# 统计代写|统计推断代写Statistical inference代考|Pivotal functions

## 统计代写|统计推断代写Statistical inference代考|Pivotal functions

Consider the following situation: if $\boldsymbol{Y}$ is a random sample from an $\mathrm{N}(\mu, 1)$ distribution and $\bar{Y}$ is the sample mean, we know that $\sqrt{n}(\bar{Y}-\mu) \sim \mathrm{N}(0,1)$ is a function of $\boldsymbol{Y}$ and $\mu$ whose distribution does not depend on $\mu$. This is an example of a pivotal function. Pivotal functions play a fundamental (or, even, pivotal) role in the construction of confidence intervals. We start with a more formal definition.
Definition 8.1.2 (Pivotal function)
Consider a sample $\boldsymbol{Y}$ and a scalar parameter $\theta$. Let $g(\boldsymbol{Y}, \theta)$ be a function of $\boldsymbol{Y}$ and $\theta$ that does not involve any unknown parameter other than $\theta$. We say that $g(\boldsymbol{Y}, \theta)$ is a pivotal function if its distribution does not depend on $\theta$.

Note that a pivotal function defines a random variable, say $W=g(\boldsymbol{Y}, \theta)$. By definition, the distribution of $W$ does not depend on $\theta$.

We illustrate the use of pivotal functions with examples. Before we start on the examples, we introduce a distribution that plays an important role in both interval estimation and hypothesis testing.
Definition 8.1.3 ( $t$-distribution)
Suppose that $Z$ has a standard normal distribution and $V$ has a chi-squared distribution on $k$ degrees of freedom, that is, $Z \sim \mathrm{N}(0,1)$ and $V \sim \chi_k^2$. Suppose also that $Z$ and $V$ are independent. If we define
$$T=\frac{Z}{\sqrt{V / k}},$$
then $T$ has a $t$-distribution on $k$ degrees of freedom, denoted $T \sim t_k$. The density function of the $t$-distribution is
$$f_T(t)=\frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k \pi} \Gamma\left(\frac{k}{2}\right)}\left(1+\frac{t^2}{k}\right)^{-(k+1) / 2} \text { for }-\infty<t<\infty .$$

## 统计代写|统计推断代写Statistical inference代考|Point estimation

Consider a scalar parameter $\theta$; here, $\theta$ is just a single unknown number. Any method for estimating the value of $\theta$ based on a sample of size $n$ can be represented as a function $h: \mathbb{R}^n \rightarrow \mathbb{R}$. When this function is applied to the observed sample it yields a point estimate, $h(\boldsymbol{y})$. This estimate is just a number. In order to gain an insight into the properties of the estimation method, we consider applying the function to the sample. The resulting random variable, $h(\boldsymbol{Y})$, is referred to as a point estimator. Notice the rather subtle distinction here between an estimator, which is a statistic (random variable), and an estimate, which is an observed value of a statistic (just a number).

It is clear that any point estimator is a statistic. In fact, this association goes in both directions, as the following definition makes clear.
Definition 8.2.1 (Point estimator)
Any scalar statistic may be taken to be a point estimator for a parameter, $\theta$. An observed value of this statistic is referred to as a point estimate.

Definition 8.2.1 seems rather loose; we do not mention anything about restricting our attention to point estimators that are likely to yield estimates close to the true value. In subsection 8.2 .1 we introduce the concepts of bias, mean squared error, and consistency. These concepts allow us to formalise “likely to yield estimates close to the true value” as a desirable property of a point estimator. Some commonly used point estimators are given in the following example.
Example 8.2.2 (Some well-known point estimators)
Consider an observed sample $y=\left(y_1, \ldots, y_n\right)^T$ that we view as an instance of the sample $\boldsymbol{Y}=\left(Y_1, \ldots, Y_n\right)^T$. An obvious statistic to calculate is the sample mean. The observed value of the sample mean is
$$\bar{y}=\frac{1}{n} \sum_{j=1}^n y_j$$

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|Pivotal functions

$$T=\frac{Z}{\sqrt{V / k}}$$

$$f_T(t)=\frac{\Gamma\left(\frac{k+1}{2}\right)}{\sqrt{k \pi} \Gamma\left(\frac{k}{2}\right)}\left(1+\frac{t^2}{k}\right)^{-(k+1) / 2} \text { for }-\infty<t<\infty .$$

## 统计代写|统计推断代写Statistical inference代考|Point estimation

$\boldsymbol{Y}=\left(Y_1, \ldots, Y_n\right)^T$. 要计算的一个明显统计量是样本均值。样本均值的观测值是
$$\bar{y}=\frac{1}{n} \sum_{j=1}^n y_j$$

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