# 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.

386 questions
Filter by
Sorted by
Tagged with
54 views

### For some $\tau=\tau(\theta)$, there exists an unbiased estimator (UMVUE), then the distribution belongs to an exponential family

I read the textbook in Cramer-Rao lower bound (CRLB). Here is a theroem For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the ...
• 611
24 views

### Why do we want to constrain E[ln(x)] in some maximum entropy models?

If we look at the table of distributions in the exponential family, we will see some sufficient statistics have $\log(x)$, which means we have put constraints on $\mathbb{E}[\log(X)]$ when formulating ...
• 1,706
1 vote
41 views

### Find out which exponential distribution the data belongs to

I'm doing a GLM homework, and I'm stuck with the following problem: Suppose that data ($Y_i$; $\mathbf{X}_i$); $i = 1, . . . , n$ are observed, where $\mathbf{X}_i$ is a p-dimensional vector for ...
26 views

### How to prove the Poisson link function is a canonical link function?

So I'm a 3rd year undergraduate doing my thesisin football score models right now. In my thesis I want to include a proof of what the link function for the Poisson distribution is and why it relates ...
1 vote
34 views

### Expectation of Fisher score not equal to 0 when parametrize Categorical distribution differently

Expectation of Fisher score should equal to zero. The prove can be found in many palces, such as wikipedia. But I tried a categorical distribution that is not parameterizatized minimally, the expected ...
• 763
1 vote
32 views

• 514
60 views

### How to check if an exponential family is regular?

A strictly $k$-parameter exponential family $$f=\exp\left(\sum \eta_i(\theta)T_i(x)-B(\theta)\right)h(x)$$ is regular if the natural parameter space $\eta(\theta)$ contains a $k$-dimensional open set. ...
115 views

### Writing exponential family in canonical form

I have the following pdf with support $x>0$: $$f_{\mu}(x)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{(x-\mu)^2}{2\mu^2x}\right)$$ This belongs to the exponential family, and I write this in ...
• 107
59 views

### Smallest threshold for hypothesis test with asymptotic level alpha

Consider a distribution with parameter $\lambda$ that has density $$f_\lambda(x)=\frac{x^4}{24\lambda^5}e^{\frac{-x}{\lambda}},x>0$$ Let $X_1,...,X_n$ be $n$ independent random variables drawn from ...
• 107
1 vote
48 views

1 vote
19 views

### Is the Hydrogen wave (probability density) function from physics define a probability density function in the exponential family?

Is the wave function from physics define a probability density function in the exponential family? Hydrogen atom  The wave functions of an electron in a Hydrogen atom are expressed in terms of ... 27 views

• 183
68 views

### GAM with opposite outcomes with different families

I'm building GAMs and I have some doubts regarding the family to use. I'm fitting GAMs because I expect some non-linear relationships between the response variable and some covariates. I've checked a ...
• 553
45 views

### UMP Test and UMVUE when there are nuisance parameters

Consider $X_1,...,X_n \sim Weibull(\theta, c)$ where $c>0$ is unknown. Several textbook examples consider when $c$ is known, but here, we consider when $c$ is unknown. Suppose now we wanted to find ...
• 89
145 views

### Is the Generalized Dirichlet Distribution an Exponential Family?

Is the Generalized Dirichlet distribution an exponential family? If so, what is its log-normalizer, sufficient statistics, and carrier measure?
• 14.4k
1 vote