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

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Hazard function of a gamma distribution

The system we are working on is biological, more specifically the distribution of specific events across a chromosome. This can be thought of as 1D array (the chromosome) across which points can be ...
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25 views

Intuitive meaning of the limit of the hazard rate of a gamma distribution

For a Gamma distribution with shape parameter $\alpha >1$ and scale parameter $\beta > 1$, one can show that its hazard rate function $h$ is increasing and satisfies \begin{equation} ...
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26 views

Square of gamma random variable

If i have a random variable with distribution $X \sim \Gamma(\alpha,\beta)$ then what would be the distribution of $Y = \lambda X^2$ (with $\lambda$ a scaling factor)? Can I say that $Y$ will follow a ...
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14 views

Must the response variable be gamma distributed to appropriately use a gamma-log model

I'm responsible for challenging a gamma model with log link. The developer claims that an assumption of the gamma-log generalized linear model is that the response variable, in this case average ...
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1answer
43 views

Attainable bounds for correlations for Gamma random variables?

I'd need to know if it's possible to reach [-1,1] bounds with Pearson's correlation with a generic pair of Gamma random variables. The problem as you may imagine is there's no known closed form for ...
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27 views

sum of gamma and normal random variables

If $X$ has gamma distribution with mean $n/\lambda$ and $Y$ is normally distributed random variable with mean $\mu$, then what is the distribution of $Z=X+Y$?
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41 views

Transform sum of gamma distribution to chi square distribution

Let's say that I have a random sample $Y_1, Y_2, \dots, Y_n \sim\Gamma(\alpha, \theta)$. I can work out using the moment generating function that the distribution of $\sum Y_i$ is $\Gamma(n\alpha, ...
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10 views

Distribution of dispersion submodel

In double generalized linear models where we assume $y$ follows an exponential dispersion model, where the mean can be modelled as $$g(\mu_i)=x_i^T\beta,$$ and the dispersion $(\phi)$ can be modelled ...
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15 views

bayes estimate possion distribution function

Let {X\, ...,Xn) be random sample from random variable which has Poisson distribution with parameter A. Assume that the prior distribution A for is Gamma(1, 1) and that you have observed sample of ...
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17 views

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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31 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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23 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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67 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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1answer
36 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
51 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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162 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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98 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
57 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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27 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
72 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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44 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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33 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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30 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
30 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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24 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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16 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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25 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
83 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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123 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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58 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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20 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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19 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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157 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
91 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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41 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
164 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
30 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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37 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
77 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
40 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
134 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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65 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 ...
3
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1answer
156 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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201 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 ...