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数学代写|统计计算作业代写Statistical Computing代考|Resampling methods

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数学代写|统计计算作业代写Statistical Computing代考|Bootstrap estimates

The basis of all resampling methods is to replace the distribution given by a model with the ’empirical distribution’ of the given data, as described in the following definition.

Definition 5.7 Given a sequence $x=\left(x_{1}, x_{2}, \ldots, x_{M}\right)$, the distribution of $X^{}=x_{K}$, where the index $K$ is random and uniformly distributed on the set ${1,2, \ldots, M}$, is called the empirical distribution of the $x_{i}$. In this chapter we denote the empirical distribution of $x$ by $P_{x}^{}$.

In the definition, the vector $x$ is assumed to be fixed. The randomness in $X^{}$ stems from the choice of a random element, with index $K \sim \mathcal{U}{1,, 2, \ldots, M}$, of this fixed sequence. Computational methods which are based on the idea of approximating an unknown ‘true’ distribution by an empirical distribution are called bootstrap methods. Assume that $X^{}$ is distributed according to the empirical distribution $P_{x}^{}$. Then we have $$P\left(X^{}=a\right)=\frac{1}{M} \sum_{i=1}^{M} \mathbb{1}{{a}}\left(x{i}\right)$$
that is under the empirical distribution, the probability that $X^{}$ equals $a$ is given by the relative frequency of occurrences of $a$ in the given data. Similarly, we have the relations $$P\left(X^{} \in A\right)=\frac{1}{M} \sum_{i=1}^{M} \mathbb{1}{A}\left(x{i}\right)$$
and
\begin{aligned} \mathbb{E}\left(f\left(X^{}\right)\right) &=\sum_{a \in\left{x_{1}, \ldots, x_{M}\right}} f(a) P\left(X^{}=a\right) \ &=\sum_{a \in\left{x_{1}, \ldots, x_{M}\right}} f(a) \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}{{a}}\left(x{i}\right)=\frac{1}{M} \sum_{i=1}^{M} f\left(x_{i}\right) \end{aligned}
Some care is needed when verifying this relation: the sums where the index $a$ runs over the set $\left{x_{1}, \ldots, x_{M}\right}$ have only one term for each element of the set, even if the corresponding value occurs repeatedly in the given data.

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Applications to statistical inference

The main application of the bootstrap method in statistical inference is to quantify the accuracy of parameter estimates.

In this section, we will consider parameters as a function of the corresponding distribution: if $\theta$ is a parameter, for example the mean or the variance, then we write $\theta(P)$ for the corresponding parameter. In statistics, there are many ways of constructing estimators for a parameter $\theta$. One general method for constructing parameter estimators, the plug-in principle, is given in the following definition.

Definition $5.12$ Consider an estimator $\hat{\theta}{n}=\hat{\theta}{n}\left(X_{1}, \ldots, X_{n}\right)$ for a parameter $\theta(P)$. The estimator $\hat{\theta}{n}$ satisfies the plug-in principle, if it satisfies the relation $$\hat{\theta}{n}\left(x_{1}, \ldots, x_{n}\right)=\theta\left(P_{x}^{}\right),$$ for all $x=\left(x_{1}, \ldots, x_{n}\right)$, where $P_{x}^{}$ is the empirical distribution of $x$. In this case, $\hat{\theta}_{n}$ is called the plug-in estimator for $\theta$.

Since the idea of bootstrap methods is to approximate the distribution $P$ by the empirical distribution $P_{x}^{*}$, plug-in estimators are particularly useful in conjunction with bootstrap methods.

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|BOOTSTRAP ESTIMATES

$$P\left(X^{} \in A\right)=\frac{1}{M} \sum_{i=1}^{M} \mathbb{1}{A}\left(x{i}\right)$$
and
\begin{aligned} \mathbb{E}\left(f\left(X^{ }\right)\right) &=\sum_{a \in\left{x_{1}, \ldots, x_{M}\right}} f(a) P\left(X^{*}=a\right) \ &=\sum_{a \in\left{x_{1}, \ldots, x_{M}\right}} f(a) \frac{1}{M} \sum_{i=1}^{M} \mathbb{1}{{a}}\left(x{i}\right)=\frac{1}{M} \sum_{i=1}^{M} f\left(x_{i}\right) . \end{aligned}

数学代写|统计计算作业代写STATISTICAL COMPUTING代考|APPLICATIONS TO STATISTICAL INFERENCE

bootstrap 方法在统计推断中的主要应用是量化参数估计的准确性。

$\hat{\theta}{n}=\hat{\theta}{n}\left(X_{1}, \ldots, X_{n}\right)$ for a parameter $\theta(P)$. The estimator $\hat{\theta}{n}$ satisfies the plug-in principle, if it satisfies the relation $$\hat{\theta}{n}\left(x_{1}, \ldots, x_{n}\right)=\theta\left(P_{x}^{}\right)$$ for all $x=\left(x_{1}, \ldots, x_{n}\right)$, where $P_{x}^{}$ is the empirical distribution of $x$. In this case, $\hat{\theta}_{n}$ is called the plug-in estimator for $\theta$.