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

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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
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• 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|Poisson process

Poisson processes are continuous time, discrete space processes that we shall analyze in detail in Chapter 5. Here, we shall distinguish between homogeneous and nonhomogeneous Poisson processes.

Definition 1.12: Suppose that the stochastic process $\left{X_{t}\right}_{t \in T}$ describes the number of events of a certain type produced until time t and has the following properties:

1. The number of events in nonoverlapping intervals are independent.
2. There is a constant $\lambda$ such that the probabilities of occurrence of events over ‘small’ intervals of duration $\Delta t$ are:
• $P$ (number of events in $(t, t+\Delta t]=1)=\lambda \Delta t+o(\Delta t)$.
• $P$ (number of events in $(t, t+\Delta t]>1)=o(\Delta t)$, where o( $\Delta t)$ is such that $o(\Delta t) / \Delta t \rightarrow 0$ when $\Delta t \rightarrow 0 .$

Then, we say that $\left{X_{t}\right}$ is an homogeneous Poisson process with parameter $\lambda$, char acterized by the fact that $X_{t} \sim P o(\lambda t)$.

For such a process, it can be proved that the times between successive events are IID random variables with distribution $\operatorname{Ex}(\lambda)$.

The Poisson process is a particular case of many important generic types of processes. Among others, it is an example of a renewal process, that is, a process describing the number of events of a phenomenon of interest occurring until a certain time such that the times between events are IID random variables (exponential in the case of the Poisson process). Poisson processes are also a special case of continuous time Markov chains, with transition probabilities $p_{i, i+1}=1, \forall i$ and $\lambda_{i}=\lambda$.
Nonhomogeneous Poisson processes
Nonhomogeneous Poisson processes are characterized by the intensity function $\lambda(t)$ or the mean function $m(t)=\int_{0}^{t} \lambda(s) \mathrm{d} s$; we consider, in general, a time-dependent intensity function but it could be space and space-time dependent as well. Note that, when $\lambda(t)=\lambda$, we have an homogeneous Poisson process. For a nonhomogeneous Poisson process, the number of events occurring in the interval $(t, t+s]$ will have a $\mathrm{Po}(m(t+s)-m(t))$ distribution.

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

The Gaussian process is continuous in both time and state spaces. Let $\left{X_{t}\right}$ be a stochastic process such that for any $n$ times $\left{t_{1}, t_{2}, \ldots, t_{n}\right}$ the joint distribution of $X_{t_{i}}, i=1,2, \ldots, n$, is $n$-variate normal. Then, the process is Gaussian. Moreover, if for any finite set of time instants $\left{t_{i}\right}, i=1,2, \ldots$ the random variables are mutually independent and $X_{t}$ is normally distributed for every $t$, we call it a purely random Gaussian process.

Because of the specific properties of the normal distribution, we may easily specify many properties of a Gaussian process. For example, if a Gaussian process is weakly stationary, then it is strictly stationary.

## 统计代写|应用随机过程代写Stochastic process代考|Inference, prediction, and decision-making

Given the key definitions and results concerning stochastic processes, we can now informally set up the statistical and decision-making problems that we shall deal with in the following chapters.

Clearly, stochastic processes will be characterized by their initial value and the values of their parameters, which may be finite or infinite dimensional.

Example 1.3: In the case of the gambler’s ruin problem of Example $1.2$ the process is parameterized by $p$, the probability of heads. More generally, for a stationary finite Markov chain model with states $1,2, \ldots, k$, the parameters will be the transition probabilities $\left(p_{11}, \ldots, p_{k, k}\right)$, where $p_{i j}$ satisfy that $p_{i j} \geq 0$ and $\sum_{j} p_{i j}=1$.

The AR(1) process of Example $1.1$ is parameterized through the parameters $\phi_{0}$ and $\phi_{1}$ –

A nonhomogeneous Poisson process with intensity function $\lambda(t)=M \beta t^{\beta-1}$, corresponding to a Power Law model, is a finite parametric model with parameters $(M, \beta)$.

A normal dynamic linear model (DLM) with univariate observations $X_{n}$, is described by
\begin{aligned} \theta_{0} \mid D_{0} & \sim \mathrm{N}\left(m_{0}, C_{0}\right) \ \theta_{n} \mid \theta_{n-1} & \sim \mathrm{N}\left(\boldsymbol{G}{n} \theta{n-1}, \boldsymbol{W}{n}\right) \ X{n} \mid \theta_{n} & \sim \mathrm{N}\left(F_{n}^{\prime} \theta_{n}, V_{n}\right) \end{aligned}
where, for each $n, F_{n}$ is a known vector of dimension $m \times 1, \boldsymbol{G}{n}$ is a known $m \times m$ matrix, $V{n}$ is a known variance, and $W_{n}$ is a known $m \times m$ variance matrix. The parameters are now $\left{\theta_{0}, \theta_{1}, \ldots\right}$.

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

1. 非重叠间隔中的事件数是独立的。
2. 有一个常数λ使得事件在“小”持续时间间隔内发生的概率Δ吨是：
• 磷（事件数(吨,吨+Δ吨]=1)=λΔ吨+这(Δ吨).
• 磷（事件数(吨,吨+Δ吨]>1)=这(Δ吨), 其中 o(Δ吨)是这样的这(Δ吨)/Δ吨→0什么时候Δ吨→0.

## 统计代写|应用随机过程代写Stochastic process代考|Inference, prediction, and decision-making

θ0∣D0∼ñ(米0,C0) θn∣θn−1∼ñ(Gnθn−1,在n) Xn∣θn∼ñ(Fn′θn,在n)

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