Questions tagged [gamma-distribution]

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

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2
votes
1answer
39 views

How to evaluate $\int_0^\infty m^{x+1}e^{-2m}dm$ as $\Gamma(x+2)\frac{1}{2}^{x+2}$?

$\int_0^\infty \frac{m^{x+1}e^{-2m}}{\Gamma(x+1)\Gamma(2)}dm =\frac{\Gamma(x+2)\frac{1}{2}^{x+2}}{\Gamma(x+1)\Gamma(2)}$ How does the left side equal the right side? I understand that the gamma ...
2
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0answers
114 views

Not sure if a gamma glm or glmm is needed

I am fitting a linear model for de CO2 dataset in r, I want to predict plant uptake (always positive) using Type, conc, and treatment, a quick look at the data ...
11
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2answers
2k views

What is the expected value of the logarithm of Gamma distribution?

If the expected value of $\mathsf{Gamma}(\alpha, \beta)$ is $\frac{\alpha}{\beta}$, what is the expected value of $\log(\mathsf{Gamma}(\alpha, \beta))$? Can it be calculated analytically? The ...
2
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0answers
471 views

Which method to use when calculating the confidence interval of GLMM Gamma Regression with the lme4 package in R

I am fitting a GLMM with family gamma using the lme4 package in R. Below is a code example to simulate the gamma GLMM fitting. ...
2
votes
1answer
401 views

Is the canonical parameter (and therefore the canonical link function) for a Gamma not unique?

Consider $Y_1, \dots, Y_n$ independent from the Gamma distribution. For $y > 0$: $$\begin{align} f(y \mid \alpha, \beta) &= \dfrac{1}{\beta^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\beta} \\ &...
2
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0answers
23 views

finding process corresponding to laplace transform

I have a positive stochastic process $X(t)$ with Laplace transforms $$ \mathbb{E}\left[\mathrm{e}^{-uX(t)}\right]=\left(\frac{a+u\mathrm{e}^{-\kappa t}}{a+u}\right)^{b} $$ One can clearly see that the ...
1
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1answer
2k views

Gamma Conjugate Prior & Poisson Process

I am analyzing daily data transaction data. I am assuming that The number of transactions in every day of length t has the Poisson distribution with mean λt The number of transactions in evert ...
1
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0answers
19 views

glmer: distribution law for number of events / time

Sorry if this is a basic question but I can't find an answer that is clear enough, so I prefer to ask. I want to model a number of events (number of gaze) that depends on the behaviour that one ...
3
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2answers
111 views

sum of $N$ gamma distributions with $N$ being a poisson distribution

I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution. I $N$ number of events start in $t$ minutes, the what will ...
1
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1answer
140 views

Exact Confidence Interval for Poisson using Gamma-Poisson Relationship

I'm reading Casella-Berger's Statistical Inference and trying to follow along in example 9.2.15, which constructs an exact confidence interval for a Poisson rate. In this example, the authors solve ...
0
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1answer
515 views

clarifying exponential-gamma conjugate prior

I'm referring to page 22 of this white paper. On page 22, it says the following: given that $s_i \sim \text{Exp}(\theta), i = 1,..,c$ $\theta \sim\text{Gamma}(k, \Theta)$, Then the posterior ...
0
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0answers
30 views

What is the motivation for the formula for the Gamma distribution? [duplicate]

In my statistics class, it was proven that the sum of independent and identically distributed random variables distributed according to an exponential distribution follows a gamma distribution using ...
3
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1answer
168 views

Probability distribution for independent time to event

I am looking at waiting times between two events from multiple patients, so I'm looking at a gamma distribution. Turns out, the model is plotting out an exponential distribution, which if I was to ...
0
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0answers
15 views

Scale mixture of normals; t-distribution; proving marginal distribution equality [duplicate]

This is a review-type homework question that I'm very much stuck on. Basically, these are the assumptions: $\theta|\lambda = N(\mu,\frac{\sigma^2}{\lambda})$ $\lambda = \rm{Gamma}(\nu/2,\nu/2)$ $\...
3
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2answers
2k views

Why is regression with Gradient Tree Boosting sometimes impacted by normalization (or scaling)?

I read that normalization is not required when using gradient tree boosting (see e.g. https://stackoverflow.com/q/43359169/1551810 and https://github.com/dmlc/xgboost/issues/357). And I think I ...
2
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1answer
282 views

Mean of truncated gamma distribution using threshold

Given a gamma distribution with PDF: $f(x;\alpha;\beta) = \frac{x^{\alpha-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^\alpha}$ with a shape parameter $\alpha > 0$, a scale parameter $\beta > ...
2
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1answer
2k views

Interpretation of coefficients from glm Gamma

I am attempting to fit a model to a dataset with frequency (Hz) is the dependent variable. Using a generalized linear model based on a gamma distribution seems appropriate since the values of the ...
2
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0answers
48 views

Hypothesis testing for generalized (three parameter) gamma distribution

I have generalized gamma distribution with the following equation: $$ f(x) = \frac{\lambda^{a\tau}\tau x^{a\tau - 1}}{\Gamma(a)}e^ {{(x\lambda)}^\tau} $$ and log-likelihood function $$ l(a, \lambda,...
1
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1answer
71 views

Strange behavior of Gamma prior in a setting with binomial likelihood

I am trying to use Bayes theorem to estimate the probability of a binary event. To give a (simplified) example: Let's say our a priori guess of the probability of the event per trial is 0.3 (and the ...
1
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0answers
361 views

How shape parameters are connected with mean, variance, skewness and kurtosis of generalized gamma distribution

I am writing a code in python that can generate probability distribution with given mean (m), variance (v), skewness (s) and kurtosis (k). In scipy library of python, there is a function named ...
5
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2answers
90 views

Did I mess up the Poisson-Gamma relationship?

Let $X_1, X_2$ be independent, exponentially distributed random variables with mean 2. So $X_1+X_2=Z$ is gamma distributed with $\alpha=2$ and $\beta=1/2$. I am trying to solve the probability that $...
8
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2answers
2k views

The identity link function does not respect the domain of the Gamma family?

I am using using a gamma generalized linear model (GLM) with an identity link. The independent variable is the compensation of a particular group. Python's statsmodels summary is giving me a warning ...
2
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1answer
810 views

Relationship between the gamma distribution and Non-central chi squared distribution?

If $Y = \sum_{i=1}^N X_i^2 $, where $X_i \sim \mathcal{N}(\mu,\sigma^2)$, i.e. all $X_i$ are i.i.d gaussian random variables of same mean and variance, then what is the resultant PDF of $Y$? How the ...
3
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1answer
64 views

Question regarding the distribution of sum of random variables

Let $X_1, ... X_n$ be i.i.d random variables that have an exponential distribution with parameter $\theta$. So we know that $\sum X_n \sim \Gamma(n, \theta)$. This makes sense by working backward. ...
9
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3answers
176 views

Independence of statistics from gamma distribution

Let $X_1,...,X_n$ be a random sample from the gamma distribution $\mathrm{Gamma}\left(\alpha,\beta\right)$. Let $\bar{X}$ and $S^2$ be the sample mean and sample variance, respectively. Then prove ...
2
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1answer
142 views

How to handle repeated measures when fitting a gamma model with log link

I am not a statistician and wanted to see if anyone could help me with some statistical modeling. I have the total medical costs for thousands of patients (total yearly cost for each patient) along ...
7
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1answer
235 views

MLE of $f(x;\alpha,\theta)=\frac{e^{-x/\theta}}{\theta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}$

Let $X_{1},X_{2},X_{3},...,X_{n}$ be a random sample from a distribution with pdf $$f(x;\alpha,\theta)=\frac{e^{-x/\theta}}{\theta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}I_{(0,\infty)}(x ),\alpha,\theta&...
3
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1answer
62 views

Probability of the generalized gamma distribution

I am trying to compute the value of $\bar F(x)=1-F(x)$ where F(X) is the generalized Gamma distribution. I found that this distribution is also called the equilibrium distribution of Weibull. Someone ...
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0answers
34 views

Variance of 1 over sum

I was given to prove that if we're given statistics of Gamma distributed random variables $X_1,X_2...X_n$ (pdf $f_{\chi}(x,\alpha,\lambda)=\frac{{\lambda}^{\alpha}x^{\alpha-1}}{\Gamma(\alpha)}e^{-{\...
1
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1answer
167 views

finding quantiles of a kernel density estimation

I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step). I used code density()...
4
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1answer
375 views

How to improve fit of distribution to data

I'm trying to fit one of common expenential distributions to data using histfit. However it seems that results aren't as good as expected - it seems that peak should be higher. Histograms presents ...
4
votes
1answer
2k views

Why do we use inverse Gamma as prior on variance, when empirical variance is Gamma (chi square)

Let $$X_i\sim \mathcal{N}(0,\sigma^2)$$ than we know that $$\sum_{i=1}^N\frac{X_i^2}{N}\sim\Gamma(\frac{N}{2},\frac{2\sigma^2}{N})$$ that the empirical variance follows a Gamma distribution. How do ...
0
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0answers
140 views

Power of a random variable gamma distributed

If a random variable $X$, follows a Gamma distribution. I know that the distribution is given by: $$F_X(x)=\frac{1}{\Gamma(\alpha)}\int_{0}^{x/\theta}u^{\alpha-1}e^{-u}du$$ Now, If I need to consider ...
0
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0answers
127 views

how to get posterior distribution of beta with gamma prior

I have $X_1, ..., X_n \sim beta(\theta,1)$ and $\theta \sim gamma(r, \lambda)$ and wish to compute the posterior distribution. Since $f(\textbf{X} | \theta) = \theta^nx^{n(\theta-1)}$ and $\pi(\...
1
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0answers
227 views

expectation of Log Noncentral Chi

Let $X$ follow non central chi distribution, the formula of $\mathbb{E}[\log(X)]$ (or equivalently $\mathbb{E}[\log(X^2)]$ where $X^2$ is Non central chi square) can be found here and here. However, ...
1
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0answers
44 views

Quotient of Pareto and Gamma random variables

I cannot find an explicit formula for the quotient of a Pareto random variable divided by a Gamma random variable. The only that I found is something like, for $P(X)$ pareto's like and $P(Y)$ Gamma's ...
3
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1answer
313 views

Generalized Gamma: log normal as limiting special case

The Generalized Gamma Distribution has p.d.f. defined as follows (see e.g. here for reference): $$ f(t; \theta, k, \beta)=\frac{\beta }{\Gamma (k)\cdot \theta }{{\left( \frac{t}{\theta } \right)}^{k\...
1
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0answers
63 views

Is my understanding of “family of distributions” correct?

As I was looking to understand the concept of "family of distributions", I stumbled upon this answer. However, I was a bit confused with answer and I'm hoping that someone may be able to clarify for ...
4
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1answer
69 views

Integrate Gamma pdf with respect to shape parameter alpha

Is there a trick to integrating a Gamma pdf with respect to the shape parameter alpha? I have yet to come up with a way to do it.
4
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1answer
46 views

Generating Matrix-Gamma distributed random variables

Is there an established algorithm for sampling from the Matrix-Gamma distribution (https://en.wikipedia.org/wiki/Matrix_gamma_distribution)? I suspect the procedure would be similar to generating from ...
0
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1answer
4k views

How to find alpha and beta from a Gamma distribution?

The task: Using the method of moments model the data (sample) as a set of 20 independent observations from a Gamma(λ, k) distribution. I have found the mean and variance but unsure how to find alpha ...
1
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1answer
188 views

Find the normalisation constant of this gamma function

I have a gamma priors defined as $$c_j \sim Ga(a_j,b_j)$$ where $b_j$ is the rate. The component likelihoods are defined to be $$L_j(c_j;x) = c_j^{r_j} exp\bigg\{-c_j \int_0^T g_j(x(t)) \, dt\...
1
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1answer
421 views

What is the expected maximum value of a gamma distribution, as a function of number of samples?

I have the following situation. I have observations that fill out a gamma distribution. (At least they seem to: the distribution of values of several thousand observations looks to the eye like a ...
5
votes
1answer
1k views

What constitutes a large KL divergence?

I have 2 gamma distributions $X_1 \sim Ga(13,1) \\ X_2 \sim Ga(3,1)$ Where we are defining our gamma distribution probability density function for a random variable $X \sim Ga(a,b)$ to be $f(X) = \...
2
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0answers
23 views

Compare the quality of distribution fits

I have two random variables $A$ and $B$ they are of different size. Both are well fitted as $\gamma$ distributions. My question is to find which one is more gamma like. Could You help me to solve ...
2
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1answer
730 views

Elastic Net for Gamma distribution

I am investigating Elastic Net method on R to build a prediction model on pricing amount. I have about 70 dummies variables and results make sense regarding variable selection, stability... However ...
1
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1answer
58 views

Fixed effects gamma estimator?

Question 18.12 in Wooldridge's Econometric Analysis of Cross Section and Panel Data describes a "fixed effects gamma estimator", which appears to be analogous to a fixed effects Poisson estimator, ...
0
votes
1answer
54 views

Determining appropriate distribution to describe wind speed data

Hey, I have a dataset of hourly wind speed data and according to literature, the weibull distribution is best suited for this case, however when I fit this distribution to the data, according to the ...
1
vote
1answer
316 views

Calculate the Kullback-Leibler Divergence for these 2 Gamma distributions

I have 2 models $P \sim Ga(115,1329.914) \\ Q \sim Ga(140,650.6775)$ and I'm looking to calculate the K-L divergence of these 2. $D_{KL}(P||Q) = \int_\infty ^\infty p(x) log \frac{p(x)}{q(x)}\,dx$...
6
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0answers
175 views

First two moments of the ratio of the geometric mean to the arithmetic mean of Gamma random variables

Let $X_1,\ldots, X_n$ be $n$ uncorrelated random variables from a Gamma distribution with different parameters: $X_i \sim Gamma(k_i, \theta_i)$. What is the distribution of $$ U=\log \left[ \dfrac{\...