### 统计代写|统计推断作业代写statistics interference代考|Relation with acceptance and rejection

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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|Relation with acceptance and rejection

There is a conceptual difference, but essentially no mathematical difference, between the discussion here and the treatment of testing as a two-decision problem, with control over the formal error probabilities. In this we fix in principle the probability of rejecting $H_{0}$ when it is true, usually denoted by $\alpha$, aiming to maximize the probability of rejecting $H_{0}$ when false. This approach demands the explicit formulation of alternative possibilities. Essentially it amounts to setting in advance a threshold for $p_{o b s}$. It is, of course, potentially appropriate when clear decisions are to be made, as for example in some classification problems. The previous discussion seems to match more closely scientific practice in these matters, at least for those situations where analysis and interpretation rather than decision-making are the focus.

That is, there is a distinction between the Neyman-Pearson formulation of testing regarded as clarifying the meaning of statistical significance via hypothetical repetitions and that same theory regarded as in effect an instruction on how to implement the ideas by choosing a suitable $\alpha$ in advance and reaching different decisions accordingly. The interpretation to be attached to accepting or rejecting a hypothesis is strongly context-dependent; the point at stake here, however, is more a question of the distinction between assessing evidence, as contrasted with deciding by a formal rule which of two directions to take.

## 统计代写|统计推断作业代写statistics interference代考|Formulation of alternatives and test statistics

As set out above, the simplest version of a significance test involves formulation of a null hypothesis $H_{0}$ and a test statistic $T$, large, or possibly extreme, values of which point against $H_{0}$. Choice of $T$ is crucial in specifying the kinds of departure from $H_{0}$ of concern. In this first formulation no alternative probability models are explicitly formulated; an implicit family of possibilities is

specified via $T$. In fact many quite widely used statistical tests were developed in this way.

A second possibility is that the null hypothesis corresponds to a particular parameter value, say $\psi=\psi_{0}$, in a family of models and the departures of main interest correspond either, in the one-dimensional case, to one-sided alternatives $\psi>\psi_{0}$ or, more generally, to alternatives $\psi \neq \psi_{0}$. This formulation will suggest the most sensitive test statistic, essentially equivalent to the best estimate of $\psi$, and in the Neyman-Pearson formulation such an explicit formulation of alternatives is essential.

The approaches are, however, not quite as disparate as they may seem. Let $f_{0}(y)$ denote the density of the observations under $H_{0}$. Then we may associate with a proposed test statistic $T$ the exponential family
$$f_{0}(y) \exp {t \theta-k(\theta)}$$
where $k(\theta)$ is a normalizing constant. Then the test of $\theta=0$ most sensitive to these departures is based on $T$. Not all useful tests appear natural when viewed in this way, however; see, for instance, Example 3.5.

Many of the test procedures for examining model adequacy that are provided in standard software are best regarded as defined directly by the test statistic used rather than by a family of alternatives. In principle, as emphasized above, the null hypothesis is the conditional distribution of the data given the sufficient statistic for the parameters in the model. Then, within that null distribution, interesting directions of departure are identified.

The important distinction is between situations in which a whole family of distributions arises naturally as a base for analysis versus those where analysis is at a stage where only the null hypothesis is of explicit interest.

Tests where the null hypotheses itself is formulated in terms of arbitrary distributions, so-called nonparametric or distribution-free tests, illustrate the use of test statistics that are formulated largely or wholly informally, without specific probabilistically formulated alternatives in mind. To illustrate the arguments involved, consider initially a single homogenous set of observations.

## 统计代写|统计推断作业代写statistics interference代考|Relation with interval estimation

While conceptually it may seem simplest to regard estimation with uncertainty as a simpler and more direct mode of analysis than significance testing there are some important advantages, especially in dealing with relatively complicated problems, in arguing in the other direction. Essentially confidence intervals, or more generally confidence sets, can be produced by testing consistency with every possible value in $\Omega_{\psi}$ and taking all those values not ‘rejected’ at level $c$, say, to produce a $1-c$ level interval or region. This procedure has the property that in repeated applications any true value of $\psi$ will be included in the region except in a proportion $1-c$ of cases. This can be done at various levels $c$, using the same form of test throughout.

Example 3.6. Ratio of normal means. Given two independent sets of random variables from normal distributions of unknown means $\mu_{0}, \mu_{1}$ and with known variance $\sigma_{0}^{2}$, we first reduce by sufficiency to the sample means $\bar{y}{0}, \bar{y}{1}$. Suppose that the parameter of interest is $\psi=\mu_{1} / \mu_{0}$. Consider the null hypothesis $\psi=\psi_{0}$. Then we look for a statistic with a distribution under the null hypothesis that does not depend on the nuisance parameter. Such a statistic is
$$\frac{\bar{Y}{1}-\psi{0} \bar{Y}{0}}{\sigma{0} \sqrt{\left(1 / n_{1}+\psi_{0}^{2} / n_{0}\right)}}$$
this has a standard normal distribution under the null hypothesis. This with $\psi_{0}$ replaced by $\psi$ could be treated as a pivot provided that we can treat $\bar{Y}_{0}$ as positive.

Note that provided the two distributions have the same variance a similar result with the Student $t$ distribution replacing the standard normal would apply if the variance were unknown and had to be estimated. To treat the probably more realistic situation where the two distributions have different and unknown variances requires the approximate techniques of Chapter $6 .$

We now form a $1-c$ level confidence region by taking all those values of $\psi_{0}$ that would not be ‘rejected’ at level $c$ in this test. That is, we take the set
$$\left{\psi: \frac{\left(\bar{Y}{1}-\psi \bar{Y}{0}\right)^{2}}{\sigma_{0}^{2}\left(1 / n_{1}+\psi^{2} / n_{0}\right)} \leq k_{1 ; c}^{}\right},$$ where $k_{1 ; c}^{}$ is the upper $c$ point of the chi-squared distribution with one degree of freedom.

Thus we find the limits for $\psi$ as the roots of a quadratic equation. If there are no real roots, all values of $\psi$ are consistent with the data at the level in question. If the numerator and especially the denominator are poorly determined, a confidence interval consisting of the whole line may be the only rational conclusion to be drawn and is entirely reasonable from a testing point of view, even though regarded from a confidence interval perspective it may, wrongly, seem like a vacuous statement.

## 统计代写|统计推断作业代写statistics interference代考|Formulation of alternatives and test statistics

$$f_{0}(y) \exp {t \theta-k(\theta)}$$相关联
，其中 $k(\theta)$ 是归一化的持续的。那么对这些偏离最敏感的$\theta=0$ 的检验是基于$T$。然而，当以这种方式查看时，并非所有有用的测试都显得自然。例如，参见示例 3.5。

## 统计代写|统计推断作业代写statistics interference代考|Relation with interval estimation

$$\frac{\bar{Y}{1}-\psi{0} \bar{Y}{0}}{\sigma{0} \sqrt{\left(1 / n_{1} +\psi_{0}^{2} / n_{0}\right)}}$$

$$\left{\psi: \frac{\left(\bar{Y}{1}-\psi \bar{Y}{0}\right)^{2}}{\ sigma_{0}^{2}\left(1 / n_{1}+\psi^{2} / n_{0}\right)} \leq k_{1 ; c}^{}\right},$$ 其中 $k_{1 ; c}^{}$ 是具有一个自由度的卡方分布的上 $c$ 点。

## 广义线性模型代考

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## MATLAB代写

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