ldots, A_{k}$ is a tree-structured partition of $\mathrm{R}^{d}$ (Devroye et al. 1996, Chap. 20). The parameter $k$ is a measure of the tree complexity. For example, trees with $k=d+1$ are such that $\overline{\operatorname{lin}(\mathscr{F})}=L^{2}\left(\mu_{X}\right)$ (Breiman 2000). Thus, in this case,
$$\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)=\inf {F \in L^{2}\left(\mu_{X}\right)} C(F)$$

## 商科代写|计量经济学代写Econometrics代考|Algorithm 1

The convergence of this algorithm rests upon the choice of the step size sequence $\left(w_{t}\right){t}$, which needs to be carefully specified. We take $w{0}>0$ arbitrarily and set
$$w_{t+1}=\min \left(w_{t},-(2 L)^{-1} \mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X)\right), \quad t \geq 0,$$
where $L$ is the Lipschitz constant of Assumption $\mathbf{A}{\mathbf{3}}$. Clearly, the sequence $\left(w{t}\right){t}$ is nonincreasing. It is also nonnegative. To see this, just note that, by definition, $$f{t+1} \in \arg \max {f \in \mathscr{F}}-\mathbb{E} \xi\left(F{t}(X), Y\right) f(X),$$
and thus, since $0 \in \mathscr{F},-\mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X) \geq 0$. The main result of this section is encapsulated in the following theorem.

Theorem 1 Assume that Assumptions $\mathbf{A}{\mathbf{1}}$ and $\mathbf{A}{\mathbf{3}}$ are satisfied, and let $\left(F_{t}\right){t}$ be defined by Algorithm 1 with $\left(w{t}\right){t}$ as in (8). Then $$\lim {t \rightarrow \infty} C\left(F_{t}\right)=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F) .$$ Proof See Supplementary Material Document. Observe that Theorem 1 holds without Assumption $\mathbf{A}{2}$, i.e., there is no need here to assume that the function $\psi(x, y)$ is strongly convex in $x$. However, whenever Assumption $\mathbf{A}_{2}$ is satisfied, there exists as in (4) a unique boosting predictor $\bar{F} \in \overline{\operatorname{lin}(\mathscr{F})}$ such that $C(\bar{F})=\inf {F \in \operatorname{lin}(\mathscr{F})} C(F)$, and the theorem guarantees that $\lim {t \rightarrow \infty} C\left(F_{t}\right)=C(\bar{F})$

The proof of the theorem relies on the following lemma, which states that the sequence $\left(C\left(F_{t}\right)\right){t}$ is nonincreasing. Since $C(F)$ is nonnegative for all $F$, we concludê thât $C\left(F{t}\right) \downarrow \inf {k} C\left(F{k}\right)$ â $t \rightarrow \infty$. This is thé kêy argumént tó prové thé convergence of $C\left(F_{t}\right)$ toward inf $F \in \operatorname{lin}(\mathscr{F}) C(F)$.

## 商科代写|计量经济学代写Econometrics代考|Some Examples

• 第一个典型的例子，在回归设置中，是让 $\psi(x, y)=(y-x)^{2}$ (平方误差损失)，它的第一个参数是 2 – 强 凸 (假设 $\mathbf{A} 2$ ) 并满足假设 $\mathbf{A 1}$ 立刻 $\mathbb{E} Y^{2}<\infty$. 也满足 $\mathbf{A} \mathbf{3}^{\prime}$ ， 和 $\partial x \psi(x, y)=2(x-y)$ 和 $L=2$.
• 回归中的另一个例子是损失 $\psi(x, y)=|y-x|$ (绝对误差损失)，它是凸的，但在其第一个参数中不是强 凸的。每当需要损失的强凸性时，一种可能的策略是通过 $L^{2}$-型惩罚，并采取
$$\psi(x, y)=|y-x|+\gamma x^{2},$$
在哪里 $\gamma$ 是一个正参数 (可能是样本量的函数 $n$ 在经验设置中) 。这种损失是 (2) – 在 $x$ 并满足 $\mathbf{A} 1$ 和 $\mathbf{A} 2$ 每当 $\mathbb{E}|Y|<\infty$ ，和 $\xi(x, y)=\operatorname{sgn}(x-y)+2 \gamma x($ 和 $\operatorname{sgn}(u)=21[u>0]-1$ 为了 $u \neq 0$ 和 $\operatorname{sgn}(0)=0$ ) 。另一方面，函数 $\psi(\cdot, y)$ 不可微分 $y$, 使平滑假设 $\mathbf{A} 3^{\prime}$ 不满意。然而，
$$\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)=\int\left(\operatorname{sgn}\left(x_{1}-y\right)-\operatorname{sgn}\left(x_{2}-y\right)\right) \mu_{Y \mid X}(\mathrm{~d} y)+2 \gamma\left(x_{1}-x_{2}\right)=$$
因此，如果我们假设例如 $\mu_{Y \mid X}$ 有一个密度（相对于 Lebesgue 测度) 为界 $B$ ，然后
$$\left|\mathbb{E}\left(\xi\left(x_{1}, Y\right)-\xi\left(x_{2}, Y\right) \mid X\right)\right| \leq 2(B+\gamma)\left|x_{1}-x_{2}\right|,$$
和假设 $\mathbf{A} 3$ 因此感到满意。当然，在经验设置中，假设 $\mu Y \mid X$ 有一个密度排除了经验测量的使用 $\mu_{n}$ 为了 $\mu_{X, Y}$. 一个安全且简单的替代方案是考虑平滑版本 $\tilde{\mu} n$ 的 $\mu n$ (基于，
• 例如，在核估计上；参见 Devroye 和 Györfi 1985)，并最小化函数
• $\$ \$$• C_{n}(F)=|int |y F(x)| \backslash tilde {\backslash m u}{n}(\backslash \operatorname{mathrm}{\sim d} x, \mathrm {\sim d} y)+\backslash gamma \backslash int F(x)^{\wedge}{2} \backslash t \mathrm{~ i l d e {} (数学 {\sim d} x) • \\ • 在函数的线性组合上 \mathscr{F}. ## 商科代写|计量经济学代写Econometrics代考|Two Algorithms 让 \operatorname{lin}(\mathscr{F}) 是函数的所有线性组合的集合 \mathscr{F} ，我们收集的基础预测变量在 L^{2}\left(\mu_{X}\right). 所以，每个 F \in \operatorname{lin}(\mathscr{F}) 有形 式 F=\sum_{j=1}^{J} \beta_{j} f_{j}. 在哪里 \left(\beta_{1}, \ldots, \beta_{J}\right) \in \mathbb{R}^{J} 和 f_{1}, \ldots, f_{J} 是元素 \mathscr{F}. 寻找泛函的下确界 C 过林 (\mathscr{F}) 是一个具 有挑战性的无限维优化问题，需要一个算法。梯度提升方法的核心思想是通过梯度下降型算法生成基本预测变量的 组合来贪婪地定位下确界。 L^{2}\left(\mu_{X}\right). 专注于基础，这可以通过两种相关但不同的策略来实现，我们将在更大的 下面的数学细节。算法 1 出现在 Mason et al. (2000 年)，而算法 2 主要归功于 Friedman（2001 年)。 在本段中隐含地假设假设 \mathbf{A 1} 很满意。我们记得在这个假设下，凸泛函 C 是局部有界的，因此是连续的。因此，特 别是，$$
\inf F \in \operatorname{lin}(\mathscr{F}) C(F)=\inf F \in \operatorname{lin}(\mathscr{F}) C(F),
$$在哪里 \overline{\lim } \overline{\operatorname{lin}(\mathscr{F})} 是 \operatorname{lin}(\mathscr{F}) 在 L^{2}(\mu X). 松散地说，寻找下确界 C 超过 \overline{\operatorname{lin}(\mathscr{F}) \text { 和寻找下确界一样 } C \text { 基函数的所 } 有 (有限) 线性组合 \mathscr{F}. 我们还注意到，如果假设 \mathbf{A} 2 满足，则存在唯一函数 \bar{F} \in \overline{\operatorname{lin}(\mathscr{F})} (我们称之为提升预测 器) 使得$$
C(\bar{F})=\inf F \in \operatorname{lin}(\mathscr{F}) C(F)
$$算法 1. 在这种方法中，我们考虑一个类 \mathscr{F} 功能 f: \mathscr{X} \rightarrow \mathbb{R} 这样 0 \in \mathscr{F}, f \in \mathscr{F} \Leftrightarrow-f \in \mathscr{F} ，和 |f|{\mu{X}}=1 为了 f \neq 0. 一个例子是集合 \mathscr{F} 其中 \pm – 二叉树 \mathbb{R}^{d} 使用轴平行切割 k 终端节点 (加零) 。每个非零 f \in \mathscr{F} 采取形式 f=\sum_{j=1}^{k} \beta_{j} 1 A j ，在哪里 \left|\beta_{j}\right|=1 和 A_{1}, \ldots, A_{k} 是一个树形结构的分区 \mathrm{R}^{d} (Devroye 等人，1996 年，第 20 章) 。参数 k 是树复杂度的度量。例如，树与 k=d+1 是这样的 \overline{l i n}(\mathscr{F})=L^{2}\left(\mu_{X}\right) (布雷曼 2000) 。因 此，在这种情况下，$$
\inf F \in \operatorname{lin}(\mathscr{F}) C(F)=\inf F \in L^{2}\left(\mu_{X}\right) C(F)
$$## 商科代写|计量经济学代写Econometrics代考|Algorithm 1 该算法的收敛取决于步长序列的选择 \left(w_{t}\right) t, 这需要仔细指定。我们采取 w 0>0 任意设置$$
w_{t+1}=\min \left(w_{t},-(2 L)^{-1} \mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X)\right), \quad t \geq 0,
$$在哪里 L 是假设的 Lipschitz 常数 \mathbf{A 3}. 显然，序列 (w t) t 是非增加的。它也是非负的。要看到这一点，请注意，根 据定义，$$
f t+1 \in \arg \max f \in \mathscr{F}-\mathbb{E} \xi(F t(X), Y) f(X),
$$因此，由于 0 \in \mathscr{F},-\mathbb{E} \xi\left(F_{t}(X), Y\right) f_{t+1}(X) \geq 0. 本节的主要结果封装在以下定理中。 定理 1 假设假设 \mathbf{A} 1 和 \mathbf{A} 3 满意，让 \left(F_{t}\right) t 由算法 1 定义 (w t) t 如（8)。然后$$
\lim t \rightarrow \infty C\left(F_{t}\right)=\inf F \in \operatorname{lin}(\mathscr{F}) C(F) .


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