### 统计代写|统计推断作业代写statistics interference代考| Formulation of objectives

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断作业代写statistics interference代考|Formulation of objectives

We can, as already noted, formulate possible objectives in two parts as follows.
Part I takes the family of models as given and aims to:

• give intervals or in general sets of values within which $\psi$ is in some sense likely to lie;
• assess the consistency of the data with a particular parameter value $\psi_{0}$;
• predict as yet unobserved random variables from the same random system that generated the data;
• use the data to choose one of a given set of decisions $\mathcal{D}$, requiring the specification of the consequences of various decisions.
Part II uses the data to examine the family of models via a process of model criticism. We return to this issue in Section 3.2.

We shall concentrate in this book largely but not entirely on the first two of the objectives in Part I, interval estimation and measuring consistency with specified values of $\psi$.

To an appreciable extent the theory of inference is concerned with generalizing to a wide class of models two approaches to these issues which will be outlined in the next section and with a critical assessment of these approaches.

## 统计代写|统计推断作业代写statistics interference代考|General remarks

Consider the first objective above, that of providing intervals or sets of values likely in some sense to contain the parameter of interest, $\psi$.

There are two broad approaches, called frequentist and Bayesian, respectively, both with variants. Alternatively the former approach may be said to be based on sampling theory and an older term for the latter is that it uses imverse probability. Much of the rest of the book is concerned with the similarities and differences between these two approaches. As a prelude to the general development we show a very simple example of the arguments involved.

We take for illustration Example 1.1, which concerns a normal distribution with unknown mean $\mu$ and known variance. In the formulation probability is used to model variability as experienced in the phenomenon under study and its meaning is as a long-run frequency in repetitions, possibly, or indeed often, hypothetical, of that phenomenon.

What can reasonably be said about $\mu$ on the basis of observations $y_{1}, \ldots, y_{n}$ and the assumptions about the model?

## 统计代写|统计推断作业代写statistics interference代考|Frequentist discussion

In the first approach we make no further probabilistic assumptions. In particular we treat $\mu$ as an unknown constant. Strong arguments can be produced for reducing the data to their mean $\bar{y}=\Sigma y_{k} / n$, which is the observed value of the corresponding random variable $\bar{Y}$. This random variable has under the assumptions of the model a normal distribution of mean $\mu$ and variance $\sigma_{0}^{2} / n$, so that in particular
$$P\left(\bar{Y}>\mu-k_{c}^{} \sigma_{0} / \sqrt{n}\right)=1-c$$ where, with $\Phi\left(\right.$.) denoting the standard normal integral, $\Phi\left(k_{c}^{}\right)=1-c$. For example with $c=0.025, k_{c}^{}=1.96$. For a sketch of the proof, see Note $1.5$. Thus the statement equivalent to (1.9) that $$P\left(\mu<\bar{Y}+k_{c}^{} \sigma_{0} / \sqrt{n}\right)=1-c,$$
can be interpreted as specifying a hypothetical long run of statements about $\mu$ a proportion $1-c$ of which are correct. We have observed the value $\bar{y}$ of the random variable $\bar{Y}$ and the statement
$$\mu<\bar{y}+k_{c}^{*} \sigma_{0} / \sqrt{n}$$
is thus one of this long run of statements, a specified proportion of which are correct. In the most direct formulation of this $\mu$ is fixed and the statements vary and this distinguishes the statement from a probability distribution for $\mu$. In fact a similar interpretation holds if the repetitions concern an arbitrary sequence of fixed values of the mean.

There are a large number of generalizations of this result, many underpinning standard elementary statistical techniques. For instance, if the variance $\sigma^{2}$ is unknown and estimated by $\Sigma\left(y_{k}-\bar{y}\right)^{2} /(n-1)$ in $(1.9)$, then $k_{c}^{*}$ is replaced by the corresponding point in the Student $t$ distribution with $n-1$ degrees of freedom.

There is no need to restrict the analysis to a single level $c$ and provided concordant procedures are used at the different $c$ a formal distribution is built up.
Arguments involving probability only via its (hypothetical) long-run frequency interpretation are called frequentist. That is, we define procedures for assessing evidence that are calibrated by how they would perform were they used repeatedly. In that sense they do not differ from other measuring instruments. We intend, of course, that this long-run behaviour is some assurance that with our particular data currently under analysis sound conclusions are drawn. This raises important issues of ensuring, as far as is feasible, the relevance of the long run to the specific instance.

## 统计代写|统计推断作业代写statistics interference代考|Formulation of objectives

• 给出区间或一般的一组值，其中ψ在某种意义上可能会撒谎；
• 评估数据与特定参数值的一致性ψ0;
• 从生成数据的同一随机系统中预测尚未观察到的随机变量；
• 使用数据来选择一组给定的决策D，要求说明各种决定的后果。
第二部分通过模型批评过程使用数据来检查模型族。我们在第 3.2 节中回到这个问题。

μ<是¯+到C∗σ0/n

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