Questions tagged [gamma-distribution]

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

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Sum of squared normals: $ \sum_{k=1}^{n}U_k^2 = \frac{1}{n}\sum_{k=1}^{n}a_k^2 \cdot \Gamma\left(\frac{n}{2},\frac{1}{2}\right)$

Assume $U_k \sim \mathcal{N}(0,a_k^2)$, where $a_k \rightarrow c > 0$ as $k \rightarrow \infty$. It follows that $U_k^2 \sim \Gamma(\frac{1}{2}, \frac{1}{2a_k^2})$. I'm interested in the exact and ...
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GAMMA vs simple linear regression write up APA style

I have been taught in my course to write up my normal linear regression results in this format: “F(1, 97) = 0.07, p = 0.794, Adjusted R2 = 0.01, f2 = 0.01, accounting for 1% of the variance.” I am ...
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Interpretation of Gamma parameters in conjugate prior

So I'm modeling a problem using a Poisson process with rate $\theta$ and this $\theta$ follows a Gamma distribution with parameters $\alpha$ and $\beta$. Say $\theta$ captures the arrival of customers ...
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1answer
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Help understanding parameterization of gamma distribution in R's glm()

I would like recover the gamma distribution parameters from a model fit in R using glm(..., family = Gamma). The first step is trying to figure out which ...
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Including dispersion parameter in prediction

I have already posted this question on r-sig-mixed-models mailing list but I received no response. I am fitting a ziGamma model using glmmTMB to predict the ...
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28 views

Unable to plot obtain realizations of Gamma distribution variant

I have the probability density function (pdf) for $\sigma^2 = 1/\phi$, where $\phi\sim \text{Gamma}(a, b)$. I am trying to simulate 1000 realizations of $\phi$ and then plot a histogram using the pdf ...
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40 views

GEE Interpretation beta coefficient

I am running a Generalized Estimating Equation in STATA with link(log) corr(independent) family(gamma) and vce(robust). The results of the command... ...
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30 views

Handling 0s in a generalized linear model---climate data

I am using a generalized linear mixed model for analyzing climate data and incidence of a disease variable. The data follows a gamma distribution. But I am getting the following error when I am ...
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1answer
32 views

Predictions from hurdle models in R

I am using hurdle models to predict a continuous cost variable that has many exact zeros. I have fitted a hurdle model with a binomial component and a gamma component, but when I am trying to combine ...
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11 views

Why do we link the rate parameter of the Gamma distribution for a Gamma GLM?

I've seen several explanations of GLMs that link the linear combination of coefficients to the rate parameter, and assume the shape parameter is constant for all values of $y_i$ (for example, here: ...
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An unbiased estimator for the 2 parameters of the gamma distribution?

Nor Maximum Likelihood Estimators (MLE) neither the Moments Matching Estimators (MME) for the two parameters $\alpha, \beta$ (shape and rate respectively) are unbiased. Is there a closed formula to ...
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How to write a gamma equation from these coefficients R?

A similar question is out there enter link description here, but I find that the answer was not comprehensive enough to cover other scenarios. ...
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Multivariate Gamma parameter estimation

Consider $X$ a d-dimensional random variable with positive values, mean $\mu\in\mathbb{R}_+^d$, and covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. If I have $n$ samples $\{ x_1, ..., x_n \}$ ...
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How do I construct a symmetric, two sided confidence interval for a given statistic?

Let $X_1,...,X_n$ be a random sample of iid random variables, $X_i\sim Exp(\lambda),\lambda>0$. Consider the statistic $T_1(X_1,...,X_n,\lambda)=2\lambda n\bar{X_n}$. The task is to construct a ...
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Checking if the variance components are chi squared

In Faraway it states that for a Gamma GLM 'The linear link, η=μ, is useful for modeling sums of squares or variance components which are $\chi^2$. If we have some data Y, what test can we use to check ...
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How do we do GEE with a dataset having a lot of zeroes? (Statistics doubts regarding exploring climate data)

I am working on relationship between climate variables available per zip code and a certain disease incidence over 3 years. I found that Gini index and Generalized estimating equations (GEE) are the ...
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Probability of X greater than Y with different types of random variable [duplicate]

My problem is the following: I have 2 random variables $X \sim Gamma(2,\mu_2)$ and $Y \sim Exp(\mu_1)$. I have to compute $P(X > Y)$. How can I do that ? Thank you
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Fitting GLMMs for RT data with different condition level distributions

I have RT data that I'm looking to analyse. RT data usually follows Gamma/Inverse-Gaussian shape distributions, thus I usually fit a glmer model (in R) specifying the family as either of those ...
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1answer
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Probability that one gamma r.v. is greater than another plus a constant

Per this answer, if $X \sim Gamma(\alpha_1, \beta_1)$ and $Y \sim Gamma(\alpha_2, \beta_2)$, then $$P[X > Y] = H_{\alpha_2, \alpha_1} \left(\frac{\beta1}{\beta1+\beta2}\right)$$ where $H$ is the ...
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2answers
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MLE of Poisson-Gamma distribution?

I am trying to create an example that applies fully parametric estimation. I am using a Gamma-Poisson distribution where the random variable is a Poisson random variable with mean $\lambda$ which has ...
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The two estimators of mean of Gamma distribution and the estimators' variances

In Casella Example 10.1.18, the author says it is not easy to calculate the mean of gamma distribution. It seems that we CAN use the easy way $\bar X=\frac{\sum X_i}n$, but the variance of the mean we ...
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Let $X_1,…,X_n\sim\text{Exp}(\beta)$. Find the moment generating function of $X_i$. Prove that $\sum_{i-1}^{n}X_i \sim \text{Gamma}(n,\beta).$

The following is a problem from Wasserman's All of Statistics Problem Let $X_1,...,X_n\sim\text{Exp}(\beta)$. Find the moment generating function of $X_i$. Prove that $\sum_{i-1}^{n}X_i \sim \text{...
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Gamma Distribution satisfying property

How can we prove that gamma random variable $X_{n}$ with parameters $(n,3)$ can satisfy the following relation for some $n$? $$P(X_{n} < n/2) > 0.999$$ I used the definition of density function ...
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Interpretetion of linear predictor of a random variable that follows a gamma distribution

Assuming that: $0 < \nu, \alpha, y < \infty$ $$f_Y(y; \nu, \alpha) = \frac{y^{\nu-1}{\alpha}^{\nu}e^{-y\alpha}}{\Gamma (\nu)} \mathbb{1}_{Y \in (0, \infty)}$$ $$ = \exp \{ -y\alpha + \nu \log \...
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Expected Fisher Information Matrix for Gamma Distribution using canonical link

How to find the fisher information matrix for a random variable $Y \sim $ Gamma$(\nu,\alpha)$? $0 < \nu, \alpha, y < \infty$ I have written: $$f_Y(y; \nu, \alpha) = \frac{y^{\nu-1}{\alpha}^{\nu}...
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Two Gamma functions with common terms produces new Gamma and Beta functions [duplicate]

Let X and Y be independent random variables, X ~ Gamma(α,λ) and Y ~ Gamma(β,λ). Prove tha S=X+Y andT =X/(X+Y) are independent ,S~Gamma(α+β,λ) and T ∼Beta(α,β).
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Scale the gamma distribution

I have a question regarding the distribution of a random variable $\sum_{i=1}^{n} X_{i}^{k}$ given that we know that $\frac{1}{\theta} \sum_{i=1}^{n} X_{i}^{k} \text { has the Gamma }(n, 1) \text { ...
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245 views

Statsmodels: how to run and interpret a Gamma regression?

I have an endogenous variable that is continuous and non-negative. From what I can gather, a Poisson regression is not appropriate because the values of the response variable are not natural number, ...
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61 views

incomplete gamma function in R (conditional mean of Weibull to the power of N)

I am trying to calculate: $$ E(w^n | \underline{w} < w < \bar{w}) $$ where $w$ follows a 2 parameter Weibull distribution $w \sim W(\lambda,k)$ From a previous question, I know the following ...
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Intuitive way to connect gamma and chi-squared distributions

I understand that a chi-squared distribution is a special case of the gamma distribution. However, I find claims of "the math just works out" to be an unhelpful in remembering or ...
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Gamma distribution what is scale and rate

I have question regarding the gamma distribution when using ...
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1answer
45 views

Is there any relationship between two normalized gamma distributions?

Consider two normalized gamma distribution functions $\frac{\Gamma(x,y)}{\Gamma(x)}$ and $\frac{\Gamma(nx,ny)}{\Gamma(nx)}$ where $n$ is a positive integer value. Is there any relationship between the ...
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Z = X1/(X1+X2) where X1 and X2 are gamma distributed [duplicate]

Suppose that $X_1 \sim \Gamma(\alpha_1,\beta)$ and $X_2 \sim \Gamma(\alpha_2,\beta)$ and let $Z = \frac{X_1}{X_1 + X_2}$ ($X_1$ and $X_2$ are assumed to be independent). I want to prove that $Z$ is ...
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2answers
109 views

Why are Poisson distribution and Exponential distribution special case of Gamma distribution?

I am aware that Gamma distribution is used as a conjugate prior distribution for various types of rate parameters such as in Poisson distribution and Exponential distribution. People say that ...
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1answer
64 views

Term for exp(beta) from a Gamma-GLM

I have read a lot about interpretation of coefficients from Gamma-GLMs (using a log-link function), e. g. from this thread How to interpret parameters in GLM with family=Gamma , and found this to be ...
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What can Ido if I get patterns in residuals vs predicted values using `lme4::glmer()` with a GAMMA distribution?

I want to model a response variable (y) as a function of two explanatory variables (x and z)....
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169 views

Interpreting results from Generalized Linear Model, gamma family, log-link

I have a small number of observation point, and the data is continuous and very skewed. I decided to analyze the data with Generalized Linear Model, gamma family, log-link. I'm having hard time ...
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510 views

Loss function in for gamma objective function in regression in XGBoost?

Suppose I want to predict $y$ from a set of predictors $x$. $y$ is gamma distributed, so I want to use gamma regression with XGBoost. The help page of XGBoost specifies, for the objective parameter (...
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The cumulative sum of the difference between dependent Gamma variables

I want to know if it is possible to find an expression for the PDF of the storage in a linear reservoir system as described below. I am aware that there are some numerical methods that would allow me ...
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1answer
81 views

How to specify Gamma parameterizations in a generalized linear model setting

I am trying to model an outcome using a generalized linear model and the Gamma distribution with a log link function using the glm() function in R. I went to ...
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1answer
111 views

Gamma GLM: why log-link is more common than canonical link

"The canonical link of Gamma GLM is $g(x)=1/x$ is often not very practical. Log-link is more appropriated in most cases." One reason I can think of is that log-link makes sure $\mu$, the ...
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Is the question requiring the use of a gamma or exponential distribution? [closed]

Incoming telephone calls to an operator are assumed to be a Poisson process with parameter $\lambda$. Find the density function of the length of time for $n$ calls to be received, and find the mean ...
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marketing hypothesis test sample size calculation

I am trying to design a coupon promotion test to measure the increase in the volume of orders from promotion versus no promotion. To do so, I want to calculate the pre-group sample size required to ...
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Sample Size Calculation - Two Gamma Distributed Mean

I have a similar question to Sample Size Calculation - Two Independent Means, in my case, I am measuring the number of orders our customer make which follows a gamma distribution I believe. How could ...
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1answer
98 views

Bayesian estimation of the variance

The mean of the Gamma distribution is $\alpha/\beta$, while the mean of the Inverse Gamma is $\beta/(\alpha-1)$. Similarly, the mode of the Gamma is $(\alpha-1)/\beta$, but the mode of the Inverse ...
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1answer
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Improving a model for a positively skewed continuous data (no zeros)

I have a positively skewed continuous data (no zeros), representing transactions by amount. Variables age and income were ...
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1answer
258 views

glmer with gamma distribution - problem fitting model

I am trying to fit the gamma distribution to my data as the residuals are not normally distributed but it has been much more difficult than I anticipated. The dependent variable is response times and ...
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1answer
67 views

sampling from $\frac{1}{1+x}$ times Gamma distribution density

I am simulating a process by drawing many random variates $X$ from a Gamma distribution with parameters $\alpha$, $\beta$, $$f_X(x) = \frac{\beta^\alpha \, x^{\alpha-1} \, e^{-\beta x}}{\Gamma(\alpha)}...
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1answer
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What length are some segments of a broken rod?

If a rod (of unit length) is broken into $n$ segments (assuming the $n-1$ breaks occur with uniform probability across the entire length) and $k$ of those segments are chosen at random and laid end to ...
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1answer
373 views

GLMM hurdle model for continuous data -Truncated negative binomial family in glmmTMB?

I am running a hurdle model using the glmmTMB function. My dependent variable is continuous and >= 0. I was looking for a function that would allow me to model the binary response in a logistic ...

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