This course extends statistical modelling from independently distributed data to modelling dependence in observed data, and develops an understanding of basic distributions and models useful in practical situations. It continues with the development of statistical methods for analysis of dependent data arising in multivariate observations, time series and spatial processes, and also covers multivariate normal distribution, Hotelling’s T-squared, Wishart distribution. It concludes with a study of time series models, stationary time series, ARMA and forecasting ARMA, spatial processes, sample semivariograms and Kriging.
这是一份unsw新南威尔士大学MATH3841代写的成功案例
– For any fixed $x \in \mathbb{R}{+}$, the following unconditional weak convergence holds, as $n$ tends to infinity $$ \left{\tilde{S}{t}^{n}: t \in \mathbb{R}{+}^{}\right} \leadsto\left{\mathcal{C}^{1 / 2} W(t): t \in \mathbb{R}{+}^{}\right}
$$
For any states $i, j \in E$ and any fixed time $x \in \mathbb{R}{+}$, we set $V{k}=\left{V_{k}^{i j}: i, j \in\right.$ $E} \in \mathbb{R}^{d^{2}}$, where $\left(V_{k}^{i j}\right){k \in \mathbb{N}^{*}}$ are random sequences on $\widetilde{\mathcal{B}}$ defined by $$ V{k}:=\left{\mathbb{1}{\left{J{k-1}=i, J_{k}=j, X_{k} \leq x\right}}-\mathbb{1}{\left{J{k-1}=i\right}} F_{i j}(x)\right} .
$$
For each $i, j \in E$ and $k \in \mathbb{N}^{}$ fixed, we have, as in $[2.8]$, $$ \mathbb{E}\left(V_{k}^{i j} \mid \mathcal{F}{k-1}\right)=0 $$ This, in turn, implies that $\left(V{k}^{n}\right){k \in \mathbb{N}^{}}$ is a $\left(\mathcal{F}{k}\right)$-martingale difference for any $n \in$ $\mathbb{N}^{}$. Let us define the double sequence $\left(V_{k}^{n}\right){k \in \mathbb{N}^{}}, n \in \mathbb{N}^{}$, on $\widetilde{\mathcal{B}}^{n}$, by $$ V{k}^{n}=\frac{V_{k}}{\sqrt{n}} .
$$
Also, for any $n \in \mathbb{N}^{}$, let us define the bootstrapped stochastic processes
$$
\widetilde{T}{t}^{i j, n}:=\sum{k=1}^{N(n t)} z_{k} V_{k}^{i j, n}=\frac{1}{\sqrt{n}} \sum_{k=1}^{N(n t)} z_{k} V_{k}^{i j}, t \in \mathbb{R}{+}^{} . $$ Making use of the notation above, we have the sequence of stochastic processes on $\mathcal{B}^{n}$ $$ \tilde{T}{t}^{n}=\left{\tilde{T}{t}^{i j, n}: i, j \in E\right}, \quad t \in \mathbb{R}{+}^{}, n \in \mathbb{N}^{*}
$$
– If $\tilde{\psi}(t)=\gamma \psi\left(\alpha t+s^{}\right)+s^{ }$, then $$ \mathbf{B}{k, \tilde{\psi}}(t)=\mathbf{D}{k, \alpha} \mathbf{B}{k, \psi}\left(\alpha t+s^{}\right) \mathbf{D}{k, \gamma},
$$
where $\mathbf{D}{k, \gamma}=\operatorname{Diag}\left(1, \gamma, \gamma^{2}, \ldots, \gamma^{k}\right)$. In particular, for $\tilde{\psi}(t)=\alpha t+s^{*},[6.33]$ becomes $\mathbf{B}{k, \tilde{\psi}}(t)=\mathbf{D}_{k, \alpha}$.
PROOF.- We compute $\widetilde{\psi}^{(j)}(t)=\gamma \alpha^{j} \psi^{(j)}\left(\alpha t+s^{}\right)$ and $B_{j, l}\left(\gamma \alpha u_{1}, \gamma \alpha^{2} u_{2}, \ldots\right.$, $\left.\gamma \alpha^{j-l+1} u_{j-l+1}\right)=\gamma^{l} \alpha^{j} B_{j, l}\left(u_{1}, u_{2}, \ldots, u_{j-l+1}\right)$. Therefore, we readily get $B_{j, l}\left(\widetilde{\psi}^{(1, j)}(t)\right)=\gamma^{l} \alpha^{j} B_{j, l}\left(\psi^{(1, j)}\left(\alpha t+s^{}\right)\right)$ for $1<l \leq j \leq k$, which in matrix form is [6.33]. For the particular case, just observe that $\psi(t)=t$ yields the constant $\operatorname{matrix} \mathbf{B}{k, \psi}(t)=\mathbf{I}{k+1}$.
本课程将统计建模从独立分布的数据扩展到观察数据的依赖性建模，并发展对实际情况中有用的基本分布和模型的理解。它继续发展用于分析多变量观察、时间序列和空间过程中出现的依赖性数据的统计方法，还包括多变量正态分布、Hotelling的T-squared、Wishart分布。它最后研究了时间序列模型、静止的时间序列、ARMA和预测ARMA、空间过程、样本半变量图和Kriging。
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