# Questions tagged [exponential-family]

A set of distributions (eg, normal, $\chi^2$, Poisson, etc) that share a specific form. Many of the distributions in the exponential family are standard, workhorse distributions in statistics, w/ convenient statistical properties.

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### Why is the EM algorithm well suited for exponential families?

I've been brushing up on the EM algorithm, and while I feel like I understand the basics, I keep seeing the claim made (e.g. here, here, among several others) that EM works particularly well for ...
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### Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional sufficient ...
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### MLE estimate of normal distribution

I am quoting this from Greene's econometrics book: The occasional statement that the properties of the MLE are only optimal in large samples is not true, however. It can be shown that when sampling ...
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### Exponential family parameter estimation and fitting, references

First of all, I want to express my apologies if the question is too broad or wrong, but I am in need of references and I have no idea whom I can ask. If you are interested, the question comes from a ...
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This is a homework problem. I've derived the following distribution from an earlier part in the problem $$f_{X_1,X_2}(x_1,x_2) = \dfrac{\Gamma(x_1+x_2+r)\alpha_1^{x_1}\alpha_2^{x_2}\theta^r}{\Gamma(r)... 0answers 2k views ### Hypothesis Testing on Exponential distributions Let X_1, \dots, X_n be independent exponential (\theta) random variables. Suppose we are interested in testing H_0: \theta = \theta_0 = 1 versus H_A: \theta = \theta_1>1. Consider two tests ... 0answers 198 views ### Confidence intervals for log normal responses I got this assignment from Generalized Linear Model class. At first glace it looked like it is an easy task, but there are a lot of subtle (at least in my opinion) things, which I would like to ... 0answers 72 views ### UMP test for exponential family when sufficient statistics T is a vector Assume we have a random sample X_1,\dots,X_n from a distribution of the form f(x_i;\theta) = h(x)g(\theta)\exp(\eta(\theta) T(x)) and we wish to test H_0: \theta \leq \theta_0, H_1: \theta > \... 0answers 129 views ### PDF of Beta Distribution written as exponential family form I am trying to write the pdf of a beta random variable in its biparametric canonic form such as: Function 1$$ f_Y(y; \theta, \phi) = exp \{ \phi[y \theta - b(\theta)] + c(y, \phi) \} \mathbb{1}_A(...
Question: Suppose that $(X_{i}, Y_{i})$, $i = 1, \dots ,n$ are sampled i.i.d. from the two-dimensional normal distribution  \begin{bmatrix} X & Y \end{bmatrix} \sim \mathcal{N}\left( \begin{...