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

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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|Starting point

We typically start with a subject-matter question. Data are or become available to address this question. After preliminary screening, checks of data quality and simple tabulations and graphs, more formal analysis starts with a provisional model. The data are typically split in two parts $(y: z)$, where $y$ is regarded as the observed value of a vector random variable $Y$ and $z$ is treated as fixed. Sometimes the components of $y$ are direct measurements of relevant properties on study individuals and sometimes they are themselves the outcome of some preliminary analysis, such as means, measures of variability, regression coefficients and so on. The set of variables $z$ typically specifies aspects of the system under study that are best treated as purely explanatory and whose observed values are not usefully represented by random variables. That is, we are interested solely in the distribution of outcome or response variables conditionally on the variables $z$; a particular example is where $z$ represents treatments in a randomized experiment.
We use throughout the notation that observable random variables are represented by capital letters and observations by the corresponding lower case letters.
A model, or strictly a family of models, specifies the density of $Y$ to be
$$f_{Y}(y: z ; \theta)$$

where $\theta \subset \Omega_{\theta}$ is unknown. The distribution may depend also on design features of the study that generated the data. We typically simplify the notation to $f_{Y}(y ; \theta)$, although the explanatory variables $z$ are frequently essential in specific applications.
To choose the model appropriately is crucial to fruitful application.
We follow the very convenient, although deplorable, practice of using the term density both for continuous random variables and for the probability function of discrete random variables. The deplorability comes from the functions being dimensionally different, probabilities per unit of measurement in continuous problems and pure numbers in discrete problems. In line with this convention in what follows integrals are to be interpreted as sums where necessary. Thus we write
$$E(Y)=E(Y ; \theta)=\int y f_{Y}(y ; \theta) d y$$
for the expectation of $Y$, showing the dependence on $\theta$ only when relevant. The integral is interpreted as a sum over the points of support in a purely discrete case. Next, for each aspect of the research question we partition $\theta$ as $(\psi, \lambda)$, where $\psi$ is called the parameter of interest and $\lambda$ is included to complete the specification and commonly called a nuisance parameter. Usually, but not necessarily, $\psi$ and $\lambda$ are variation independent in that $\Omega_{\theta}$ is the Cartesian product $\Omega_{\psi} \times \Omega_{\lambda}$. That is, any value of $\psi$ may occur in connection with any value of $\lambda$. The choice of $\psi$ is a subject-matter question. In many applications it is best to arrange that $\psi$ is a scalar parameter, i.e., to break the research question of interest into simple components corresponding to strongly focused and incisive research questions, but this is not necessary for the theoretical discussion.

## 统计代写|统计推断作业代写statistics interference代考|Role of formal theory of inference

The formal theory of inference initially takes the family of models as given and the objective as being to answer questions about the model in the light of the data. Choice of the family of models is, as already remarked, obviously crucial but outside the scope of the present discussion. More than one choice may be needed to answer different questions.

A second and complementary phase of the theory concerns what is sometimes called model criticism, addressing whether the data suggest minor or major modification of the model or in extreme cases whether the whole focus of the analysis should be changed. While model criticism is often done rather informally in practice, it is important for any formal theory of inference that it embraces the issues involved in such checking.

## 统计代写|统计推断作业代写statistics interference代考|Some simple models

General notation is often not best suited to special cases and so we use more conventional notation where appropriate.

Example 1.1. The normal mean. Whenever it is required to illustrate some point in simplest form it is almost inevitable to return to the most hackneyed of examples, which is therefore given first. Suppose that $Y_{1}, \ldots, Y_{n}$ are independently normally distributed with unknown mean $\mu$ and known variance $\sigma_{0}^{2}$. Here $\mu$ plays the role of the unknown parameter $\theta$ in the general formulation. In one of many possible generalizations, the variance $\sigma^{2}$ also is unknown. The parameter vector is then $\left(\mu, \sigma^{2}\right)$. The component of interest $\psi$ would often be $\mu$

but could be, for example, $\sigma^{2}$ or $\mu / \sigma$, depending on the focus of subject-matter interest.

Example 1.2. Linear regression. Here the data are $n$ pairs $\left(y_{1}, z_{1}\right), \ldots,\left(y_{n}, z_{n}\right)$ and the model is that $Y_{1}, \ldots, Y_{n}$ are independently normally distributed with variance $\sigma^{2}$ and with
$$E\left(Y_{k}\right)=\alpha+\beta z_{k} .$$
Here typically, but not necessarily, the parameter of interest is $\psi=\beta$ and the nuisance parameter is $\lambda=\left(\alpha, \sigma^{2}\right)$. Other possible parameters of interest include the intercept at $z=0$, namely $\alpha$, and $-\alpha / \beta$, the intercept of the regression line on the $z$-axis.

Example 1.3. Linear regression in semiparametric form. In Example $1.2$ replace the assumption of normality by an assumption that the $Y_{k}$ are uncorrelated with constant variance. This is semiparametric in that the systematic part of the variation, the linear dependence on $z_{k}$, is specified parametrically and the random part is specified only via its covariance matrix, leaving the functional form of its distribution open. A complementary form would leave the systematic part of the variation a largely arbitrary function and specify the distribution of error parametrically, possibly of the same normal form as in Example 1.2. This would lead to a discussion of smoothing techniques.

Example 1.4. Linear model. We have an $n \times 1$ vector $Y$ and an $n \times q$ matrix $z$ of fixed constants such that
$$E(Y)=z \beta, \quad \operatorname{cov}(Y)=\sigma^{2} I,$$
where $\beta$ is a $q \times 1$ vector of unknown parameters, $I$ is the $n \times n$ identity matrix and with, in the analogue of Example 1.2, the components independently normally distributed. Here $z$ is, in initial discussion at least, assumed of full rank $q<n$. A relatively simple but important generalization has $\operatorname{cov}(Y)=$ $\sigma^{2} V$, where $V$ is a given positive definite matrix. There is a corresponding semiparametric version generalizing Example 1.3.

Both Examples $1.1$ and $1.2$ are special cases, in the former the matrix $z$ consisting of a column of $1 \mathrm{~s}$.

Example 1.5. Normal-theory nonlinear regression. Of the many generalizations of Examples $1.2$ and 1.4, one important possibility is that the dependence on the parameters specifying the systematic part of the structure is nonlinear. For example, instead of the linear regression of Example $1.2$ we might wish to consider
$$E\left(Y_{k}\right)=\alpha+\beta \exp \left(\gamma z_{k}\right)$$

F是(是:和;θ)

## 广义线性模型代考

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