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6 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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17 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 ...
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0answers
39 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
65 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. ...
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2answers
459 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
112 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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0answers
30 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
45 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
22 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? ...
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1answer
67 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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34 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
189 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
140 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
202 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) - ...
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0answers
39 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
256 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? ...
3
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0answers
135 views

Show 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 $z=y^\lambda$? I know the form I need to express it in is $\exp\lbrace(y.\theta - ...
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1answer
224 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 ...
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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?
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65 views

calculate the bias and variance of the MLE in an exponential family

I'm trying to calculate the bias and variance of $\hat{\lambda}_{MLE}$ where $X \sim exponential(\lambda)$. Secondly, does it attain the CRLB? Attempt: $$f_x(X) = \lambda e^{-\lambda x}$$ ...
2
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1answer
698 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) = - ...
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0answers
47 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
109 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
79 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 ...
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0answers
52 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
79 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) ...
3
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0answers
57 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
95 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) = ...
2
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0answers
118 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
204 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 ...
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2answers
101 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 ...
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1answer
1k 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
331 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
120 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 ...
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0answers
102 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
308 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, ...
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0answers
209 views

Two-Parameter Exponential Family Conjugate Prior

A probability distribution is a member of the two-parameter exponential family if the distribution can be expresses in the following form: $$ h(\theta, \phi) \text{exp}\left[\sum t(x_{i})\psi(\theta, ...
2
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0answers
72 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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344 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 ...
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3answers
2k views

Poisson is to exponential as Gamma-Poisson is to what?

A Poisson distribution can measure events per unit time, and the parameter is $\lambda$. The exponential distribution measures the time until next event, with the parameter $\frac{1}{\lambda}$. One ...
4
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1answer
130 views

How to test whether button pressing behaviour is random or is in response to a stimulus?

I have a set of univariate data, where the response variable in a trial is a sequence of time intervals between button presses (my experimental subjs have to press a button intermittently to react to ...
4
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2answers
468 views

Sample size to detect exponential distribution

I'm not a statistician, so I hope this question makes sense. I'm an EMT and we have patients known to be "frequent flyers". Example, we have a diabetic that we have seen 100+ times in 3 years. I have ...
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0answers
612 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 = ...
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3answers
1k views

KullbackÔÇôLeibler divergence between two gamma distributions

Choosing to parameterize the gamma distribution $\Gamma(b,c)$ by the pdf $g(x;b,c) = \frac{1}{\Gamma(c)}\frac{x^{c-1}}{b^c}e^{-x/b}$ The Kullback-Leibler divergence between $\Gamma(b_q,c_q)$ and ...