统计代写|统计模型作业代写Statistical Modelling代考|Small Sample Refinement: Saddlepoint Approximations

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统计代写|统计模型作业代写Statistical Modelling代考|Small Sample Refinement: Saddlepoint Approximations

统计代写|统计模型作业代写Statistical Modelling代考|Small Sample Refinement: Saddlepoint Approximations

Saddlepoint approximation is a technique for refined approximation of densities and distribution functions when the normal (or $\chi^{2}$ ) approximation is too crude. The method is typically used on distributions for

  • Sums of random variables.
  • ML estimators.
  • Likelihood ratio test statistics,
    and for approximating structure functions. These approximations often have surprisingly small errors, and in some remarkable cases they yield exact results. We confine ourselves to the basics, and consider only density approximation for canonical statistics $t$ (which involves the structure function $g(t)$ ), and for the MLE in any parameterization of the exponential family.
    The saddlepoint approximation for probability densities (continuous distribution case) is due to Daniels (1954). Approximations for distribution functions, being of particular interest for test statistics (for example the so-called Lugannani-Rice formula for the signed log-likelihood ratio), is a more complicated topic, because it will require approximation of integrals of densities instead of just densities, see for example Jensen (1995).

Suppose we have a regular exponential family, with canonical statistic $\boldsymbol{t}$ (no index $n$ here) and canonical parameter $\theta$. If the family is not regular, we restrict $\boldsymbol{\theta}$ to the interior of $\Theta$. Usually $t=t(y)$ has some sort of a sum form, even though the number of terms may be small. We shall derive an approximation for the density $f\left(t ; \theta_{0}\right)$ of $t$, for some specified $\theta=\theta_{0}$. One reason for wanting such an approximation in a given exponential family is the presence of the structure function $g(t)$ in $f\left(t ; \theta_{0}\right)$, which is the factor, often complicated, by which $f\left(t ; \boldsymbol{\theta}{0}\right)$ differs from $f\left(\boldsymbol{y} ; \boldsymbol{\theta}{0}\right)$. Another reason is that we might want the distribution of the ML estimator, and $t=\hat{\boldsymbol{\mu}}_{t}$ is one such MLE.

统计代写|统计模型作业代写Statistical Modelling代考|Extensions and further refinements.

The same technique can at least in principle be used outside exponential families to approximate a density $f(t)$. The idea is to embed $f(t)$ in an exponential family by exponential tilting, see Example $2.12$ with $f(t)$ in the role of its $f_{0}(y)$, For practical use, $f$ must have an explicit and manageable Laplace transform, since the norming constant $C(\boldsymbol{\theta})$ will be the Laplace transform of the density $f$, cf. (3.9).
For iid samples of size $n$ one might guess the error is expressed by a factor of magnitude $1+O(1 / \sqrt{n})$. It is in fact a magnitude better, $1+O(1 / n)$, and this holds uniformly on compact subsets of $\Theta$. The error factor can be further refined by using a more elaborate Edgeworth expansion instead of the simple normal density in the second line of (4.6). An Edgeworth expansion is typically excellent in the centre of the distribution, where it would be used here (for $\boldsymbol{\theta}=\hat{\theta}(t))$. On the other hand, it can be terribly bad in the tails and is generally not advisable for a direct approximation of $f\left(t ; \boldsymbol{\theta}_{0}\right)$. See for example Jensen (1995, Ch. 2) for details and proofs. Such a refinement of the saddlepoint approximation was used by Martin-Löf to prove Boltzmann’s theorem, see Chapter $6 .$

Further refinement is possible, at least in principle, through multiplication by a norming factor such that the density approximation becomes a proper probability density. Note, this factor will typically depend on $\theta_{0}$. Even if the relative approximation error is often remarkably small, it generally depends on the argument $\boldsymbol{t}$. For the distribution of the sample mean it is constant only for three univariate densities of continuous distributions: the normal density, the gamma density, and the inverse Gaussian density (Exercise 4.6). The representation is exact for the normal, see Exercise 4.4, and the inverse Gaussian. For the gamma density the error factor is close to 1, representing the Stirling’s formula approximation of $n$ ! (Section B.2.2). Correction by a norming factor, as mentioned in the previous item, would of course eliminate such a constant relative error, but be pointless in this particular case.

统计代写|统计模型作业代写Statistical Modelling代考|Saddlepoint approximation for the MLE

The saddlepoint approximation for the density $f\left(\hat{\psi} ; \psi_{0}\right)$ of the $M L$ estimator $\hat{\psi}=\hat{\psi}(t)$ in any smooth parameterization of a regular exponential family is
f\left(\hat{\psi} ; \psi_{0}\right) \approx \frac{\sqrt{\operatorname{det} I(\hat{\psi})}}{(2 \pi)^{k / 2}} \frac{L\left(\psi_{0}\right)}{L(\hat{\psi})}
Here $I(\hat{\psi})$ is the Fisher information for the chosen parameter $\psi$, with inserted $\hat{\psi}$.

Proof We transform from the density for $t=\hat{\mu}{t}$ to the density for $\hat{\psi}(t)$. To achieve this we only have to see the effect of a variable transformation on 4.2 Small Sample Refinement: Saddlepoint Approximations 73 the density. When we replace the variable $\hat{\mu}$, by $\hat{\psi}$, we must also multiply by the Jacobian determinant for the transformation that has Jacobian matrix $\left(\frac{\partial \hat{\mu}{t}}{\partial \bar{\phi}}\right)$. However, this is also precisely what is needed to change the square root of $\operatorname{det} I\left(\mu_{t}\right)=\operatorname{det} V_{t}^{-1}=1 / \operatorname{det} V_{t}$ to take account of the reparameterization, see the Reparameterization lemma, Proposition 3.14. Hence the form of formula (4.9) is invariant under reparameterizations.

Proposition $4.7$ works even much more generally, as shown by BarndorffNielsen and others, and the formula is usually named Barndorff-Nielsen’s formula, or the $p^{*}$ ( ‘ $p$-star’) formula, see Reid (1988) and Barndorff-Nielsen and Cox (1994). Pawitan (2001, Sec. 9.8) refers to it as the ‘magical formula’. The formula holds also after conditioning on distribution-constant (= ancillary) statistics (see Proposition $7.5$ and Remark 7.6), and it holds exactly in transformation models, given the ancillary configuration statistic. It is more complicated to derive outside the exponential families, however, because we cannot then start from (4.7) with the canonical parameter $\boldsymbol{\theta}$ as the parameter of interest.

By analogy with the third item of Remark 4.6 about approximation (4.7), we may here refine (4.9) by replacing the approximate normalization constant $1 / \sqrt{2 \pi}$ by the exact one, that makes the density integrate to 1 .

统计代写|统计模型作业代写Statistical Modelling代考|Small Sample Refinement: Saddlepoint Approximations


统计代写|统计模型作业代写Statistical Modelling代考|Small Sample Refinement: Saddlepoint Approximations

鞍点近似是一种在正态(或χ2) 近似太粗略。该方法通常用于分布

  • 随机变量的总和。
  • 机器学习估计器。
  • 似然比检验统计量,
    用于近似结构函数。这些近似值通常具有令人惊讶的小误差,并且在某些非凡的情况下,它们会产生精确的结果。我们仅限于基础知识,仅考虑规范统计的密度近似吨(其中涉及结构函数G(吨)),对于指数族的任何参数化中的 MLE。
    概率密度的鞍点近似(连续分布情况)归功于 Daniels (1954)。分布函数的近似值,对检验统计特别感兴趣(例如所谓的有符号对数似然比的 Lugannani-Rice 公式),是一个更复杂的主题,因为它需要密度积分的近似值,而不仅仅是密度,例如参见 Jensen (1995)。

假设我们有一个正则指数族,具有典型统计量吨(无索引n这里)和规范参数θ. 如果家庭不规律,我们限制θ到内部θ. 通常吨=吨(是)具有某种求和形式,即使项的数量可能很少。我们将推导出密度的近似值F(吨;θ0)的吨, 对于一些指定的θ=θ0. 在给定的指数族中需要这种近似的一个原因是结构函数的存在G(吨)在F(吨;θ0),这是一个因素,通常很复杂,通过它F(吨;θ0)不同于F(是;θ0). 另一个原因是我们可能想要 ML 估计器的分布,并且吨=μ^吨就是这样一种 MLE。

统计代写|统计模型作业代写Statistical Modelling代考|Extensions and further refinements.

至少原则上可以在指数族之外使用相同的技术来近似密度F(吨). 这个想法是嵌入F(吨)通过指数倾斜在指数族中,请参见示例2.12和F(吨)在其作用F0(是), 为了实际使用,F必须有一个显式且可管理的拉普拉斯变换,因为规范常数C(θ)将是密度的拉普拉斯变换F,参见。(3.9)。
对于大小的 iid 样本n有人可能会猜测误差是由一个数量级表示的1+这(1/n). 它实际上是一个数量级更好,1+这(1/n), 这在θ. 误差因子可以通过使用更精细的 Edgeworth 展开而不是 (4.6) 第二行中的简单正态密度来进一步细化。Edgeworth 展开通常在分布的中心非常出色,这里将使用它(例如θ=θ^(吨)). 另一方面,尾部可能非常糟糕,通常不建议直接近似F(吨;θ0). 有关详细信息和证明,请参见 Jensen (1995, Ch. 2)。Martin-Löf 使用鞍点近似的这种改进来证明玻尔兹曼定理,参见第6.

至少在原则上,进一步的细化是可能的,通过乘以一个规范因子,使得密度近似成为一个适当的概率密度。请注意,此因素通常取决于θ0. 即使相对近似误差通常非常小,它通常也取决于参数吨. 对于样本均值的分布,它仅对连续分布的三个单变量密度是恒定的:正态密度、伽马密度和逆高斯密度(练习 4.6)。表示法是精确的,见习题 4.4,和逆高斯。对于伽马密度,误差因子接近 1,表示斯特林公式近似n!(第 B.2.2 节)。如上一项所述,通过规范因子进行校正当然会消除这种恒定的相对误差,但在这种特殊情况下毫无意义。

统计代写|统计模型作业代写Statistical Modelling代考|Saddlepoint approximation for the MLE

这里一世(ψ^)是所选参数的 Fisher 信息ψ, 插入ψ^.

证明 我们从密度变换为吨=μ^吨密度为ψ^(吨). 为了实现这一点,我们只需要查看变量变换对 4.2 小样本改进:鞍点近似 73 密度的影响。当我们替换变量时μ^, 经过ψ^,我们还必须乘以雅可比行列式来进行具有雅可比矩阵的变换(∂μ^吨∂φ¯). 然而,这也正是改变平方根所需要的这⁡一世(μ吨)=这⁡在吨−1=1/这⁡在吨要考虑重新参数化,请参见重新参数化引理,命题 3.14。因此,公式(4.9)的形式在重新参数化下是不变的。

主张4.7正如 BarndorffNielsen 和其他人所展示的那样,它的工作原理更加普遍,并且该公式通常被命名为 Barndorff-Nielsen 公式,或者p∗ ( ‘ p-star’) 公式,参见 Reid (1988) 和 Barndorff-Nielsen 和 Cox (1994)。Pawitan (2001, Sec. 9.8) 将其称为“神奇公式”。该公式在以分布常数(=辅助)统计为条件后也成立(见命题7.5和备注 7.6),并且在给定辅助配置统计的情况下,它完全适用于转换模型。然而,在指数族之外推导更复杂,因为我们不能从 (4.7) 开始使用规范参数θ作为感兴趣的参数。

类推 4.6 中关于近似 (4.7) 的第三项,我们可以在这里通过替换近似归一化常数来细化 (4.9)1/2圆周率由确切的一个,这使得密度积分为 1 。

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