The exponential-family tag has no wiki summary.
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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 ...
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1answer
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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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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 ...
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1answer
146 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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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, ...
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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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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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4answers
832 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 ...
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1answer
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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
283 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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357 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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683 views
Kullback–Leibler 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 ...
