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Questions tagged [gamma-distribution]

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

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Finding a confidence interval for shifted exponential distribution

Let $X_1,\ldots, X_n$ are i.i.d. random variables such that: $$f(x;\sigma ,\theta)=\frac{1}{\sigma}e^{\frac{-(x-\theta)}{\sigma}}, x\gt \theta$$ where $\sigma \gt 0 $ and $\theta \in R$ . a) if ...
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Model validation in R - Gamma GLMM

I'm trying to model a response variable y with respect to a nested variable x in R. First of all, I fitted a linear mixed model (LMM) as it follows: ...
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Poisson and Gamma distribution for testing randomness

In genetics I want to test whether InDel (insertion and deletion in DNA) sizes occurs with the same probability. I heard that I should gamma distribution to model it. I found ...
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59 views

Gamma Distribution CDF Intuition; Are My Parameters Correct?

I'm learning about gamma distributions and I want to make sure my reasoning is correct for an example I found here. Suppose you are fishing and you expect to get a fish once every 1/2 hour. ...
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Optimal distribution for Acceptance Rejection Sampling

For some project I have been sampling from the Gamma distribution. I have been using the exponential distribution intensively. One method I have employed is the Acceptance rejection sampling, ...
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47 views

Finding method of moments estimator of $\theta$ in $\Gamma(\theta,\theta)$ distribution

Please refer to the question in image I have tried to find $ E(x) $ but i ended up with $\overline x $ = $\frac{\theta + 1}{\theta} $ which statisfies no option , i also tried to find $ E(x-1)^2 ...
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Should I subtract lower bound from Gamma distributed data before estimating distribution parameters?

I have some real world data that reflects waiting time in a system. As it's about waiting times I assume it's Gamma distributed and visual check (histogram overlaid by a fitted Gamma PDF) shows no ...
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49 views

When a probability density function is defined to be finite?

In "Pattern recognition and machine learning" by Cristopher Bishop, Chapter 2.3.6 (pag. 100) says that The gamma distribution has a finite integral if $a>0$, and the distribution itself is ...
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interpreting Gamma regression coefficients - individuals not totalling sum of parts

I have fit a gamma regression to a dataset, and like with traditional linear regression, I would like to calculate the contribution of each independent variable across the entire model. The problem I ...
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79 views

Gamma distribution parameters estimation [closed]

I have a set of samples taken from a population distributed with a Gamma distribution, so \begin{equation} f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} \end{equation} I should ...
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47 views

MLE for Beta distribution, with $\beta$ = 3

I'm trying to calculate the Maximum-Likelihood Estimator for $\alpha$, using the beta distribution with $\beta = 3$. I'm kind of stuck at the last bit. Perhaps I've made a mistake somewhere, or this ...
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How to measure trends in the frequency of events

for my master thesis i want to figure out which website visitors increased / decreased the intervals between their visits during 3 months. Eventually, I want to segment them into increased and ...
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Marginal Distribution of Hierarchal Model Normal distribution with unknown mean and precision

I am trying to use a Hierarchical model where there I have a normal distribution with random mean and precision: $$ y \sim N(\mu, \tau)\\ \mu \sim N(M, T)\\ \tau \sim Gamma(\alpha, \beta) $$ I'm ...
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How was the pdf of the generalized gamma distribution in GAMLSS reparametrized?

I am trying to bring together the definition of the Generalized Gamma distribution in R-package GAMLSS by Rigby and Stasinopoulos and the general definition of the Generalized Gamma distribution, ...
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Probability question in Mat

My teacher give me this question: Using MATLAB, generate 10000 Random Vectors of size 500 with the PDF of Gamma distribution. Find the PDF of maximum and minimum of the generated Random vectors. (Use ...
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42 views

PDF of human population weight distribution

I have a lot of data on human weight coupled with information about gender, age, country and platform (desktop or mobile user). Looking something like this: ...
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Hypothesis testing for the Gamma distribution - Rejection region

Suppose under $H_0$, a test statistic $T$ has a gamma distribution with paramaters $\theta$ and $k$. Suppose also that the distribution of $T$ under $H_1$ is unknown. What is the appropriate rejection ...
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Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
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Which gamma regression model to use for extrapolation?

I'm looking for a regression model which would satify these requirements: My target variable follows the exponential distribution, so to my understanding I should use gamma loss function. I have ...
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133 views

Truncated Gamma Distribution

The Gamma distribution is the conjugate prior of Poisson distribution. What about the Truncated Gamma distribution? Is it still the conjugate prior of Poisson distribution?
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167 views

Numerically/approximately integrating over independent gamma variables

Problem Statement For a problem in biology, I am testing out a joint distribution of the form: $$ X \sim Multinomial(\frac{\theta_1}{\sum \theta_i}, ...,\frac{\theta_n}{\sum{\theta_i}}) \\ \theta_i \...
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In an MCMC model, what happens if the observed data contains negative and positive observations, but a strictly positive model (Gamma) is used?

Suppose that we observe data that range from negative to positive support. Now, suppose we use a likelihood specification that is a Gamma, or some other distribution with strict positive support. I ...
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MLE For Gamma Distribution [duplicate]

The $(\alpha, \lambda)$ Gamma distribution has density function $f(x| \alpha, \lambda) = \frac{\lambda^\alpha x^{\alpha-1} e^{-\lambda x}}{\Gamma(\alpha)}$ for $x>0$ How can we obtain an MLE ...
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FITDIST: Why am I getting this shape for the gamma distribution?

I am using fitdist to fit a gamma distribution based on this link: How to draw fitted graph and actual graph of gamma distribution in one plot? My code is as follows: ...
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Poisson Gamma Distribution in R - Creating Enrollment Modeling Curve

I'm trying to create an enrollment curve for a clinical trial based on the following variables: Country start up timelines (staggered), Number of sites, Number of total subjects needed, Number of ...
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Do mismatches in areas of peak density affect the KS-test more than mismatches in low-density areas?

In the following plot you see my empirical data (black) plotted against a hypothesised distribution (blue). However, a KS-test shows that there is no indication that my sample follows this ...
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Gamma glm log link - what does predicted values mean

Does the predict function in R for gamma glm with log link predict the actual values or the mean value? There is a gamma glm model in R with log link. Using predict(model,data,type = 'response') to ...
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Shift data to use Gamma glm with “identity” link?

I am trying to fit a glm using the Gamma family and the "identity" link. As I am analyzing nutrient concentrations in coastal waters (that can not be negative and are unlikely to be zero) I assume the ...
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1answer
126 views

Gamma Distribution Sufficient Statistics

I've been asked to show the gamma distribution can be written in the form $p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$ where $T(x)$ is a sufficient statistic. .... I have ...
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65 views

Inverse of Gamma Distribution [closed]

I am using python to calculate Inverse of a CDF of gamma distribution (using scipy. stat.gamma.fit). But for probability value 1, it is coming infinite. If it is replaced from 1 to 0.99 it works but ...
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42 views

two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
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How much data is considered “sparse” for fitting a mixed (Beta Geometric) distribution with 4 shape parameters?

I'm using CamDavidsonPhillips Customer Lifetime Value library to calculate CLV, and it uses a distribution based on Peter Fader's work on the subject that fits a Gamma distribution to model customer ...
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1answer
27 views

Point mass at zero and a chi square distribution with one degree of freedom

I am unclear about the critical value of a point mass at zero and a chi square distribution with one degree of freedom. How to find this?
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34 views

Apply 3 sigma formula in gamma distribution?

Let say i have some data that follows gamma distribution, and i calculated the Mean and Standard deviation of the gamma distribution. I also know that there are some outliers(Noise) in the data i ...
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2answers
135 views

What are the assumptions of a Gamma GLM or GLMM for hypothesis testing?

What are the assumptions when doing hypothesis testing using a Gamma GLM or GLMM? Are the residuals suppose to be normally distributed and is heteroscedasticity a concern like the Gaussian (normal) ...
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130 views

Expected value of last Gamma RV in a sum

I've got a sum of $X_i \sim \text{Gamma}(k, \theta)$ i.i.d. random variables. I'm trying to find the expected value of the final $X_i$ that takes the sum above a certain value, i.e., to find the value ...
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What is the best point forecast for gamma distributed data?

I believe that the values I am forecasting are gamma distributed with shape $k>0$ and scale $\theta>0$. I need a point forecast (i.e., a one-number summary) that minimizes the expected error. ...
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What is the problem in my CDF derivation?

Let $Z = \frac{XY}{aX+bY+c}$ where the random variable $X$ and $Y$ follows gamma distribution such that $X\sim G(\lambda_x,\theta_x)$ and $Y\sim G(\lambda_y,\theta_y)$ The CDF of $Z$ can be ...
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Gamma family GLM fails after removal of highly influential point

I have some data where distribution seems light a rough Gamma distribution. I am therefore investigating the relationship in r using a Gamma family Generalized Linear Model. When I investigate the ...
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84 views

glmer model convergence question

We are working with a longitudinal dataset, with three variables: WAIP, BPSRRI and group. WAIP and BPSRRI are measured repeatedly for 10 times and group refers to the group assignment of our subjects ...
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2answers
116 views

Independence of ratios of independent variates

If $X= x_1/(x_1+x_2)$ and $Y= (x_1+x_2)/(x_1+x_2+x_3)$ where $x_1,x_2,x_3$ independent chi-square variates with d.f $n_1,n_2,n_3$ respectively, are $X$ & $Y$ independent? I know the condition ...
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Is my data suited for an ANOVA?

I have two types of "media" (surface and sediment). I have six toxin variants "variant". All the toxins originate in the surface water, but I want to know if certain toxins are more likely to ...
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480 views

Gamma distribution as a member of exponential family

In my lecture notes I have that the distribution of a random variable $Y$ is said to be in the exponential family if it can be written as $f(y;\theta)=exp(a(y)b(\theta)+c(\theta)+d(y))$, where $a,b,c$ ...
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50 views

Gamma distribution

We consider the gamma pdf of a random variable $Y$ that is given by: $$f(y) = (s^a \Gamma (a))^{-1} y^{a-1}e^{-\frac{y}{s}},$$ where $y \geq 0$, $s$ is the scale parameter and $a$ the shape ...
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48 views

Is the convolution of independent normal and gamma also a Pearson distribution?

The answer here What is the convolution of a normal distribution with a gamma distribution? gives the pdf of convolution of normal and gamma random variables. Is there a known random variable to which ...
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Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
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165 views

Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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Generate an autocorrelated Gamma sample of length N

How does one simulate an autocorrelated Gamma sample of length $N$? All I found online was about generation correlated variables and not an autocorrelated sample.
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In a gamma regression, how can i interpret coefficients?

My question is pretty simple, i have done a bayesian gamma regression with an inverse link, so: $\eta_i$=$\beta_0+\beta_1x_{i1}+\dots+\beta_px_{ip}$ < using an inverse link, mu is the ...
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136 views

hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...