A non-negative continuous probability distribution indexed by two strictly positive parameters.

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Distribution of the duration of a markov-process in a specified state during a specified time

I have a continuous time markov chain with two states $A$ and $B$. The transition rate $A\rightarrow B$ is $\lambda$ and $B\rightarrow A$ is $\mu$. Imagine that $P(X{t_0}=A)=1$ (the process starts in ...
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15 views

Posterior predictive for Gamma distribution with unknown scale and shape

I have a question that needs clarification. The posterior predictive distribution can be described as the distribution that a new i.i.d. data point $\tilde{x}$ would have, given a set of $N$ existing ...
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17 views

Poisson/Negative Binomial/Gamma log-link for continuous dependent variable (scale DV)

In my research about sport injuries in football, I am trying to obtain Incidence Rate Ratios (IRR) comparing my categories with the reference category. I have number of days a player was absent due to ...
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1answer
59 views

Gamma Distribution and Life of Component?

I came across an old exam question as follows: If the life of one computer component (in years) has Gamma distribution with mean $6$ and variance $18$, how can we find the probability that this ...
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24 views

Beginner level: What is the technique for calculating distribution for the minimum of random variables? [duplicate]

Based on the comments, this Question appears to be a duplicate of Distribution of Minimum of RVs. But, the link does not mention how to find the pdf. Will the pdf of the minimum of r.vs the same as ...
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20 views

How to calculate implied volatility for a Variance Gamma option pricing model

I need to calculate the implied volatility for an option that has been priced using the Variance-Gamma model to produce a volatility smile. I can use Excel or VBA to do this. If possible, any ...
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1answer
32 views

Independence-Metropolis-Hastings Algorithm

IMHA is an importance-sampling version of MCMC, where the proposal is drawn from a fixed distribution g. Usually, g is chosen to be an approximation to f. The acceptance probability becomes ...
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1answer
34 views

Zero-inflated Poisson and Gibbs sampling, proofs and sampling

I am trying to figure out zip-inflated Poisson (ZIP) model. In this model, random data $X_1, .., X_n$ are of the form $X_i=R_iY_i$, where the $Y_i$'s have Poisson distribution ($\lambda$) and the ...
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1answer
105 views

Distribution of the quotient of two gamma random variables with different rate parameters?

I have a question about how to derive the distribution of the quotient of two random gamma variables drawn from two different Gamma distributions with the same shape, but different rates. For example, ...
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56 views

Most Powerful Test and Rejection Region of Gamma Distribution

Let $X_1,\ldots,X_n$ be a random sample from a Gamma $(\alpha,\beta)$ population, where $\beta>0$ is a known constant. The rejection region of the most powerful test for $H_0:\alpha=1$ against ...
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1answer
42 views

Test if two gamma distributed populations are different

I have data from two populations of different sizes. Both have Gamma distributions with different shapes and scales (as estimated in r): fitdistr(x/10000, "gamma") #186 members shape 0.586219900 ...
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22 views

combining samples from different gamma distributions normalized by means

I'm trying to figure out what to do with some experimental data from a psychoacoustic experiment. The data come from multiple subjects and are approximately gamma distributed; I am treating them as ...
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1answer
66 views

Construct the likelihood if measurement uncertainties have a Gamma distribution

I want to construct my likelihood. General case: If my data do come from a line of the form $y = mx + b$ and the uncertainties are normally distributed with mean zero and known variance ...
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0answers
31 views

Modeling discrete count data with a gamma distribution

I've encountered a statistical model in which discrete count data are modeled with a gamma distribution (supported on nonnegative reals). The model relies on the property of the gamma that a sum of ...
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0answers
31 views

Compound Poisson Distributions: When, Why, and How To Split the Problem

I've just stumbled upon the Compound Poisson Distribution (CPD) and it seems to be precisely what I need. For the purposes of this post, let's suppose I have a store that sells many items of ...
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20 views

Fitting gamma distribution to data set with one zero observation

I am using maximum likelihood estimation to fit a gamma distribution to shelf life data. Specifically, the data I have is the time (in days) between the day a product was sold and the day the first ...
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12 views

likelihood and gibbs for univariate UE model

This is the first time when i post something here. I would like to ask how can i compute the likelihood of the following model? i put only the product of the densities and that is it? I think the ...
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1answer
23 views

GLM with Gamma distribution of errors: negative residuals?

I'm trying to understand how the Gamma distribution, which is always positive, is used to describe errors when using a GLM. In practice, errors can be negative, as I get negative residuals when ...
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17 views

When to pick Scale or Rate for Gamma/ Inverse Gamma parameters? Picking up the correct conjugate prior

I am very confused about the following problem. I think my question is both theoretical and applied. When I model a Gamma distribution, when do I use this model, so I guess Inverse Gamma: ...
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1answer
15 views

Gamma Distribution with Percentages

I am dealing with a set of data that appears to follow a gamma distribution or a lognormal distribution but the only issue is that the data set is in percentages and both of these distributions don't ...
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19 views

Bivariate gamma distribution with one parameter marginal

Suppose two random variables X,Y>0, have a joint gamma distribution. Its marginal for X is G(a,b) and that for Y is G(a,c). For identification purposes, normalization is needed.does anyone know if ...
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1answer
80 views

Expected value of Y = (1/X) where $X \sim Gamma$

I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$ I want to find ...
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1answer
86 views

Sufficient statistic for a Gamma distribution

I am confused about the steps I need in order to solve the equation below. I must use conditional distribution (and NOT the factorization theorem). Q: $X_1, . . . , X_n$ is a random sample from a ...
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51 views

How to find the appropriate family for glm models?

As suggested over at stackoverflow I poste the question here instead: I have a data frame with three variables, where "Resp" is my response variable (count data), F1 is a categorical predictor (4 ...
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19 views

To obtain Pearson type III parameters and shift value

How to obtain Pearson type III parameters and shift value? I am using R and if you can give me an instruction, it would be helpful. I have used pearsonFitML function from PearsonDS package, but I can ...
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1answer
17 views

Prior for gamma distribution in “mean form”

I need to specify priors for the parameters of a gamma distribution. Normally the gamma distribution is parametrized in either the "rate-form'': ...
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124 views

How to Transform a Folded Normal Distribution into a Gamma Distribution?

Let the random variable $X$ have the folded Normal pdf $$f(x)=\frac{2}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$$ with $0\lt x \lt \infty$. What is the transformation $g(X)=Y$ and values of $\alpha$ and ...
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1answer
81 views

Proving a distribution is a member of the simple exponential family

Does anyone have any tips/ideas/method for proving that a distribution is a member of the simple exponential family (SEF)? Or is the process unique to each distribution? For example, I am trying to ...
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39 views

How can I sample multivariate binary variables such that sum of them follows a gamma distribution?

Edit: Since the original question was confusing as whuber pointed out, let me rephrase the question with a Poisson distribution instead of a gamma distribution. The energy term of a Poisson ...
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1answer
19 views

Test to determine correlation between organism size and concentration?

I have a gamma distribution. I am looking to determine if higher concentrations of a toxin are correlated to larger organism size. I think a non-parametric test would be best, but I'm not sure which ...
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1answer
141 views

Is a gamma distribution bounded between 0 and 1 the same as a beta distribution? [closed]

After making the assumption that monetary losses could be well represented by a gamma distribution (Boland, 2007), mostly negatively skewed, and being interested in loss ratios (ie. lost value / total ...
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1answer
27 views

Estimate parameters of three parameters gamma distribution

I need to estimate parameters of three parameters gamma distribution. Can anybody please give me a clue in which software and by which commands I can do it? Thank you
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33 views

Help me out with real-life Gamma distributions

I'm working with some data relating to maintenance and parts failures. I've got a pure math background with only a little bit of probability, and I'm currently learning by doing. I've got a list of ...
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1answer
70 views

KL divergence between a gamma distribution and a lognormal distribution?

Is there a closed-form formula for the following KL divergence? $D_{KL}(X,Y)$ where $X \sim \mathrm{Gamma}(k,\theta)$ and $Y \sim \mathrm{LogNormal}(\mu,\sigma^2)$
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1answer
73 views

If $X$ takes on a Gamma Distribution, how can I find $X^2$, $X^3$, etc?

I am trying to take consecutive powers of a Gamma Distribution. For example, if $X \sim \textrm{Gamma}(k, \theta)$, I would like to find $X^2$, $X^3$, and in general $X^m$ for $m>0$. The pdf ...
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27 views

How do you approach transformations when modeling?

I'm working with a simple univariate dataset and I've built several models for it. Some I think are fairly decent given that datas structure. In order to get a decent model I had to do some ...
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2answers
39 views

Hierarchical Gamma-Poisson CDF?

What is the most computationally efficient way to evaluate the CDF $$P(X \leq x | r,v)$$ where $$ X \sim Poisson(\lambda)$$ and $$ \lambda \sim Gamma(r,v)$$ I can't see the next obvious step after ...
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3answers
120 views

Hypothesis test for correlation between Gamma random variables

I have two Gamma random variables. I need a hypothesis test to detect a possible correlation between them.
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52 views

How to sample the degrees of freedom of a Wishart distribution?

SHORT VERSION: Given K precission matrices drawn from a single Wishart distribution, I try to infer the degrees of freedom of this Wishart. How can I do it? Is there some place where this derivation ...
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1answer
147 views

Convergence from Gamma to Normal Distribution

I came across this problem: Problem If I have $X_1, X_2, ..., X_n$ $n$ iid random variables which pdf is $$ f_X(x) = \begin{cases} \dfrac{x^{\mu-1} e^{-x}}{\Gamma{(\mu)}} &0<x<\infty, ...
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1answer
37 views

Gamma GLM predicting the second parameter of the Gamma

The gamma distribution has two parameters, I understand that the linear predictor predicts $\mu = g^{-1}(X\beta)$ where $g$ is the link function but how does the linear predictor specify the second ...
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Question on Inverse-Wishart Distribution when reading Peter Hoff's book

I have a couple of questions when reading the chapter 7 The Multivariate Normal Model of Peter Hoff's "A First Course in Bayesian Statistical Methods". First, could anyone give me any resource about ...
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1answer
184 views

Maximize the profit of a product given a Gamma distribution

I'm having trouble translating this problem into a workable form: A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters ...
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35 views

How to prove NegativeBinomial(r,p) converges to Gamma(r,1) as p->0

Let $X\sim NegBin(r,p)$ and $Y\sim Gamma(r,1)$. How can I prove that $pX \overset{dist}\to Y$ as $p\to 0$. Is this statement the same as $X\overset{dist}\to Gamma(r,1/p)$. Thanks.
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41 views

How to estimate Extreme value distribution parameter

Assume that I have non-negative Gamma random variables $\{X_i,i=1\dots n\}$ and I want to find $M_n=\max\{ X_i\}$. I want to apply generalized extreme value distribution (GEV), but how to find $\mu$, ...
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1answer
159 views

goodness of fit gamma distribution matlab

I have a set of observations obs. If I plot the histogram of the observation I see that they could come from a gamma distribution ...
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65 views

Gamma vs tweedie distribution for large productivity dataset

I'm running some GAMs using the mgcv R package on a dataset with ~8.5k observations, where productivity is the response and environmental conditions are the covariates. However I am unsure of which ...
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1answer
276 views

Need help calculating poisson posterior distribution given prior

I have been attempting to figure this out for hours, but gamma distribution is somehow beyond me. I have a question where we are given α=5 and ...
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11 views

Performing OLS with gamma transformation

In some specific areas it is common to perform OLS regresion with beta distribution transformation. The α and b parameters are calculated by the sample's μ and σ^2. While the transformed dependent ...
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1answer
107 views

When is distribution of $|X+Y|^2 $ equivalent to $|X|^2+|Y|^2$?

I am trying to compute the distribution of the following $$Z=\bigl(X+Y\bigl)^2$$ BUT I have that both $X$ and $Y$ are Nakagami with parameter $m$. (A Nakagami random variable is the square root of a ...