Questions tagged [gamma-distribution]

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

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207 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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32 views

Deviance for Gamma GLM

I was wondering why the Gamma deviance formula is given as: $$2 \sum [ -log(\frac{y_i}{\mu_i}) + \frac{y_i-\mu_i}{\mu_i} ] $$ Shouldn't the 2nd term become zero after the summation is conducted?
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F test and T test in GLM gamma Log link

We get both t-test and F-test for the estimation of GLM Gamma with a log link model. F-test: T-test: From my knowledge for ANOVA, I can report the main effect of League to use the F test(F4,53699=20,...
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Integration of gamma random variable

How can I integrate a random variable $X\sim \text{Gamma}(n,p)$ from $X=0$ to $X=c$? The integration of a gamma random variable is done by the help of the gamma function when the limits of integration ...
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If I have three sets of continuous data which are not normally distributed, can I still compare them with ANOVA?

I have three sets of data from some experiments. I fitted each set to different distributions, and each one fits a different distribution. For example, Gamma, Weibull, and Lognormal. If I want to ...
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Generalized Gamma Distribution in Lifetime Models

It seems to me that a version of generalized gamma distribution is often used in lifetime models. For example, survival analysis uses this distribution. Is there some intuition as to why this ...
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Identifying the best distribution to this data?

I'm trying to fit an appropriate distribution to a data with 216 values and estimate parameters. From Cullen and Frey graph, it looks like lognormal could be a good fit. From q-q plot, Weibull seems ...
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22 views

Box-Cox vs GLM for skewed non homoskedastic data

I need to do a regression on a variable Y that is skewed right (non normal) and heteroskedastic and therefore violates two assumptions of the normal linear model. The data is non negative (some 0 ...
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How do I show that $Y=2\sqrt{X_1X_2}\sim$Gamma$(2p,1)$?

Suppose that $X_1\sim \operatorname{Gamma}(p,1)$ and independently, $X_2\sim \operatorname{Gamma}(p+1/2,1)$. Show that $Y=2\sqrt{X_1X_2}\sim\operatorname{Gamma}(2p,1)$. This problem followed a section ...
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208 views

Is there a Poisson-Gamma-Gamma model?

An example to elucidate my problem: The total claim amount can be modelled by a Poisson-Gamma model as it is assumed that the events (e.g. accidents) are Poisson distributed and the claims are Gamma ...
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QQ Plot for GLM Regression Model in R with Family NOT Gaussian

I apologize if this has been asked and answered elsewhere -- I've tried to find the answer and could not. In R, if you use the normal plot() command on a ...
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Conditional expectation of a Gamma Distribution

$X$ a Gamma Distributed random variable. Calculate $E[X\mid X \in [a,b]]$, where $a>0$, $b>0$. Is there a closed form solution for this, and if so, how can I calculate it?
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Gamma distribution and applications

I'm looking for references to read about gamma distribution and applications in industry or in quality control. I had a look at statistical methods for reliability data (Meeker and Escobar). It has ...
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Citation for Gamma Probability Density Function?

Wondering if anyone has a reference for the Gamma PDF? I cannot find any leads. Thanks
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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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How to compute categorical probability given a continuous distribution?

I have a continuous target column that consists of IQ scores for kids of age 10. My main purpose is to forecast the probability that a kid is genius given 3 covariates. If a kid has IQ over 160 she/he ...
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1answer
52 views

Iterative solution to Gamma distribution MLE problem

I'm trying to follow the derivation for the MLE parameters of the gamma distribution in [1]. The standard approach is to derive an expression for the log likelihood, differentiate with respect to ...
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12 views

fitting multiplicative non linear model in R

I have samples per month over several years (assuming stationary behavior). My data contains an independent variable: hours (representing available hours per month). The model to be fitted should ...
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1answer
655 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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Normal distribution with known mean and unknown variance (product of two variables)

Assume there is a data point $x$ sampled from a Normal distribution: $$\begin{align} x \sim \mathcal{N}(\mu,\frac{1}{yz}) \propto (yz)^{1/2} \exp [-\frac{1}{2} (x-\mu)^2yz] \end{align}$$ where $\mu$ ...
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1answer
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Computing expected frequencies for gamma distribution R VS Python

I am tryng to "translate" from R to Python a code that computed expected frequencies from the Gamma distribution. More specifically, I have this line of code: ...
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3k views

Data transformation to fit gamma distribution in R

I'm having trouble to fit and simulate a gamma distribution using the fitdistr function from the ...
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1answer
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How much better is the best Moment Bound?

I've been looking at Gabor Lugosi's wonderful notes on concentration of measure inequalities. On page 7 of the notes the exercise asks you to show that $$ min_q\mathbb{E}(X^q)t^{-q} \leq inf_{s\geq ...
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How do I choose a prior for this hierarchical model? (Kruschke book)

I am working through Kruschke's "Doing Bayesian Data Analysis", currently working on the Hierarchical models chapter. The book uses JAGS for MCMC. One of the exercises asks the reader to compare two ...
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700 views

What kind of distribution does this have?

Currently I'm trying to figure out the distribution of the following: $X \sim \frac{\sqrt{n}}{\sqrt{Gamma(n,\beta)}}$ where the denominator follows a $Gamma(n,\beta)$ distribution. I've checked out ...
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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 ...
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product of two probability density functions

I have on one hand a z variable following a standard normal distribution N(0,1) and on the other hand a variable X following a Nakagami distribution. I am supposed to find a Gaussian distribution when ...
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What is the distribution of these functions of Nakagami random variables?

I am new to this forum and hope I can get help. A Nakagami random variable $X$ with parameter $m$ has the following pdf $$X\sim \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ ...
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What is the distribution of a mixture of exponential distributions whose rate parameters follow a gamma distribution?

I want to know the theoretical distribution of a mixture of exponential distributions whose rate parameters are distributed according to a gamma distribution: $$ y\sim\text{Exp}(\theta), \quad\text{...
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Do I use the Poisson or the Gamma distribution to answer this question involving a Poisson process?

On average, 1 customer arrives every 2 minutes to purchase a ticket to an event. After 100 customers, tickets sell out. What is the probability that tickets sell out in less than 2 hours? I am ...
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Survivor function for generalized gamma from flexsurvreg output

I am trying to plot/generate a survival curve in Excel using the output from flexsurvreg in R. The below is a snapshot from R with the corresponding estimates (y axis values) for the time (x axis ...
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Highly fluctuating value of test statistic in Anderson Darling test

I am trying to verify whether my sampled data is from a population with a Gamma density function. My plan is to do this by means of the Anderson-Darling test. Before doing so, I thought it would be ...
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Gamma distribution and hyperparameters

The formula for mean and variance of a gamma distribution is given by a/b and a/b^2 (hyperparameters) respectively.Are they estimates of the posterior gamma distribution? Can prior, likelihood and ...
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Comparing Gamma GLMS using Analysis of Deviance

Im new to stats/R and have just started a course on generalized linear models and am a little confused. Im comparing two gamma glms using analysis of deviance to see whether m0 is a reasonable ...
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How to scale the probability in the gamma probability density function for modeling infectious disease transmission's latent period?

I like to model infectious disease transmission between people over time. Usually there is a latent period (or delay) between the initial infection of person A and when person A becomes infectious to ...
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Mean of nearest neighbour distance in a clustered distribution

What would be the expected value of distances to the nearest neighbor in a set of points in 2-dim space that have a clustered (not random) spatial distribution? If the distribution is random the ...
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1answer
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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 ...
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Density Function for Time until Second Arrival with Poisson Process [Solved]

Problem Statement: The number of arrivals $N$ at a supermarket checkout counter in the time interval from $0$ to $t$ follows a Poisson distribution with mean $\lambda t.$ Let $U$ denote the time until ...
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Differentiating the Log-Normalizer of the Gamma distribution

Converting the Gamma distribution to the Exponential family form, you get: $$f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} = e^{(\alpha-1)\ln(x) -\beta x -(\ln(\Gamma(\alpha)-\...
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Using Python - scipy.optimize to fit a weighted sum of two distributions

I have been working at finding a statistical function that fits a set of empirical data in a project I'm working on. After having tried a number of statistical distributions by using scipy.stats I ...
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How to calculate variance of Y explained by my predictor X when using a gamma GLMM and an identity link?

I want to assess how well my variable X explains Y. As far as I know, I have to use mixed-effects models since what I have is data over time from 6 individuals. Below I show the relationship between X ...
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Obtaining distribution parameters from GLM

My question is part statistical and part coding. Say I have fitted a gamma log-link GLM. doc example ...
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1answer
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Gamma distribution: ratio of 2 CSS not containing $\beta$

Let $X_1,...,X_n$ be iid and follow $Gamma(\alpha, \beta)$, where $$f(x,\alpha, \beta)=\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$ I already showed that $\overline{X}$ and $X^*=\...
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Conditional distribution of complete sufficient statistics being ancillary of $\alpha$

Regarding the distribution and statistics as described here, I need to show that the conditional distribution of $\overline{X}$ given $X^*=x^*$ does not depend on $\alpha$. I remember my professor ...
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What is the distribution of a r.v. if the reciprocal is distributed gamma?

I want to find the posterior distribution of σ^2 when (X1, X2, ..., Xn) ~ N(μ,σ^2), μ is known, and 1/(σ^2) ~ Gamma(α,β), but I'm not sure how to find the prior of σ^2 given the prior of 1/(σ^2). ...
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Application of Pareto/NBD and Pareto/GGG models for customer lifetime value estimates in high churn setting

I have been attempting to estimate customer lifetime value in the context of online classifieds (high churn context) using probabilistic models, chiefly the Pareto/NBD and Pareto/GGG techniques ...
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1answer
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Comparing generalized linear mixed models (varying the distribution & link function)

I have some questions on performing mixed models on multi-rater data when residuals are heteroskedastic. I've found some of the info on Cross Validated confusing and quite technical-- would be very ...
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How do you compute ICC in a gamma loglink glmm?

I have fitted a gamma generalised linear model with a log link in r using glmer from the package lmer. The goal is to make prediction models for age in old trees for conservation purposes. The choice ...
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1answer
88 views

Conditional distributions derivation

I am trying to solve the following problem: Given $n$ independent observations $Y_i$ from a Normal$(\theta, \tau^{-1})$ distribution with unknown mean $\theta$ and unknown precision $\tau$, i.e $$...
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What is the expected value of x log(x) of the gamma distribution?

Let $w(x) = x \log{x}$ $x \sim Gamma(\alpha = 3.7, \lambda = 1)$ Find $E[w(x)]$ I have set up the following integral: $\int_0^{\infty} x\log{x} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha -1}...

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