### 统计代写|统计模型作业代写Statistical Modelling代考|Extension to higher-order moments

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## 统计代写|统计模型作业代写Statistical Modelling代考|Extension to higher-order moments

Remark 3.9. Extension to higher-order moments. Repeated differentiation of $C(\boldsymbol{\theta})$ yields higher-order raw moments, after normalization by $C(\boldsymbol{\theta})$ itself, generalizing (3.7) and (3.8). Alternatively they can be obtained from the moment-generating function (the Laplace transform) for the distribution of $t$, with argument $\psi$,
$$E_{\theta}\left(e^{\psi^{x} t}\right)=\frac{C(\theta+\psi)}{C(\theta)},$$
or, if preferable, from the characteristic function, that can be written
$$E_{\theta}\left(e^{i \psi^{x} t}\right)=\frac{C(\theta+i \psi)}{C(\boldsymbol{\theta})}$$
For example, for component $t_{j}$ we have
$$E_{\theta}\left(t_{j}^{r}\right)=\frac{\partial^{r} C(\boldsymbol{\theta})}{\partial \theta_{j}^{r}} \frac{1}{C(\boldsymbol{\theta})}$$
Repeated differentiation of $\log C(\theta)$ yields other expressions in terms of higher-order moments. They can be precisely characterized. It is often simpler to work with $\log C(\theta)$ than with $C(\theta)$. The corresponding logarithm of the moment-generating function is generally called the cumulant function, and the moment expressions derived by differentiating the cumulant function are called the cumulants (or semi-invariants). The first cumulants, up to order three, are the mean, the central second-order moments (that is the variance-covariance matrix) and the central third-order moments. For higher cumulants, see for example Cramér (1946, Sec. 15.10). Now, for exponential families, differentiation of the cumulant function $\log C(\theta+\psi)-$ $\log C(\theta)$ with respect to $\psi$ in the point $\psi=\mathbf{0}$ evidently yields the same result as differentiation of $\log C(\theta)$ in the point $\theta$. Hence, $\log C(\theta)$ can be interpreted as the cumulant function for the distribution of $t$.

## 统计代写|统计模型作业代写Statistical Modelling代考|Likelihood and Maximum Likelihood

We now turn to the likelihood function $L(\theta)$, that is, the density or probability $f(\boldsymbol{y} ; \boldsymbol{\theta})$ for the data $\boldsymbol{y}$ (or $\boldsymbol{t}$, since $\boldsymbol{t}$ is sufficient), regarded as a function of the canonical $\theta$. The importance of the concepts of likelihood and maximum likelihood in statistical inference theory should be known to the reader, but see also Appendix A. There is not enough space here for a comprehensive introduction, but some basic statements follow, additionally making our terminology and notation clear. More conveniently than the likelihood itself, we will study $\log L(\theta)$, given by (3.11). Not only for exponential families, but generally under some weak smoothness conditions, automatically satisfied for exponential families, the first derivative (the gradient) $U(\boldsymbol{\theta})=D \log L(\boldsymbol{\theta})$ of the log-likelihood function exists and has nice and useful properties. It is called the (Fisher) score function or score vector. The expected score vector is zero, if the score and the density in this expected value use the same $\boldsymbol{\theta}$, that is, $E_{\theta}{U(\boldsymbol{\theta})}=\mathbf{0}$. Setting the actual score vector to zero yields the likelihood equation system, to which the maximum likelihood (ML) estimator $\hat{\boldsymbol{\theta}}$ is usually a root. The variance-covariance matrix for the score vector is the Fisher information matrix, $I(\theta)$. The Fisher information can alternatively be calculated as the expected value of $-D^{2} \log L(\boldsymbol{\theta})$, that is, minus the second-order derivative (the Hessian) of the log-likelihood. This function, $J(\boldsymbol{\theta})=-D^{2} \log L(\boldsymbol{\theta})$ is called the observed information. Hence, the Fisher information is the expected value of the observed information. The observed information $J(\boldsymbol{\theta})$ tells the curvature of the likelihood function, and is of particular interest in the ML point $\boldsymbol{\theta}=\hat{\boldsymbol{\theta}}$, where the inverse of the (observed or expected) information matrix asymptotically equals the variance-covariance matrix of the maximum likelihood estimator, see Chapter 4. For the general theory of likelihood in more detail, see for example Cox and Hinkley (1974), Pawitan (2001) and Davison (2003, Ch. 4).

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

Remark 3.12. Extension. For a nonregular family the result of this proposition holds if we exclude the boundary points by restricting consideration to the interior of $\boldsymbol{\Theta}$ and the corresponding image under $\boldsymbol{\mu}_{t}$. This follows from the proof given.

An observed $t$ in a regular family will usually be found to satisfy the requirement $t \in \boldsymbol{\mu}{t}(\boldsymbol{\Theta})$ of Proposition 3.11. When it does not, the observed $t$ typically suggests some degenerate distribution as having a higher likelihood, a distribution which falls outside the exponential family but corresponds to a boundary value of the parameter in some alternative parameterization. An example is provided by the binomial distribution family: If we observe $t=0$ successes, which has a positive probability for any $n$ and $\theta$, the ML estimate of the success probability $\pi{0}, 0 \leq \pi_{0} \leq 1$, is $\hat{\pi}{0}=0$. The value $\pi{0}=0$ has no corresponding canonical $\theta$-value in $\Theta=\mathbb{R}$. It is generally possible (at least for discrete families) to extend the definition of the model, so that the ML estimation always has a solution (Lauritzen, 1975 ; Barndorff-Nielsen, 1978, Sec. 9.3). This is perhaps not a very fruitful approach, however, since we would lose some of the nice regularity properties possessed by regular exponential families.

Suppose we have a sample of scalar $y$-values from a distribution in a family with a $p$-dimensional parameter. The classic moment method of estimation equates the sample sums of $y, y^{2}, \ldots, y^{p}$ to their expected values. The form of the likelihood equation for exponential families shows that the

ML estimator in exponential families can be regarded as a sort of a generalized, more sophisticated moment estimator. Proposition $1.2$ namely tells that the $j$-th component of the $t$-vector is a sum of a function $t_{j}(y)$ over the sample $y$-values, so the function $t_{j}(y)$ replace the simple moment $y^{j}$.

The set of possible outcomes of $t$, for which the ML estimate exists, can be characterized in a simple way for a regular exponential family. This result is due to Barndorff-Nielsen:

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