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# 数学代写|统计计算作业代写Statistical Computing代考|Applications to statistical inference

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## 数学代写|统计计算作业代写Statistical Computing代考|Point estimators

A point estimator (or an estimator in short) for the parameter $\theta$ is any function of the random sample $X$ with values in $\Theta$. Typically, we write $\hat{\theta}=\hat{\theta}(X)=\hat{\theta}\left(X_{1}, \ldots, X_{n}\right)$ to denote an estimator for a parameter $\theta$. The value of the estimator for the observed data $x$, that is $\hat{\theta}(x)$ is called a point estimate (or an estimate) for $\theta$.

While the definition of an estimator does not refer to the ‘true’ value $\theta$, useful estimators will have the property that $\hat{\theta}$ is close to $\theta$. How close the estimate is to the exact value determines the quality of an estimator. This is measured by quantities like the bias and the standard error. In simple cases, for example when the data consist of independent, normally distributed values or in the limit $n \rightarrow \infty$, it is possible to determine the bias and standard error of estimators analytically. In more complicated cases, exact expressions for the bias and the standard error are no longer available. Here we will illustrate how Monte Carlo estimation can be used in these cases, to obtain numerical approximations for the bias and the standard error of an estimator.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Confidence intervals

If the set $\Theta$ of all possible parameter values is one-dimensional, that is if $\Theta \subseteq \mathbb{R}$, we can draw inference about the unknown parameter $\theta$ using confidence intervals. Confidence intervals serve a similar purpose as point estimators but, instead of returning just one ‘plausible’ value of the parameter, they determine a range of possible parameter values, chosen large enough so that the true parameter value lies inside the range with high probability.

A confidence interval with confidence coefficient $1-\alpha$ for a parameter $\theta$ is a random interval $[U, V] \subset \mathbb{R}$ where $U=U(X)$ and $V=V(X)$ are functions of the random sample $X=\left(X_{1}, \ldots, X_{n}\right)$, such that
$$P_{\theta}(\theta \in[U(X), V(X)]) \geq 1-\alpha$$
for all $\theta \in \Theta$. The subscript $\theta$ on the probability $P$ indicates that the random sample $X=\left(X_{1}, \ldots, X_{n}\right)$, for the purpose of computing the probability in (3.24), is distributed according to the distribution with parameter $\theta$.

## 数学代写|统计计算作业代写STATISTICAL COMPUTING代考|Hypothesis tests

In hypothesis testing, inference about an unknown parameter is restricted to the question of whether or not the parameter satisfies a given ‘hypothesis’ $H_{0}$. Such a hypothesis about the parameter $\theta$ could, for example, be a statement like ‘ $\theta=0$ ‘ or ‘ $\theta>0$ ‘. The alternative hypothesis, that is the hypothesis that $\theta$ does not satisfy $H_{0}$, is denoted by $H_{1}$.While the dichotomy between $H_{0}$ and $H_{1}$ is symmetric, it transpires that in most situations any given statistical test can only determine for one of the hypotheses whether it is likely to be true, whereas the other hypothesis can only be shown to be likely to be wrong. Traditionally the names are chosen such that $H_{0}$ is the hypothesis which can only be disproved (called the null hypothesis) and $H_{1}$ is the hypothesis which can be proved. The two possible outcomes of a statistical test are then ‘ $H_{0}$ has been rejected’ and ‘ $H_{0}$ has not been rejected’.