统计代写|统计模型作业代写Statistical Modelling代考|Regularity Conditions and Basic Properties

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等楖率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|统计模型作业代写Statistical Modelling代考|Regularity Conditions and Basic Properties

统计代写|统计模型作业代写Statistical Modelling代考|Order and minimality

The order of an exponential family is the lowest dimension of the canonical statistic for which the family of distributions can be represented in the form (1.5) of Definition 1.1, with $\theta$ in some set of parameters. The representation is said to be minimal if its canonical statistic $t$ has this lowest dimension.
What can it look like when the representation is not minimal? One possibility is that the components of $t$ are linearly (affinely 1 ) dependent. An example is if we write the probability function for the multinomial model (see Example 2.4) on the form
f(y ; \pi)=\frac{n !}{\prod y_{j} !} \prod_{j=1}^{k} \pi_{j}^{y_{j}}=\frac{n !}{\prod y_{j} !} e^{\sum_{j=1}^{k} y_{j} \log \pi_{j}} .
and take $\boldsymbol{t}=\left(y_{1}, \ldots, y_{k}\right), \boldsymbol{\theta}=\left(\log \pi_{1}, \ldots, \log \pi_{k}\right)$. Since $\sum y_{j}=n$ (given), the components of $t$ vary in a $(k-1)$-dimensional hyperplane, so we do

not need more than $k-1$ of the $k$ components to represent the distribution family. Correspondingly, the components of $\boldsymbol{\theta}$ are functionally dependent, $\sum_{j=1}^{k} e^{\theta_{j}}=1$, so $\theta$ is also in effect $(k-1)$-dimensional. Such situations can always be avoided by reduction of $t$ in combination with a reparameterization of $\theta$, see Example $2.4$ as a typical illustration. An extension is when in a two-dimensional multiplicative Poisson table (Example 2.5) we reduce the table to the set of all row sums and the set of all column sums. These two sets are linearly connected by both of them summing to the table total. For symmetry reasons we might want, in some cases, to retain the redundant dimension in these examples.

Another possibility is that the components of $\boldsymbol{\theta}$ are linearly (affinely) dependent, $\sum c_{j} \theta_{j}=c_{0}$. After reparameterization we can assume that $\theta_{k}$ has a fixed value. Then $t_{k}$ is redundant and should simply be omitted from $\boldsymbol{t}$, whose dimension is reduced. As soon as we want to test a hypothesis that is linear in the canonical parameters, we have this kind of situation, with a hypothesis model of reduced dimension, see Example $2.8$ for an illustration. Exceptionally, however, this situation may occur unintended, in spite of a maximal $\boldsymbol{\Theta}$. As an example, admittedly artificial, let
f(y ; \theta) \propto \exp \left(\theta_{1} y+\theta_{2} y^{2}+\theta_{3} y^{3}\right)
for $y$ on the whole real line. For $\theta_{3}=0$ and $\theta_{2}<0$ we have a Gaussian density, cf. Example $2.8$, but for $\theta_{3} \neq 0$ the density is not integrable. Hence the maximal canonical parameter space $\Theta$ is a two-dimensional subspace of $\mathbb{R}^{3}$, and the component $y^{3}$ is meaningless and should be omitted from $t$.
Hence, in order to check that a given representation of an exponential family is minimal,

  • check that there is no linear (affine) dependency between the components of $t$, and
  • check that the parameter space is not such that there is a linear (affine) dependency between the components of $\theta$.

统计代写|统计模型作业代写Statistical Modelling代考|Full families

A minimal exponential family is called full if its parameter space is maximal, that is, equals the canonical space $\boldsymbol{\Theta}$.

Most of the theory to follow will only hold for full exponential families. When the family is not full, it is typically because the parameter space is a lower-dimensional ‘curved’ subset of the maximal, canonical space $\boldsymbol{\Theta}$ (mathematically a lower-dimensional manifold). Such families are called curved exponential families. Their treatment will be postponed to Chapter 7 , because the statistical theory for curved families is much more complicated than for the full families. As a simple example, let the coefficient of variation of a normal distribution be assumed known, equal to 1 say. The model distribution is then $\mathrm{N}\left(\mu, \mu^{2}\right), \mu>0$, and its parameter space is represented by a curve in the canonical parameter set, the curve $\theta_{2}=-\theta_{1}^{2} / 2$ in the notations of Example 2.8. Even though this is a one-dimensional subspace of $\mathbb{R}^{2}$, the minimal sufficient statistic for this model is the same two-dimensional one as in the full model, as a consequence of Proposition 3.3.

统计代写|统计模型作业代写Statistical Modelling代考|Minimal sufficiency of t

If $y$ has the exponential family model (1.5), and the representation is minimal, then the statistic $\boldsymbol{t}=\boldsymbol{t}(\boldsymbol{y})$ is minimal sufficient for $\boldsymbol{\theta}$.

Proof That $t$ is sufficient follows immediately from the factorization criterion, see e.g. Pawitan (2001, Sec. 3.1), since the density $f$ is the product of the parameter-free factor $h(\boldsymbol{y})$ and a factor that depends on data through $t$ alone. For the minimality we must show that a ratio of the density in two points $\boldsymbol{y}$ and $z$, that is, $f(\boldsymbol{y} ; \boldsymbol{\theta}) / f(z ; \boldsymbol{\theta})$, cannot be independent of $\boldsymbol{\theta}$ unless $\boldsymbol{t}(y)=\boldsymbol{t}(z)$. This ratio is
\frac{f(y ; \theta)}{f(z ; \theta)}=\frac{h(y)}{h(z)} e^{\boldsymbol{\theta}^{T}{t(y)-t(z)}} .
The minimality of the exponential family representation guarantees that there is no affine dependence between the components of $\boldsymbol{\theta}$, and hence the ratio cannot be independent of $\theta$ unless $t(y)=t(z)$, which was to be proved.

A converse of Proposition $3.3$ also holds, actually. We only sketch the theory. The property was conjectured by Fisher 1935 and soon afterwards proved independently by Darmois, Koopman and Pitman, whose names have therefore sometimes been used to name the exponential families. Suppose we have a sample of size $n$ from a distribution family only known to satisfy a number of regularity conditions. The most important condition is that all distributions have the same support. This demand is of course satisfied by an exponential family – its support is the parameter-independent set where $h(y)>0$.

统计代写|统计模型作业代写Statistical Modelling代考|Regularity Conditions and Basic Properties


统计代写|统计模型作业代写Statistical Modelling代考|Order and minimality

指数族的阶是规范统计量的最低维度,其分布族可以用定义 1.1 的形式 (1.5) 表示,其中θ在一组参数中。如果其典型统计量,则表示该表示是最小的吨有这个最低维度。
当表示不是最小的时候它会是什么样子?一种可能性是吨是线性(仿射 1 )相关的。一个例子是,如果我们将多项式模型的概率函数(参见示例 2.4)写在表格上
并采取吨=(是1,…,是ķ),θ=(日志⁡圆周率1,…,日志⁡圆周率ķ). 自从∑是j=n(给定),的组件吨变化在一个(ķ−1)维超平面,所以我们这样做

不需要超过ķ−1的ķ表示分布族的组件。相应地,组件θ在功能上依赖,∑j=1ķ和θj=1, 所以θ也有效(ķ−1)维。这种情况总是可以通过减少吨结合重新参数化θ, 见例子2.4作为一个典型的例证。扩展是在二维乘法泊松表(示例 2.5)中,我们将表简化为所有行和的集合和所有列和的集合。这两个集合是线性连接的,它们都加到表格总数中。出于对称原因,在某些情况下,我们可能希望在这些示例中保留冗余维度。

另一种可能性是θ是线性(仿射)相关的,∑Cjθj=C0. 重新参数化后,我们可以假设θķ有一个固定值。然后吨ķ是多余的,应该简单地从吨, 其维数减少。一旦我们想要测试一个在规范参数中线性的假设,我们就会遇到这种情况,使用降维的假设模型,请参见示例2.8插图。然而,在特殊情况下,这种情况可能会意外发生,尽管最大θ. 举个例子,诚然是人为的,让
为了是在整条实线上。为了θ3=0和θ2<0我们有一个高斯密度,cf。例子2.8, 但对于θ3≠0密度不可积。因此最大规范参数空间θ是一个二维子空间R3, 和分量是3是无意义的,应该从吨.

  • 检查组件之间是否存在线性(仿射)依赖关系吨, 和
  • 检查参数空间是否不存在组件之间的线性(仿射)依赖关系θ.

统计代写|统计模型作业代写Statistical Modelling代考|Full families


要遵循的大部分理论仅适用于全指数族。当族不完整时,通常是因为参数空间是最大规范空间的低维“弯曲”子集θ(数学上是一个低维流形)。这样的族称为弯曲指数族。它们的处理将推迟到第 7 章,因为曲线族的统计理论比全族复杂得多。举个简单的例子,假设正态分布的变异系数已知,等于 1。模型分布为ñ(μ,μ2),μ>0,其参数空间由规范参数集中的一条曲线表示,曲线θ2=−θ12/2在示例 2.8 的符号中。尽管这是一个一维子空间R2,由于命题 3.3,该模型的最小足够统计量与完整模型中的二维统计量相同。

统计代写|统计模型作业代写Statistical Modelling代考|Minimal sufficiency of t


证明吨是充分的紧随分解标准,参见例如 Pawitan (2001, Sec. 3.1),因为密度F是无参数因子的乘积H(是)和一个依赖于数据的因素吨独自的。对于最小我们必须证明两点的密度比是和和, 那是,F(是;θ)/F(和;θ), 不能独立于θ除非吨(是)=吨(和). 这个比例是

命题的反面3.3实际上也成立。我们只勾画理论。该属性由 Fisher 1935 年推测,不久之后由 Darmois、Koopman 和 Pitman 独立证明,因此有时使用他们的名字来命名指数族。假设我们有一个大小的样本n来自一个只知道满足一些规律性条件的分布族。最重要的条件是所有发行版都具有相同的支持。这个需求当然由指数族满足——它的支持是参数无关的集合,其中H(是)>0.

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