### 统计代写|统计推断代写Statistical inference代考|MAST20005

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断代写Statistical inference代考|Cumulant-generating functions and cumulants

It is often convenient to work with the log of the moment-generating function. It turns out that the coefficients of the polynomial expansion of the log of the momentgenerating function have convenient interpretations in terms of moments and central moments.
Definition 3.5.7 (Cumulant-generating function and cumulants)
The cumulant-generating function of a random variable $X$ with moment-generating function $M_{X}(t)$, is defined as
$$K_{X}(t)=\log M_{X}(t)$$
The $r^{\text {th }}$ cumulant, $K_{r}$, is the coefficient of $t^{r} / r !$ in the expansion of the cumulantgenerating function $K_{X}(t)$ so
$$K_{X}(t)=\kappa_{1} t+\kappa_{2} \frac{t^{2}}{2 !}+\ldots+\kappa_{r} \frac{t^{r}}{r !}+\ldots=\sum_{j=1}^{\infty} K_{j} \frac{t^{j}}{j !}$$
It is clear from this definition that the relationship between cumulant-generating function and cumulants is the same as the relationship between moment-generating function and moments. Thus, to calculate cumulants we can either compare coefficients or differentiate.

1. Calculating the $r^{\text {th }}$ cumulant, $\kappa_{r}$, by comparing coefficients:
$$\text { if } K_{X}(t)=\sum_{j=0}^{\infty} b_{j} t^{j} \text { then } K_{r}=r ! b_{r} \text {. }$$
2. Calculating the $r^{\text {th }}$ cumulant, $\kappa_{r}$, by differentiation:
$$K_{r}=K_{X}^{(r)}(0)=\left.\frac{d^{r}}{d t^{r}} K_{X}(t)\right|_{t=0}$$
Cumulants can be expressed in terms of moments and central moments. Particularly useful are the facts that the first cumulant is the mean and the second cumulant is the variance. In order to prove these results we will use the expansion, for $|x|<1$,
$$\log (1+x)=x-\frac{1}{2} x^{2}+\frac{1}{3} x^{3}-\ldots+\frac{(-1)^{j+1}}{j} x^{j}+\ldots$$

## 统计代写|统计推断代写Statistical inference代考|Distribution and mass/density for g(X)

Suppose that $X$ is a random variable defined on $(\Omega, \mathcal{F}, \mathrm{P})$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a well-behaved function. We would like to derive an expression for the cumulative distribution function of $Y$, where $Y=g(X)$. Some care is required here. We define $g$ to be a function, so for every real number input there is a single real number output. However, $g^{-1}$ is not necessarily a function, so a single input may have multiple outputs. To illustrate, let $g(x)=x^{2}$, then $g^{-1}$ corresponds to taking the square root, an operation that typically has two real outputs; for example, $g^{-1}(4)={-2,2}$. So, in general,
$$\mathrm{P}(Y \leq y)=\mathrm{P}(g(X) \leq y) \neq \mathrm{P}\left(X \leq g^{-1}(y)\right)$$
Our first step in deriving an expression for the distribution function of $Y$ is to consider the probability that $Y$ takes values in a subset of $\mathbb{R}$. We will use the idea of the inverse image of a set.
Definition 3.6.1 (Inverse image)
If $g: R \rightarrow R$ is a function and $B$ is a subset of real numbers, then the inverse image of $B$ under $g$ is the set of real numbers whose images under $g$ lie in $B$, that is, for all $B \subseteq \mathbb{R}$ we define the inverse image of $B$ under $g$ as
$$g^{-1}(B)={x \in \mathbb{R}: g(x) \in B}$$
Then for any well-behaved $B \subseteq \mathbb{R}$,
\begin{aligned} \mathrm{P}(Y \in B) &=\mathrm{P}(g(X) \in B)=\mathrm{P}({\omega \in \Omega: g(X(\omega)) \in B}) \ &=\mathrm{P}\left(\left{\omega \in \Omega: X(\omega) \in g^{-1}(B)\right}\right)=\mathrm{P}\left(X \in g^{-1}(B)\right) \end{aligned}
Stated loosely, the probability that $g(X)$ is in $B$ is equal to the probability that $X$ is in the inverse image of $B$. The cumulative distribution function of $Y$ is then
\begin{aligned} F_{Y}(y) &=\mathrm{P}(Y \leq y)=\mathrm{P}(Y \in(-\infty, y])=\mathrm{P}(g(X) \in(-\infty, y]) \ &=\mathrm{P}\left(X \in g^{-1}((-\infty, y])\right) \ &= \begin{cases}\sum_{{x: g(x) \leq y}} f_{X}(x) & \text { if } X \text { is discrete, } \ \int_{{x: g(x) \leq y}} f_{X}(x) d x & \text { if } X \text { is continuous. }\end{cases} \end{aligned}
In the discrete case, we can use similar reasoning to provide an expression for the mass function,
$$f_{Y}(y)=\mathrm{P}(Y=y)=\mathrm{P}(g(X)=y)=\mathrm{P}\left(X \in g^{-1}(y)\right)=\sum_{{x: g(x)=y}} f_{X}(x)$$

## 统计代写|统计推断代写Statistical inference代考|Sequences of random variables and convergence

Suppose that $x_{1}, x_{2}, \ldots$ is a sequence of real numbers. We denote this sequence $\left{x_{n}\right}$. The definition of convergence for a sequence of real numbers is well established.
Definition 3.7.1 (Convergence of a real sequence)
Let $\left{x_{n}\right}$ be a sequence of real numbers and let $x$ be a real number. We say that $x_{n}$ converges to $x$ if and only if, for every $\varepsilon>0$, we can find an integer $N$ such that $\left|x_{n}-x\right|<\varepsilon$ for all $n>N$. Under these conditions, we write $x_{n} \rightarrow x$ as $n \rightarrow \infty$.
This definition is based on an intuitively appealing idea (although in the formal statement given above, this might not be obvious). If we take any interval around $x$, say $[x-\varepsilon, x+\varepsilon]$, we can find a point, say $N$, beyond which all elements of the sequence fall in the interval. This is true for an arbitrarily small interval.

Now consider a sequence of random variables $\left{X_{n}\right}$ and a random variable $X$. We want to know what it means for $\left{X_{n}\right}$ to converge to $X$. Using Definition 3.7.1 is not possible; since $\left|X_{n}-X\right|$ is a random variable, direct comparison with the real number $\varepsilon$ is not meaningful. In fact, for a random variable there are many different forms of convergence. We define four distinct modes of convergence for a sequence of random variables.

## 统计代写|统计推断代写Statistical inference代考|Cumulant-generating functions and cumulants

ķX(吨)=日志⁡米X(吨)

ķX(吨)=ķ1吨+ķ2吨22!+…+ķr吨rr!+…=∑j=1∞ķj吨jj!

1. 计算rth 累积，ķr，通过比较系数：
如果 ķX(吨)=∑j=0∞bj吨j 然后 ķr=r!br.
2. 计算rth 累积，ķr，通过微分：
ķr=ķX(r)(0)=drd吨rķX(吨)|吨=0
累积量可以用矩和中心矩来表示。特别有用的是第一个累积量是平均值，第二个累积量是方差。为了证明这些结果，我们将使用展开式，对于|X|<1,
日志⁡(1+X)=X−12X2+13X3−…+(−1)j+1jXj+…

## 统计代写|统计推断代写Statistical inference代考|Distribution and mass/density for g(X)

G−1(乙)=X∈R:G(X)∈乙

\begin{对齐} \mathrm{P}(Y\inB) &=\mathrm{P}(g(X)\inB)=\mathrm{P}({\omega\in\Omega: g(X (\ omega))\in B})\&=\mathrm{P}\left(\left{\omega\in\Omega:X(\omega)\in g^{-1}(B)\right} \right )=\mathrm{P}\left(X\in g^{-1}(B)\right)\end{aligned}\begin{对齐} \mathrm{P}(Y\inB) &=\mathrm{P}(g(X)\inB)=\mathrm{P}({\omega\in\Omega: g(X (\ omega))\in B})\&=\mathrm{P}\left(\left{\omega\in\Omega:X(\omega)\in g^{-1}(B)\right} \right )=\mathrm{P}\left(X\in g^{-1}(B)\right)\end{aligned}

F是(是)=磷(是≤是)=磷(是∈(−∞,是])=磷(G(X)∈(−∞,是]) =磷(X∈G−1((−∞,是])) ={∑X:G(X)≤是FX(X) 如果 X 是离散的，  ∫X:G(X)≤是FX(X)dX 如果 X 是连续的。

F是(是)=磷(是=是)=磷(G(X)=是)=磷(X∈G−1(是))=∑X:G(X)=是FX(X)

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