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How to derive the conjugate prior of an exponential family distribution

I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the ...
2
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1answer
47 views

ML estimate of exponential distribution (with censored data)

In Survival Analysis, you assume the survival time of a r.v. $X_i$ to be exponentially distributed. Considering now that I have $x_1,\dots,x_n$ "outcomes" of i.i.d r.v.'s $X_i$. Only some proportion ...
4
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1answer
55 views

Exponential family distribution with high-order statistics

An exponential family distribution in its simplest form is given by $p(x|\theta) = \exp(\theta^\top T(x) - A(\theta))$ where $T(x)$ is a vector of sufficient statistics, $\theta$ is its natural ...
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0answers
26 views

Families of distributions with bounded variations

For what family of probability distributions $f(x)$ do we have the following property? $$ \forall x, \quad \int f(u) - f(u+x) du \leq L\| x \| $$ for some $L$. Can we say anything about the ...
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0answers
23 views

Scaling of a Weibull variate

I am currently working on an inter-event time dataset. These times are Weibull distributed (I verified my fit with a KS-Test). However, I need to subdivide them into different phases, that are in turn ...
5
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1answer
127 views

Why the mixtures of conjugate priors is important?

I have a questions about the mixture of conjugate priors. I learnt and say the mixture of conjugate priors a couple of times when I am learning bayesian. I am wondering why this theorem is such ...
1
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1answer
39 views

How to prove Bernoulli distribution belongs to the exponential family

According to a book, a distribution belongs to the exponential family if it can be written in the form of I wrote the Bernoulli distribution as $\exp\Big(y \log\,[{\mu}/{(1-\mu)}] + ...
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1answer
46 views

MLE for two-parameter exponential distribution

I have to find the parameters of a two-parameter exponential distribution using the MLE. But imposing FOC I do not find enough conditions to found both the paramenters. Any hint? Thanks!
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21 views

log-linear models and exponential models?

What is usually referred to as "log-linear models"? Is a log-linear model an exponential model where the normalization constant is 1? (since its logarithm needs to be a linear function.) Or is there ...
9
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1answer
280 views

When if ever is a median statistic a sufficient statistic?

I came across a remark on The Chemical Statistician that a sample median could often be a choice for a sufficient statistic but, besides the obvious case of one or two observations where it equals the ...
2
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0answers
81 views

“weight” input in glm.nb function in R. How exactly does the weight affect the likelihood?

I would like to understand how the weight argument of glm.nb is affecting the likelihood function. I understand that glm.nb find the MLE in an alternating iteration process where for a given theta the ...
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0answers
92 views

What is relationship between Fisher Information and Variance in natural exponential Family?

I know that $Var(\hat\theta)\geq 1/I(\theta)$ where $I(\theta)$ is Fisher information. Let take an example of natural exponential family with density $f(x)=\lambda\exp(-\lambda x)$. In this case we ...
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0answers
9 views

Is $A(\eta)$ function in Canonical exponential family differentiable?

I'm studying canonical exponential family, and I'm stuck in differentiability of $A(\eta)$. There's no assumption or comments about $A(\eta)$ in canonical exponential family. (I struggled to search ...
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0answers
20 views

What is a convex support? (Bickel&Doksum, Mathematical Statistics, Basic ideas…Vol1)

Bickel&Doksum, Mathematical Statistics, Basic ideas...Vol1 page 122, just above Cor2.3.1, it says: Define the convex support of a probability P to be the smallest convex set C such that ...
8
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1answer
66 views

Is the negative binomial not expressible as in the exponential family if there are 2 unknowns?

I had a homework assignment to express the negative binomial distribution as an exponential family of distributions given that the dispersion parameter was a known constant. This was fairly easy, but ...
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2answers
108 views

Best Fit for Exponential Data

I'm trying to better understand some of the theory behind fitting models that have a nonlinear link between the response and the predictors. ...
8
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2answers
472 views

What is meant by the term 'exponential family'? Why it is named so?

I have come across the term exponential family. The Bernoulli, Gaussian and many more distributions come under this exponential family. What would be the commonalities between them?
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1answer
193 views

Exponential Family: Observed vs. Expected Sufficient Statistics

My question arises from reading reading Minka's "Estimating a Dirichlet Distribution", which states the following without proof in the context of deriving a maximum-likelihood estimator for a ...
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1answer
66 views

The distribution of a sufficient statistic

If I understand correctly, a distribution in the exponential family... $$\underline X\sim f_{\underline\theta}(\underline x) = exp\{\sum\limits_{i}\eta_i(\underline\theta)T_i(\underline ...
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1answer
58 views

How to show linear model corresponds to exponential family?

I am confronted with the exercise below. I have no given solution, so I hope someone can tell me whether my solution is right or wrong. I want to show the deterministic linear model $\quad ...
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0answers
24 views

Understanding Conjugate families

I'm trying to understand conjugate families of distributions, and am hoping someone could give me a hint or direct to a good paper (or book) covering basics. Or perhaps, simple example would help? ...
2
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1answer
70 views

What the dimension of an exponential family tell us about that family?

In Wikipedia it is stated that: A vector exponential family is said to be curved if the dimension of $$ {\boldsymbol \theta} = \left (\theta_1, \theta_2, \ldots, \theta_d \right )^T$$ is ...
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42 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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2answers
216 views

A stochastically increasing exponential family for which $\lim_{\theta\rightarrow\inf\Theta}\mbox{P}_\theta(X\leq x)\neq 1$

Question A little something that I've been wondering about for a while: Let $P_\theta$ be a stochastically increasing (one-parameter) exponential family on the sample space $\mathcal{X}$ with ...
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0answers
164 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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1answer
242 views

Do the mean and the variance always exist for exponential family distributions?

Assume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f. $$ f_X(x|\boldsymbol \theta) = h(x) \exp\left(\sum_{i=1}^s \eta_i({\boldsymbol \theta}) T_i(x) - ...
1
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0answers
40 views

Sequence of Bernouilli trials with diminishing returns

Say I have a sequence of yes-no trials where the probability of winning decays over time as the supply of wins is gradually exhausted. Assume that for every trial the probability of replacement is not ...
8
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1answer
358 views

Does log likelihood in GLM have guaranteed convergence to global maxima?

My questions are: Are generalized linear models (GLMs) guaranteed to converge to a global maximum? If so, why? Furthermore, what constraints are there on the link function to insure convexity? ...
4
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1answer
226 views

Show that Weibull distribution can be transformed to exponential family

How do I show the Weibull distribution $f:(y; \lambda, \rho)$ can be transformed to the exponential family using the transformation $z=y^\lambda$? I know the form I need to express it in is ...
10
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1answer
238 views

Does a canonical link function always exist for a Generalized Linear Model (GLM)?

In GLM, assuming a scalar $Y$ and $\theta$ for the underlying distribution with p.d.f. $$f_Y(y | \theta, \tau) = h(y,\tau) \exp{\left(\frac{\theta y - A(\theta)}{d(\tau)} \right)}$$ It can be shown ...
0
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0answers
11 views

SNEP distribution fitting and parameter estimates

I'm looking to try and fit a SNEP distrib to some data but am not sure how to go about this - can anyone give me some derivations for parameter estimates etc?
2
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1answer
1k views

How to find the expected Fisher information for a parameter of exponential distribution in an iid sample?

I am unclear to on how to find the Fisher information of $\lambda$ in the iid sample $X_1,...,X_n$. Attempt: I know that the fisher information for $X$ (not in a sample) would be: $I(\lambda) = - ...
3
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0answers
53 views

Exponential family where set of natural parameters has empty interior

In my math-stat class we have a theorem that goes: Let $\{P_\theta : \theta \in \Theta\}$ be a $k$ parameter exponential family (i.e. the density of a member of this family can be written as ...
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0answers
133 views

Most Powerful Test; Two-Parameter Normal Distribution

Is it possible to show that the two-parameter Normal distribution has monotone likelihood ratio? EDIT: This is actually part of a larger problem. We have a random sample from $\mathcal N(\mu, ...
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0answers
89 views

Hypothesis testing with exponential family

I'm interested in running hypothesis tests for a variety of members of the exponential family with continuous support, for different values of the parameter/s, for a sample of n i.i.d random variables ...
1
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0answers
62 views

Computing non-central moments and normalizer of a quartic exponential distribution

Consider a random variable $X$ which has quartic exponential distribution: $$X \sim P(x)=\frac{1}{Z}e^{ax + bx^2 + cx^3 + dx^4}$$ How can one compute $Z$ or non-central moments $E X^k$ given that they ...
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0answers
89 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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0answers
23 views

normalising constant on exponential of exponential [duplicate]

I have a distribution of the form $\exp(-\exp(-x^2))$. Is this a known family of distributions. Otherwise how would you find/approximate the normalising constant. The domain is $x \in (-\infty,\infty) ...
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0answers
58 views

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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1answer
110 views

Checking whether a density is exponential family

Trying to prove that this doesn't belong to exponential family. $f(y|a)=4\frac{(y+a)}{(1+4a)} ; 0 < y < 1 , a>0$ Here is my approach: $$f(y|a) = 4(y+a)e^{-log(1+4a)}$$ $$f(y|a) = ...
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0answers
126 views

3 parameter Exponential Family and sufficient statistics

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) = ...
3
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0answers
258 views

Is a normalized version of an exponential family distribution still an exponential family distribution?

Is a normalized version of an exponential family distribution still an exponential family distribution? Here "normalized" means making its mean zero and variance one. According to the following ...
3
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2answers
102 views

What to do in logistic regression if you have a huge amount of variables?

I am dealing with logistic regression, trying to identify variables which have a causal relationship with a binary response. The way I usually do it is to try variables one by one and visualize the ...
2
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1answer
2k views

How should you express a negative binomial distribution in an exponential family form?

When $X$ is $X_1$,...$X_n$, how do you express the following negative binomial distribution in an exponential family form? $$ ...
2
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0answers
356 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 ...
0
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1answer
124 views

Time series and multiple variables

I created a time series in Excel (not ideal) using Holts-Winters to forecast daily loan values in a month and it works very well. I've been asked to build a similar model that integrates other ...
2
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0answers
109 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 ...
0
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1answer
362 views

Log - likelihood function, why does the summation sign vanish?

I have the log-likelihood function: $$l(p_i,y_i) = \sum_{i = 1}^n \left( \ln(p_i) + y_i \ln(1 - p_i) \right) $$ And I need to calculate the maximum likelihood estimator of $p_i$. When I do this, ...
3
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0answers
81 views

Smooth expectations outside the exponential family

At page 85-86 of Young and Smith "Essentials of Statistical Inference" there is an interesting result. If $X$ is a r.v. distributed according to the exponential family and $\phi(x)$ is a bounded (but ...
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1answer
390 views

Gaussian Like distribution with higher order moments

For the Gaussian distribution with unknown mean and variance, the sufficient statistics in the standard exponential family form is $T(x)=(x,x^2)$. I have a distribution that has ...