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# 数学代写|统计计算作业代写Statistical Computing代考|Studying models via simulation

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## 数学代写|统计计算作业代写Statistical Computing代考|observation underlying

When studying statistical models, analytical calculations often are only possible under assumptions such as independence of samples, normality of samples or large sample size. For this reason, many problems occurring in ‘real life’ situations are only approximately covered by the available analytical results. This chapter presents an alternative approach to such problems, based on estimates derived from computer simulations instead of analytical calculations.

The fundamental observation underlying the methods discussed in this and the following chapters is the following: if we can simulate a statistical model on a computer, then we can generate a large set of samples from the model and then we can learn about the behaviour of the model by studying the computer-generated set of samples instead of the model itself. We give three examples for this approach:

• As a consequence of the law of large numbers (see theorem A.8), the expected value of a random variable can be approximated by generating a large number of samples of the random variable and then considering the average value.
• The probability of an event can be approximated by generating a large number of samples and then considering the proportion of samples where the event occurs.
• The quality of a method for statistical inference can be assessed by repeatedly generating synthetic data with a known distribution and then analysing how well the inference method recovers the (known) properties of the underlying distribution from the synthetic data sets.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Poisson distribution

Since many interesting questions can be reduced to computing the expectation of some random variable, we will mostly restrict our attention to the problem of computing expectations of the form $\mathbb{E}(f(X))$ where $X$ is a random sample from the system under consideration and $f$ is a real-valued function, determining some quantity of interest in the system. There are several different methods to compute such an expectation:
(a) Sometimes we can find the answer analytically. For example, if the distribution of $X$ has a density $\varphi$, we can use the relation
$$\mathbb{E}(f(X))=\int f(x) \varphi(x) d x$$
to obtain the value of the expectation (see Section A.3). This method only works if we can solve the resulting integral.
(b) If the integral in (3.1) cannot be solved analytically, we can try to use numerical integration to get an approximation to the value of the integral. When $X$ takes values in a low-dimensional space, this method often works well, but for higher dimensional spaces numerical approximation can become very expensive and the resulting method may no longer be efficient. Since numerical integration is outside the topic of statistical computing, we will not follow this approach here.
(c) The approach we will study in this chapter is called Monte Carlo estimation or Monte Carlo integration. This technique is based on the strong law of large numbers: if $\left(X_{j}\right){j \in \mathbb{N}}$ is a sequence of i.i.d. random variables with the same distribution as $X$, then $$\mathbb{E}(f(X))=\lim {N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^{N} f\left(X_{j}\right)$$
with probability $1 .$
Our aim for this chapter is to study approximations for $\mathbb{E}(f(X))$ based on Equation (3.2). While the exact equality in (3.2) holds only in the limit $N \rightarrow \infty$, we can use the approximation with fixed, large $N$ to get the following approximation method.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|OBSERVATION UNDERLYING

• 由于大数定律s和和吨H和○r和米一种.8，随机变量的期望值可以通过生成大量随机变量的样本然后考虑平均值来近似。
• 一个事件的概率可以通过生成大量样本然后考虑事件发生的样本的比例来近似。
• 统计推断方法的质量可以通过重复生成具有已知分布的合成数据，然后分析推断方法恢复数据的能力来评估。到n○在n来自合成数据集的基础分布的属性。

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|POISSON DISTRIBUTION

b如果积分在3.1无法解析求解，我们可以尝试使用数值积分来获得积分值的近似值。什么时候X在低维空间中取值，这种方法通常效果很好，但对于高维空间，数值逼近可能变得非常昂贵，并且得到的方法可能不再有效。由于数值积分超出了统计计算的主题，因此我们不会在这里采用这种方法。
C我们将在本章中研究的方法称为蒙特卡洛估计或蒙特卡洛积分。该技术基于强数定律：如果 \mathbb{E}(f(X))=\lim {N \rightarrow \infty} \frac{1}{N} \sum{j=1}^{N} f\left(X_{j}\right)