### 统计代写|应用随机过程代写Stochastic process代考|Branching processes

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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|应用随机过程代写Stochastic process代考|Branching processes

The Bienaymé-Galton-Watson branching process was originally introduced as a model for the survival of family surnames over generations and has later been applied in areas such as survival of genes. The process is defined as follows. Assume that at time 0 , a population consists of a single individual who lives for a single time unit and then dies and is replaced by his offspring. These offspring all survive for a further single time unit and are then replaced by their offspring, and so on.

Formally, define $Z_{n}$ to be the population after time $n$. Then, $Z_{0}=1$. Also let $X_{i j}$ be the number of offspring born to the $j$ th individual in generation $i$. Assume that the $X_{i j}$ are all independent and identically distributed variables, $X_{i j} \sim X$, with some distribution $P(X=x)=p_{x}$ for $x=0,1,2, \ldots$ where we assume that $p_{0}>0$. Then,
$$Z_{n+1}=\sum_{j=1}^{Z_{s}} X_{n j}$$
Interest is usually focused on the probability $\gamma$ of extinction,
$$\gamma=P\left(Z_{n}=0, \text { for some } n=1,2, \ldots\right)$$
It is well known that extinction is certain if $\theta=E[X] \leq 1$. Otherwise, $\gamma$ is the smallest root of the equation $G(s)=s$, where $G(s)$ is the probability generating function of $X$ (see Appendix B). Obviously, if the initial population is of size $k>1$, then the probability of eventual extinction is $\gamma^{k}$.
Inference for branching processes is provided in Section 3.4.4.

## 统计代写|应用随机过程代写Stochastic process代考|Hidden Markov models

Hidden Markov models (HMMs) have been widely applied to the analysis of weakly dependent data in diverse areas such as econometrics, ecology, and signal processing. A hidden Markov model is defined as follows. Observations $Y_{n}$ for $n=0,1,2, \ldots$ are generated from a conditional distribution $f\left(y_{n} \mid X_{n}\right)$ with parameters that depend on an unobserved or hidden state, $X_{n} \in{1,2, \ldots K}$. The hidden states follow a Markov

chain with transition matrix $\boldsymbol{P}$ and an initial distribution, usually assumed to be the equilibrium distribution, $\pi(\cdot \mid \boldsymbol{P})$, of the underlying Markov chain.

The architecture of this process can be represented by an influence diagram as in Figure 3.1, with arrows denoting conditional dependencies.In the preceding text, we are assuming that the hidden state space of the HMM is discrete. However, it is straightforward to extend the definition to HMMs with a continuous state space. A simple example is the dynamic linear model described in Section 2.4.1. Inference for HMMs is overviewed in Section 3.4.5.

## 统计代写|应用随机过程代写Stochastic process代考|Inference for first-order, time homogeneous, Markov chains

In this section, we study inference for a first-order, time homogeneous, Markov chain, $\left{X_{n}\right}$, with state space ${1,2, \ldots, K}$ and (unknown) transition matrix $\boldsymbol{P}$.

Initially, we consider the simple experiment of observing $m$ successive transitions of the Markov chain, say $X_{1}=x_{1}, \ldots, X_{m}=x_{m}$, given a known initial state $X_{0}=x_{0}$. In this case, the likelihood function is
$$l(\boldsymbol{P} \mid \mathbf{x})=\prod_{i=1}^{K} \prod_{j=1}^{K} p_{i j}^{n_{i j}},$$
where $n_{i j} \geq 0$ is the number of observed transitions from state $i$ to state $j$ and $\sum_{i=1}^{K} \sum_{j=1}^{K} n_{i j}=m .$

Given the likelihood function (3.3), it is easy to show that the classical, maximum likelihood estimate for $\boldsymbol{P}$ is $\hat{\boldsymbol{P}}$ with $i, j$ th element equal to the proportion of transitions from state $i$ that go to state $j$, that is,
$$\hat{p}{i j}=\frac{n{i j}}{n_{i}}, \quad \text { where } \quad n_{i},=\sum_{j=1}^{K} n_{i j}$$
However, especially in chains where the number $K$ of states is large and, therefore, a very large number $K^{2}$ of transitions are possible, it will often be the case that there are no observed transitions between various pairs, $(i, j)$, of states and thus $\hat{p}_{i j}=0$.

## 统计代写|应用随机过程代写Stochastic process代考|Branching processes

Bienaymé-Galton-Watson 分支过程最初是作为家族姓氏世代生存的模型引入的，后来被应用于基因生存等领域。该过程定义如下。假设在时间 0 ，人口由一个个体组成，该个体生活了一个时间单位，然后死亡并被他的后代所取代。这些后代都存活了一个时间单位，然后被它们的后代取代，依此类推。

C=磷(从n=0, 对于一些 n=1,2,…)

## 统计代写|应用随机过程代写Stochastic process代考|Inference for first-order, time homogeneous, Markov chains

l(磷∣X)=∏一世=1ķ∏j=1ķp一世jn一世j,

p^一世j=n一世jn一世, 在哪里 n一世,=∑j=1ķn一世j

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