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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Can I build deviance residuals from an XGBoost model that learns an exponential family parameter?

I'm taking a course on GLMs after a few years of using machine learning models. The good about GLMs is how the probabilistic model ties in with the estimation and evaluation. So I'm trying to transfer ...
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Interpretation of concentration of posteriors in the limit of infinitely many independent versus dependent random variables

Disclaimer: the setup and specific example may not be a minimal example to illustrate the point, but I am not well-versed in these topics enough to construct a smaller example without accidentally ...
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Reparameterizing PDF to get components of exponential family

For the following problem, I am trying to identify the components of the exponential family in the form: $\exp(y\theta - b(\theta))/a(\phi) + c(y; \phi)$ Namely, I need to identify the $\theta, b(\...
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479 views

Extreme Value (Gumbel) distribution a member of exponential family

This is a question for discussion in my Linear Model class. I am having a hard time showing that the distribution belongs to the exponential family PDF: $f(y; \theta) = 1/\varphi \exp([y − \theta]/\...
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History of the curved exponential family

Does anyone know the first person who introduced the curved exponential family and also which paper it was first presented? I vaguely remember that it might be Fisher who wrote about it in a paper on ...
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UMVUE of Bernoulli random variables

Let $X_1, X_2..... X_n$ be a random sample from a Bernoulli population with parameter $p$. A sufficient statistic is $\sum_{i=1}^{n}X_i$. If we define $$ U(X_1,X_2,\ldots,X_n)= \begin{cases}1/2n &\...
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24 views

What is the conjugate prior for the hypoexponential distribution?

Can't find it anywhere. I know Gamma is the conjugate prior for the exponential distribution (one parameter) but for the sum of exponential distributions (the hypoexponential distribution), I can't ...
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232 views

Division of Multivariate Normal Distributions

$$ \newcommand{\vect}[1]{\boldsymbol{\mathbf{#1}}} \newcommand{\nc}[2]{\newcommand{#1}{#2}} \nc{\vx}{\vect{x}} \nc{\vmu}{\vect{\mu}} \nc{\vSigma}{\vect{\Sigma}} \nc{\vtheta}{\vect{\theta}} $$ ...
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259 views

Bayesian Linear Regression and the Exponential Family

In a straight forward linear regression model, assuming a fixed input $\mathbf{x}$, and additive noise with unit variance we can write: \begin{equation} p(y\mid \mathbf{x,w})=\frac{1}{\sqrt{2\pi}\...
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112 views

Exponential family and efficient estimator

In my lecture notes there is the notion of efficiency related to the exponential family. More precisely, the lecturer stated that for an exponential family an efficient estimator always exists. How is ...
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39 views

Percentages in utilization of different distributions

I'm sure that this question has been asked before on CV but, in drilling through many pages of previous CV questions, no matches surfaced. Regardless, I'm confident some observant participant will be ...
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170 views

Does the family of binomial distributions conditioned on $X > 0$ belong to the exponential family?

Does the family of distributions where $p(x,\theta)$ is the conditional frequency function of a binomial $\mathcal{B}(n,\theta)$, variable $X$, given that $X > 0$, belong to the exponencial family? ...
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Exponential family and geometric distribution: how do we prove the sum of independent geometric random variables has negative binomial distribution?

Let $X$ be the number of failures before the first success in a sequence of Bernoulli trials with probability of success $\theta$. Then $P_{\theta}[X = k] = (1-\theta)^{k}\theta$, $k = 1,2,\ldots$ ...
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143 views

Binomial distribution in exponential form

I am trying to find out if I am doing this right: I have started with: $f(y)=\binom{n}{y}(\frac{\mu}{n})^y(1-\frac{\mu}{n})^{n-y}$ This is what I get: Result: $exp[ylog(\frac{\mu}{n-\mu}) - nlog(1+e^...
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158 views

Proof that one distribution is a GLM (general linear model)

Given $Y_1, Y_2,...,Y_n$ i.i.d random variables, where $Y_i|x_i \sim N(\mu_i, \sigma^2)$ and $\mu_i = \beta_0 + \log(\beta_1 + \beta_2x_i)$. How do I proof that the distribution is a GLM (general ...
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Mixed parameterization of sample from normal distribution

I am studying exponential families and mixed parameterizations. Now, I am told that $$ \mathbf{\theta} = \begin{bmatrix}\mu\\ -\frac{1}{2\sigma^2}\end{bmatrix} $$ is the parameter in a variation-...
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308 views

Sufficient Statistic and MLE

Suppose $X_1, \dots, X_n \sim B(1,p)$. Show that a sufficient statistic for $\theta = (1-p)^2$ is $T(x) = \sum X_i$ and that the MLE for $\theta$ is $(1-\frac{1}{n}T)^2$. I am having a lot of ...
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334 views

Convert exponential to Bernoulli

If I have a binary variable x, with distribution p(x) = exp{Cx}, how do I put this into the canonical Bernoulli form so as to get the probability p that x=1 that I ...
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570 views

Minimal sufficient statistic for multivariate normal

I have the following iid. variables $X_1,..,X_n,Y_1,..,Y_m$ with distribution $X_i\sim N(\mu_1,\sigma_1^2), Y_j\sim N(\mu_2,\sigma_2^2)$. How do I find the minimal sufficient statistic for $(\mu_1,\...
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142 views

Family use in glm in R for index's

I have a set of fluctuating asymmetry indices data sheets and would like to run a GLM using this index as the dependent variable and the other characteristics of the sheet as predictor variables (...
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263 views

Show that Sufficient statistic is complete

I have the following Gamma$(2,\theta)$ distribution: $$f_\theta(x) = \theta^2x e^{-\theta x}\mathbb{1}_{[0,+\infty[}(x) $$ I am asked the two following questions: ...
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Estimating parameters of random exp. independent variable based on desired distribution properties of dependent variable in logistic regression model

I have a logistic regression model describing the behaviour of customers of a call centre - relation between waiting time in a queue (independent variable $W$) and probability of hanging up before ...
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548 views

example of uniformly most powerful test

let X be a single observation from the density $f(x;\theta)$ =$ \theta x^{\theta -1} I_{(0,1)}(x)$ is there a UMP size-$\alpha$ test for testing $H_0 :\theta \ge \frac{1}{2} $ V/S $ H_1 : \theta <...
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How to approach expectation of order statistic conditioned on known minimum?

Say you have an $n \times n$ matrix whose entries follow a standard exponential distribution, $A_{i,j} \sim \text{Exp}(1)$ for $1 \le i,j \le n$. You find the minimum value, $x_{i:n^2}$, and remove ...
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is Pareto distribution exponential dispersion family and the form is unique?

I am trying to show that Pareto distribution with $$f(y;\alpha)=\alpha y^{-\alpha-1} $$ is exponential dispersion (ED) family which means that it can be rewritten as: $$f(y;\theta,\phi)={\rm exp}\{\...
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Time to second failure of independent exponential lifetimes

This maybe an easy question, but as I am a beginner, I need help. Suppose that an electronic system contains $n$ similar components that function independently of each other and that are connected ...
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48 views

GLM - exponential form

i stumbled upon the following formula in Kevin Murphy's machine learning book: I am familiar wiht the following formula for the exponential family: $$ 1/Z(\theta) h(x) \exp \left( \theta^T \phi(x)...
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54 views

Subdistributions and superdistributions of exponential families

Suppose you have an n-parameter distribution f(x) such that, for some particular set of parameter values, it is an m < n parameter distribution g(x) which is a member of an exponential family, and ...
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558 views

sufficient statistic and KL-divergence: Confusion with an equation

I am reading a paper, which talks about minimising KL-divergence of any arbitrary distribution over a family of exponential distribution. So, given a distribution $p$, we want to compute its ...
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928 views

Sample size for a non-normal distribution

I'm quite new in this field. I hope my question makes sense. I have a database that stores information for around 10.000.000 projects. Each project has several features (let's call them X) like number ...
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327 views

Normalising Constant for exponentiated function

What would the normalising constant be of the following, or atleast an approximation? I would like to avoid sampling. $$f(\theta)=\exp(-k_1e^{-k_2\theta^2}-\theta^2)\qquad\theta\in(-\infty,\infty), \...
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2k views

Uniform minimum variance unbiased estimator

Let $X_1, X_2, ..., X_n$ be an iid random sample from a Poisson$(\lambda)$ distribution: a) find the UMVUE of $\theta$ = $\lambda^k$ for $k > 0$ a known integer b) find the UMVUE of $\tau = e^{-\...
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Log Likelihoods of Exponential Families

How can one derive the log-likelihood of the saturated model of an exponential family in general? Differentiating the log likelihood w.r.t $\theta$ gives $y_i=\hat{\mu_i}$ but I don't think replacing ...

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