### 统计代写|统计模型作业代写Statistical Modelling代考|What Is an Exponential Family

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• Statistical Computing 统计计算
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计模型作业代写Statistical Modelling代考|Exponential family

A parametric statistical model for a data set $y$ is an exponential family (or is of exponential type), with canonical parameter vector $\theta=\left(\theta_{1}, \ldots, \theta_{k}\right)$ and canonical statistic $\boldsymbol{t}(\boldsymbol{y})=\left(t_{1}(\boldsymbol{y}), \ldots, t_{k}(\boldsymbol{y})\right)$, if $f$ has the structure
$$f(\boldsymbol{y} ; \boldsymbol{\theta})=a(\boldsymbol{\theta}) h(\boldsymbol{y}) e^{\theta^{\tau} t(\boldsymbol{y})}$$
where $\boldsymbol{\theta}^{T} \boldsymbol{t}$ is the scalar product of the $k$-dimensional parameter vector and a $k$-dimensional statistic $\boldsymbol{t}$, that is,
$$\boldsymbol{\theta}^{T} \boldsymbol{t}=\sum_{j=1}^{k} \theta_{j} t_{j}(y)$$
and $a$ and $h$ are two functions, of which $h$ should (of course) be measurable.
It follows immediately that $1 / a(\boldsymbol{\theta})$, to be denoted $C(\boldsymbol{\theta})$, can be interpreted as a normalizing constant, that makes the density integrate to 1 ,
$$C(\boldsymbol{\theta})=\int h(y) e^{\theta^{\tau} t(y)} \mathrm{d} y$$
or the analogous sum over all possible outcomes in the discrete case. Of course $C(\boldsymbol{\theta})$ or $a(\boldsymbol{\theta})$ are well-defined only up to a constant factor, which can be borrowed from or lent to $h(y)$.

In some literature, mostly older, the canonical parameterization is called the natural parameterization. This is not a good term, however, because the canonical parameters are not necessarily the intuitively natural ones, see for example the Gaussian distribution above.

We think of the vector $t$ and parameter space $\boldsymbol{\Theta}$ as in effect $k$-dimensional (not $<k$ ). This demand will later be shown to imply that $t$ is minimal sufficient for $\theta$. That $t$ is really $k$-dimensional means that none of its components $t_{j}$ can be written as a linear expression in the others. Unless otherwise explicitly told, $\boldsymbol{\Theta}$ is taken to be maximal, that is, comprising all $\boldsymbol{\theta}$ for which the integral (1.6) or the corresponding sum is finite. This maximal parameter space $\boldsymbol{\Theta}$ is called the canonical parameter space. In Section $3.1$ we will be more precise about regularity conditions.

Before we go to many more examples in Chapter 2, we look at some simple consequences of the definition. Consider first a sample, that is, a set of independent and identically distributed (iid) observations from a distribution of exponential type.

## 统计代写|统计模型作业代写Statistical Modelling代考|The structure function for repeated Bernoulli trials

If the sequence $y=\left(y_{1}, \ldots, y_{n}\right)$ is the realization of $n$ Bernoulli trials, with common success probability $\pi_{0}$, with $y_{i}=1$ representing success, the probability function for the sequence $y$ represents an exponential family,
$$f\left(y ; \pi_{0}\right)=\pi_{0}^{t}\left(1-\pi_{0}\right)^{n-t}=\left(1-\pi_{0}\right)^{n} e^{t \log \frac{x_{0}}{1-\pi_{0}}}$$
where $t=t(y)=\sum y_{i}$ is the number of ones. The structure function $g(t)$ is found by summing over all the equally probable outcome sequences having $t$ ones and $n-t$ zeros. The well-known number of such sequences is $g(t)=$ $\left(\begin{array}{l}n \ t\end{array}\right)$, cf. the binomial example (1.3) above. The distribution for the statistic $t$, induced by the Bernoulli distribution, is the binomial, $\operatorname{Bin}\left(n ; \pi_{0}\right)$. $\Delta$
The distribution for $t$ in the Gaussian example requires more difficult calculations, involving $n$-dimensional geometry and left aside here.

The conditional density for data $y$, given the statistic $t=t(y)$, is obtained by dividing $f(y ; \theta)$ by the marginal density $f(t ; \theta)$. We see from ( $1.5)$ and (1.8) that the parameter $\theta$ cancels, so $f(y \mid t)$ is free from $\theta$. This is the general definition of $\boldsymbol{t}$ being a sufficient statistic for $\theta$, with the interpretation that there is no information about $\boldsymbol{\theta}$ in primary data $\boldsymbol{y}$ that is not already in the statistic $\boldsymbol{t}$. This is formalized in the Sufficiency Principle of statistical inference: Provided we trust the model for data, all possible outcomes $y$ with the same value of a sufficient statistic $t$ must lead to the same conclusions about $\boldsymbol{\theta}$.

A sufficient statistic should not be of unnecessarily high dimension, so the reduction of data to a sufficient statistic should aim at a minimal sufficient statistic. Typically, the canonical statistic is minimal sufficient, see

Proposition 3.3, where the mild additional regularity condition for this is specified.

In statistical modelling we can go a reverse way, stressed in Chapter $6 .$ We reduce the data set $\boldsymbol{y}$ to a small-dimensional statistic $\boldsymbol{t}(\boldsymbol{y})$ that will take the role of canonical statistic in an exponential family, and thus is all we need to know from data for the inference about the parameter $\theta$.

The corresponding parameter-free distribution for $\boldsymbol{y}$ given $\boldsymbol{t}$ is used to check the model. Is the observed $\boldsymbol{y}$ a plausible outcome in this conditional distribution, or at least with respect to some aspect of it? An example is checking a normal linear model by use of studentized residuals (i.e. variance-normalized residuals), e.g. checking for constant variance, for absence of auto-correlation and time trend, for lack of curvature in the dependence of a linear regressor, or for underlying normality.

The statistical inference in this text about the parameter $\boldsymbol{\theta}$ is frequentistic in character, more precisely meaning that the inference is primarily based on the principle of repeated sampling, involving sampling distributions of parameter estimators (typically maximum likelihood), hypothesis testing via $p$-values, and confidence regions for parameters with prescribed degree of confidence. Appendix A contains a summary of inferential concepts and principles, intended to indicate what is a good background knowledge about frequentistic statistical inference for the present text.

## 统计代写|统计模型作业代写Statistical Modelling代考|Structure function for Poisson and exponential samples

Calculate these structure functions by utilizing well-known distributions for $t$, and characterize the conditional distribution of $y$ given $t$ :
(a) Sample of size $n$ from the Poisson $\operatorname{Po}(\lambda)$. Use the reproducibility property for the Poisson, that $\sum y_{i}$ is distributed as $\operatorname{Po}\left(\sum \lambda_{i}\right)$.
(b) Sample of size $n$ from the exponential with intensity $\lambda$. Use the fact that $t=\sum y_{i}$ is gamma distributed, with density
$$f(t ; \lambda)=\frac{\lambda^{n} t^{n-1}}{\Gamma(n)} e^{-\lambda t},$$
and $\Gamma(n)=(n-1)$ ! (Section B.2.2). See also Example $2.7$ below.
(c) Note that the conditional density for $y$ is constant on some set, $Y_{t}$ say. Characterize $Y_{t}$ for the Poisson and the exponential by specifying its form and its volume or cardinality (number of points).
$\Delta$

## 统计代写|统计模型作业代写Statistical Modelling代考|Exponential family

F(是;θ)=一种(θ)H(是)和θτ吨(是)

θ吨吨=∑j=1ķθj吨j(是)

C(θ)=∫H(是)和θτ吨(是)d是

## 统计代写|统计模型作业代写Statistical Modelling代考|The structure function for repeated Bernoulli trials

F(是;圆周率0)=圆周率0吨(1−圆周率0)n−吨=(1−圆周率0)n和吨日志⁡X01−圆周率0

## 统计代写|统计模型作业代写Statistical Modelling代考|Structure function for Poisson and exponential samples

(a) 样本大小n从泊松后⁡(λ). 使用泊松的再现性属性，即∑是一世分布为后⁡(∑λ一世).
(b) 大小样本n从指数强度λ. 利用这个事实吨=∑是一世是伽马分布的，有密度
F(吨;λ)=λn吨n−1Γ(n)和−λ吨,

(c) 注意条件密度是在某些集合上是恒定的，是吨说。表征是吨通过指定其形式及其体积或基数（点数）来计算泊松和指数。
Δ

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## MATLAB代写

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