### 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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## 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|Order and minimality

F(是;圆周率)=n!∏是j!∏j=1ķ圆周率j是j=n!∏是j!和∑j=1ķ是j日志⁡圆周率j.

F(是;θ)∝经验⁡(θ1是+θ2是2+θ3是3)

• 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|Minimal sufficiency of t

F(是;θ)F(和;θ)=H(是)H(和)和θ吨吨(是)−吨(和).

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