统计代写|STAT209 Generalized Linear Models

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STAT209课程简介

Theory, methods, and applications of generalized linear statistical models; review of linear models; binomial models for binary responses (including logistical regression and probit models); log-linear models for categorical data analysis; and Poisson models for count data. Case studies drawn from social, engineering, and life sciences.

Course description and background: This is a graduate-level course on the theory, methods and applications of Generalized Linear Models (GLMs). Emphasis will be placed on statistical modeling, building from standard normal linear models, extending to GLMs, and briefly covering more specialized topics. With regard to inference, prediction, and model assessment, we will study both likelihood and Bayesian methods. In particular, within the Bayesian modeling framework, we will discuss practically important hierarchical extensions of the standard GLM setting.

Prerequisites 

Note that this is a course on methods for GLMs, rather than a course on using software for data analysis with GLMs. Students will be expected to be familiar with statistical software R, which will be used to illustrate the methods with data examples as part of the homework assignments (and exam). For data analysis problems involving Bayesian GLMs, you will be expected to write your own programs to fit Bayesian models using Markov chain Monte Carlo posterior simulation methods (R will suffice for this).

STAT209 Generalized Linear Models HELP（EXAM HELP， ONLINE TUTOR）

The list below includes a number of distributions, providing in each case the probability density or mass function, support, and parameter space. Determine whether each of the distributions belongs to the exponential dispersion family. Similarly for the two-parameter exponential family of distributions. In both cases, justify your answers.

(a) The double exponential distribution does belong to the exponential dispersion \& family. To see this, we can write the density function as follows:
$$
f(y \mid \theta, \sigma)=\frac{1}{2 \sigma} \exp \left(-\frac{|y-\theta|}{\sigma}\right)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{1}{2}\left(\frac{y-\theta}{\sigma / \sqrt{2}}\right)^2\right) \exp \left(-\frac{1}{2}\left(\frac{-y+\theta}{\sigma / \sqrt{2}}\right)^2\right)
$$
which is in the form of the exponential dispersion family with natural parameter
$\$ \backslash$ theta $\$$, dispersion parameter $\$ \backslash$ phi $=\backslash$ sigma^2/2\$, and sufficient statistic $\$ T(\mathrm{y})=$
$(y,-y)$ \$. Therefore, the double exponential distribution is a member of the exponential
dispersion family.

(b) The uniform distribution does not belong to the exponential dispersion family. To see this, note that the density function can be written as follows:
$$
f(y \mid \theta, \sigma)= \begin{cases}\frac{1}{2 \sigma} & \theta-\sigma<y<\theta+\sigma \ 0 & \text { otherwise }\end{cases}
$$
This is not in the form of the exponential dispersion family, since the density does not factorize as a product of an exponential term and a function of the sufficient statistic. Specifically, there is no way to write the density as $\$ \mathrm{f}(\mathrm{y} \backslash \mathrm{mid} \backslash$ theta, $\backslash$ sigma) $=\backslash \exp \backslash \backslash$ left $\backslash$ frac ${y \backslash$ theta $\left.} \backslash \operatorname{sigma}^{\wedge} 2\right}-\backslash f r a c \backslash \backslash$ theta$\left.^{\wedge} 2\right}\left{2 \backslash\right.$ sigma^ $\left.^{\wedge} 2\right}+h(y, \backslash$ sigma) $\backslash$ right $) \$$ for some function $\$ h \$$. Therefore, the uniform distribution is not a member of the exponential dispersion family.

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