### 统计代写|统计推断作业代写statistics interference代考|More complicated situations

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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|General remarks

The previous frequentist discussion in especially Chapter 3 yields a theoretical approach which is limited in two senses. It is restricted to problems with no nuisance parameters or ones in which elimination of nuisance parameters is straightforward. An important step in generalizing the discussion is to extend the notion of a Fisherian reduction. Then we turn to a more systematic discussion of the role of nuisance parameters.

By comparison, as noted previously in Section 1.5, a great formal advantage of the Bayesian formulation is that, once the formulation is accepted, all subsequent problems are computational and the simplifications consequent on sufficiency serve only to ease calculations.

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

The argument outlined in Section $1.5$ for inference about the mean of a normal distribution can be generalized as follows. Consider the model $f_{Y \mid \Theta}(y \mid \theta)$, where, because we are going to treat the unknown parameter as a random variable, we now regard the model for the data-generating process as a conditional density. Suppose that $\Theta$ has the prior density $f_{\Theta}(\theta)$, specifying the marginal

distribution of the parameter, i.e., in effect the distribution $\Theta$ has when the observations $y$ are not available.

Given the data and the above formulation it is reasonable to assume that all information about $\theta$ is contained in the conditional distribution of $\Theta$ given $Y=y$. We call this the posterior distribution of $\Theta$ and calculate it by the standard laws of probability theory, as given in (1.12), by
$$f_{\Theta \mid Y}(\theta \mid y)=\frac{f_{Y \mid \Theta}(y \mid \theta) f_{\Theta}(\theta)}{\int_{\Omega_{\theta}} f_{Y \mid \Theta}(y \mid \phi) f_{\Theta}(\phi) d \phi} .$$
The main problem in computing this lies in evaluating the normalizing constant in the denominator, especially if the dimension of the parameter space is high. Finally, to isolate the information about the parameter of interest $\psi$, we marginalize the posterior distribution over the nuisance parameter $\lambda$. That is, writing $\theta=(\psi, \lambda)$, we consider
$$f_{\Psi \mid Y}(\psi \mid y)=\int f_{\Theta \mid Y}(\psi, \lambda \mid y) d \lambda$$
The models and parameters for which this leads to simple explicit solutions are broadly those for which frequentist inference yields simple solutions.

Because in the formula for the posterior density the prior density enters both in the numerator and the denominator, formal multiplication of the prior density by a constant would leave the answer unchanged. That is, there is no need for the prior measure to be normalized to 1 so that, formally at least, improper, i.e. divergent, prior densities may be used, always provided proper posterior densities result. A simple example in the context of the normal mean, Section $1.5$, is to take as prior density element $f_{M}(\mu) d \mu$ just $d \mu$. This could be regarded as the limit of the proper normal prior with variance $v$ taken as $v \rightarrow \infty$. Such limits raise few problems in simple cases, but in complicated multiparameter problems considerable care would be needed were such limiting notions contemplated. There results here a posterior distribution for the mean that is normal with mean $\bar{y}$ and variance $\sigma_{0}^{2} / n$, leading to posterior limits numerically identical to confidence limits.

In fact, with a scalar parameter it is possible in some generality to find a prior giving very close agreement with corresponding confidence intervals. With multidimensional parameters this is not in general possible and naive use of flat priors can lead to procedures that are very poor from all perspectives; see Example 5.5.

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

One approach to simple problems is essentially that of Section $2.5$ and can be summarized, as before, in the Fisherian reduction:

• find the likelihood function;
• reduce to a sufficient statistic $S$ of the same dimension as $\theta$;
• find a function of $S$ that has a distribution depending only on $\psi$;
• place it in pivotal form or alternatively use it to derive $p$-values for null hypotheses;
• invert to obtain limits for $\psi$ at an arbitrary set of probability levels.
There is sometimes an extension of the method that works when the model is of the $(k, d)$ curved exponential family form. Then the sufficient statistic is of dimension $k$ greater than $d$, the dimension of the parameter space. We then proceed as follows:
• if possible, rewrite the $k$-dimensional sufficient statistic, when $k>d$, in the form $(S, A)$ such that $S$ is of dimension $d$ and $A$ has a distribution not depending on $\theta$;
• consider the distribution of $S$ given $A=a$ and proceed as before. The statistic $A$ is called ancillary.
There are limitations to these methods. In particular a suitable A may not exist, and then one is driven to asymptotic, i.e., approximate, arguments for problems of reasonable complexity and sometimes even for simple problems.
We give some examples, the first of which is not of exponential family form.
Example 4.1. Uniform distribution of known range. Suppose that $\left(Y_{1}, \ldots, Y_{n}\right)$ are independently and identically distributed in the uniform distribution over $(\theta-1, \theta+1)$. The likelihood takes the constant value $2^{-n}$ provided the smallest

and largest values $\left(y_{(1)}, y_{(n)}\right)$ lie within the range $(\theta-1, \theta+1)$ and is zero otherwise. The minimal sufficient statistic is of dimension 2 , even though the parameter is only of dimension 1 . The model is a special case of a location family and it follows from the invariance properties of such models that $A=Y_{(n)}-Y_{(1)}$ has a distribution independent of $\theta$.

This example shows the imperative of explicit or implicit conditioning on the observed value $a$ of $A$ in quite compelling form. If $a$ is approximately 2 , only values of $\theta$ very close to $y^{}=\left(y_{(1)}+y_{(n)}\right) / 2$ are consistent with the data. If, on the other hand, $a$ is very small, all values in the range of the common observed value $y^{}$ plus and minus 1 are consistent with the data. In general, the conditional distribution of $Y^{}$ given $A=a$ is found as follows. The joint density of $\left(Y_{(1)}, Y_{(n)}\right)$ is $$n(n-1)\left(y_{(n)}-y_{(1)}\right)^{n-2} / 2^{n}$$ and the transformation to new variables $\left(Y^{}, A=Y_{(n)}-Y_{(1)}\right)$ has unit Jacobian. Therefore the new variables $\left(Y^{}, A\right)$ have density $n(n-1) a^{(n-2)} / 2^{n}$ defined over the triangular region $\left(0 \leq a \leq 2 ; \theta-1+a / 2 \leq y^{} \leq \theta+1-a / 2\right)$ and density zero elsewhere. This implies that the conditional density of $Y^{}$ given $A=a$ is uniform over the allowable interval $\theta-1+a / 2 \leq y^{} \leq \theta+1-a / 2$.

Conditional confidence interval statements can now be constructed although they add little to the statement just made, in effect that every value of $\theta$ in the relevant interval is in some sense equally consistent with the data. The key point is that an interval statement assessed by its unconditional distribution could be formed that would give the correct marginal frequency of coverage but that would hide the fact that for some samples very precise statements are possible whereas for others only low precision is achievable.

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

Fθ∣是(θ∣是)=F是∣θ(是∣θ)Fθ(θ)∫ΩθF是∣θ(是∣φ)Fθ(φ)dφ.

FΨ∣是(ψ∣是)=∫Fθ∣是(ψ,λ∣是)dλ

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

• 找到似然函数；
• 减少到足够的统计量小号尺寸相同θ;
• 找到一个函数小号它的分布仅取决于ψ;
• 把它放在关键的形式，或者用它来推导p- 零假设的值；
• 反转以获得限制ψ在任意一组概率水平上。
当模型属于(到,d)弯曲指数族形式。那么充分的统计量是有维度的到比…更棒d，参数空间的维数。然后我们进行如下操作：
• 如果可能，重写到维充分统计量，当到>d，形式为(小号,一种)这样小号有维度d和一种分布不依赖于θ;
• 考虑分布小号给定一种=一种并像以前一样进行。统计数据一种称为辅助。
这些方法有局限性。特别是一个合适的 A 可能不存在，然后一个人被驱使对合理复杂的问题，有时甚至是简单问题进行渐近的，即近似的论证。
我们给出一些例子，其中第一个不是指数族形式。
例 4.1。已知范围的均匀分布。假设(是1,…,是n)在均匀分布中独立同分布(θ−1,θ+1). 可能性取常数值2−n提供最小的

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## MATLAB代写

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