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# 经济代写|计量经济学代写Introduction to Econometrics代考|ECON471 Examples of Random Sets Defined by Random Points

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## 经济代写|计量经济学代写Introduction to Econometrics代考|Examples of Random Sets Defined by Random Points

Example $1.2$ (Random singleton) Random elements in $\mathfrak{X}$ are defined as measurable maps from $(\Omega, \mathfrak{A}, \mathbf{P})$ to the space $\mathfrak{X}$ equipped with its Borel $\sigma$-algebra $\mathcal{B}(\mathfrak{X})$. Then the singleton $\boldsymbol{X}={\boldsymbol{x}}$ is a random closed set. Indeed,
$${X \cap K \neq \emptyset}={x \in K} \in \mathfrak{A}$$
for each compact set $K$.
Example 1.3 (Random half-line) If $\boldsymbol{x}$ is a random variable on the real line $\mathbb{R}$, then the half-lines $\boldsymbol{X}=[\boldsymbol{x}, \infty)$ and $\boldsymbol{Y}=(-\infty, \boldsymbol{x}]$ are random closed sets on $\mathfrak{\notin}=\mathbb{R}$. Indeed,
\begin{aligned} &{X \cap K \neq \emptyset}={x \leq \sup K} \in \mathfrak{Q}, \ &{\boldsymbol{Y} \cap K \neq \emptyset}={x \geq \inf K} \in \mathfrak{A}, \end{aligned}
for each compact set $K$. This example is useful for relating the classical notion of the cumulative distribution function of random variables to more general concepts arising in the theory of random sets.

Example 1.4 (Random ball) Let $\mathfrak{X}$ be equipped with a metric d. A random ball $\boldsymbol{X}=B_{\boldsymbol{y}}(\boldsymbol{x})$ with center $\boldsymbol{x}$ and radius $\boldsymbol{y}$ is a random closed set if $\boldsymbol{x}$ is a random vector and $y$ is a non-negative random variable. Then
$${\boldsymbol{X} \cap K \neq \emptyset}={\boldsymbol{y} \geq \mathbf{d}(\boldsymbol{x}, K)},$$
where $\mathbf{d}(\boldsymbol{x}, K)$ is the distance from $\boldsymbol{x}$ to the nearest point in $K$. Since both $\boldsymbol{y}$ and $\mathbf{d}(\boldsymbol{x}, K)$ are random variables, it is immediately clear that ${\boldsymbol{X} \cap K \neq \emptyset} \in \mathfrak{A}$. If the joint distribution of $(\boldsymbol{x}, \boldsymbol{y})$ depends on a certain parameter, we obtain a parametric family of distributions for random balls.

## 经济代写|计量经济学代写Introduction to Econometrics代考|Random Sets Related to Deterministic and Random Functions

Example $1.8$ (Deterministic function at random level) Let $f: \mathbb{R}^d \mapsto \mathbb{R}$ be a deterministic function, and let $\boldsymbol{x}$ be a random variable. If $f$ is continuous, then $\boldsymbol{X}={x: f(x)=\boldsymbol{x}}$ is a random closed set called the level set of $f$.
If $f$ is upper semicontinuous, i.e.,
$$f(u) \geq \limsup {v \rightarrow u} f(v)$$ for all $x$, then $\boldsymbol{Y}={u: f(u) \geq \boldsymbol{x}}$ is closed and defines a random closed set (called the upper excursion set). Indeed, $${\boldsymbol{Y} \cap K \neq \emptyset}=\left{\sup {u \in K} f(u) \geq \boldsymbol{x}\right} \in \mathfrak{A},$$
since $\boldsymbol{x}$ is a random variable. The distributions of $\boldsymbol{X}$ and $\boldsymbol{Y}$ are determined by the distribution of $\boldsymbol{x}$ and the choice of $f$. Both $\boldsymbol{X}=f^{-1}({\boldsymbol{x}})$ and $\boldsymbol{Y}=$ $f^{-1}([x, \infty))$ can be obtained as inverse images. Note that $f$ is called lower semicontinuous if $(-f)$ is upper semicontinuous.

Example 1.9 (Excursions of random functions) Let $\boldsymbol{x}(t), t \in \mathbb{R}$, be a realvalued stochastic process. If this process has continuous sample paths, then ${t: \boldsymbol{x}(t)=c}$ is a random closed set for each $c \in \mathbb{R}$. For instance, if $\boldsymbol{x}(t)=$ $z_n t^n+\cdots+z_1 t+z_0$ is the polynomial of degree $n$ in $t \in \mathbb{R}$ with random coefficients, then $X={t: \boldsymbol{x}(t)=0}$ is the random set of its roots.

If $\boldsymbol{x}$ has almost surely lower semicontinuous sample paths, then the lower excursion set $\boldsymbol{X}={t: \boldsymbol{x}(t) \leq c}$ and the epigraph
$$\boldsymbol{Y}=\text { epi } \boldsymbol{x}={(t, s) \in \mathbb{R} \times \mathbb{R}: \boldsymbol{x}(t) \geq s}$$
are random closed sets. For instance,
$${\boldsymbol{X} \cap K \neq \emptyset}=\left{\inf _{t \in K} \boldsymbol{x}(t) \leq c\right} \in \mathfrak{A}$$
In view of this, statements about the supremum of a stochastic process can be formulated in terms of the corresponding excursion sets. The same construction works for random functions indexed by multidimensional arguments. Lower excursion sets appear as solutions to inequalities or systems of inequalities in partial identification problems (see Section 5.2).

## 经济代写|计量经济学代写Introduction to Econometrics代考|Examples of Random Sets Defined by Random Points

$$X \cap K \neq \emptyset=x \in K \in \mathfrak{A}$$

$$X \cap K \neq \emptyset=x \leq \sup K \in \mathfrak{Q}, \quad \boldsymbol{Y} \cap K \neq \emptyset=x \geq \inf K \in \mathfrak{A},$$

$$\boldsymbol{X} \cap K \neq \emptyset=\boldsymbol{y} \geq \mathbf{d}(\boldsymbol{x}, K),$$

## 经济代写|计量经济学代写Introduction to Econometrics代考|Random Sets Related to Deterministic and Random Functions

$$f(u) \geq \lim \sup v \rightarrow u f(v)$$

〈left 缺少或无法识别的分隔符

$$\boldsymbol{Y}=\text { epi } \boldsymbol{x}=(t, s) \in \mathbb{R} \times \mathbb{R}: \boldsymbol{x}(t) \geq s$$

〈left 缺少或无法识别的分隔符

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