Questions tagged [gamma-distribution]

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

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12
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722 views

What is the intuition behind the expected transaction value for a customer in the gamma-gamma model?

Background and Motivation: I was reading the paper RFM and CLV: Using Iso-Value Curves for Customer Base Analysis by Peter S. Fader, Bruce G. S. Hardie and Ka Lok Lee, in an attempt to gain some ...
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3k views

How do I identify the “Long Tail” portion of my distribution?

I have a number of series that would typically be described as normal skewed or Gamma distributed. For example, say I have a group of customers and have calculated their spend over a fixed length of ...
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180 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{\...
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558 views

Hypothesis testing for Gamma distribution

I have a sample $X_1,...,X_n \sim \Gamma(\alpha, \beta)$, where $\alpha, \beta$ - unknown parameters of Gamma distribution. How to build a test for testing $H_0:\alpha=1$ against $H_1:\alpha>1$?
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84 views

Gamma distribution different derivations

According to this link - http://cnx.org/contents/2d28fe6a-5000-454e-a2b9-6fbca9e9b56c@3/THE_GAMMA_AND_CHI-SQUARE_DISTR the waiting time of the $k$th event in a poisson process is gamma distributed. ...
5
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646 views

How do I sample from the posterior distribution with gamma likelihood with unknown alpha and beta?

I realize that this Wikipedia page provides the proportional form of the conjugate prior to the gamma distribution with unknown $\alpha$ and $\beta$ parameters, as well as the posterior values of $p$, ...
4
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500 views

Showing that a Gamma distribution converges to a Normal distribution

Consider $G = \operatorname{Gamma}(p)$. As $p$ goes to $\infty$, the Gamma becomes more and more bell-shaped. How do I show that $\frac{G - p}{\sqrt{p}} \to Z \sim N(0,1)$ as $p \to \infty$? I ...
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6k views

Relationship between the Gamma and Beta distributions

I was looking at a proof of the following fact Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then $$...
4
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500 views

Basic idea of zero inflated two part models(hurdel) and application to big data (machine learning)

I'm currently working on the data which has 90% 0s in response variable. Based on my research, it seems zero inflated models could be a solution to this. However, while I was reading related documents,...
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2k views

Neyman-Pearson lemma: critical region and hypothesis testing

Let $X_1,X_2,...,X_n$ be i.i.d r.v's with common p.d.f. $$ \mbox f(x)=\frac{x^5e^{-x/\theta}}{5!\theta^6} $$ where $\theta$ > 0. Show that the Neyman-Pearson lemma produces a test of $H_0: \...
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156 views

Is there anything special about Gamma distribution with the shape parameter k=e?

Is there any unique property of $\mathrm{Gamma}(k=e, \text{ scale})$ or a Negative binomial distribution with $r=e$? Here, $e$ is Euler's number, $e \approx 2.71828$. The reason I'm asking is that ...
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3k views

Gamma distribution and Cramér-Rao bound

There are two definitions of the GAMMA distribution: I did the ML estimation, generated the Fisher Information, compared it to the Variance and the Cramer Lower Bound was reached, so the estimator ...
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52 views

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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29 views

Possible to work backward from Convolution of Distributions?

So, having discovered distribution convolution, which is a method for deriving the density of a sum of individual probability distribution densities, $$S = X_{first\_distribution} + Y_{second\...
3
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31 views

Can the logarithm of the sum of random variables be decomposed into the sum of independent random variables?

Suppose $X1$ and $X2$ are two independent Gamma random variables, both follow Gamma distribution but different scales (or rates), their shape can be same or different. Let $Z$ be another random ...
3
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70 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 ...
3
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1answer
67 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 ...
3
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551 views

How to generate random numbers form a gamma-type distribution with small shape parameter

I need to generate a time series of random numbers. I want to do this such that I obtain a stationary Markov Chain with a $\Gamma[\alpha, p]$ marginal distribution, the probability density function ...
3
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1answer
68 views

Inference using Gibbs sampling

Suppose there is a one-dimensional normal distribution $\mathcal{N}(\mu, \sigma)$ for which we want to infer the joint distribution of the parameters using Gibbs sampling. Let $D$ be the data, ...
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151 views

Sum of truncated Gammas and degenerate

I have a variable $X$ which I am modelling with a mixture model: $$\begin{aligned} (X|A) &\sim \mathbb{1}_{0 \leq x < w \cdot m} \cdot \frac{\text{Gamma}(\alpha,0,\beta / m)}{k_1} \\ (X|B) &...
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113 views

Is the Gaussian distribution the only statistical distribution fully determined by the mean and variance?

I've read that the Gaussian marginal is fully determined by the mean and variance. What does this mean in reality? If we consider a Gaussian marginal PDF is given by $$ \pi_G(\xi|\mu,\sigma) = {1\...
3
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436 views

Are kurtosis and skewness meaningful for comparing distributions such as gamma distributions with very pronounced shape parameters?

Are kurtosis and skewness meaningful for comparing distributions such as gamma distributions with very pronounced shape parameters? For instance, take the red distribution in the first plot here: ...
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214 views

Distribution of the ratio of two shifted generalized gamma random variable

$X \sim \mathrm{GG}\left(p,d,\theta_{1},\mu\right)$ where $p$ is power, $d$ is shape, $\theta_1$ is scale and $\mu$ is location parameter. Also Consider $Y \sim \mathrm{GG}\left(p,d,\theta_{2},\mu\...
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226 views

Fitting a non-linear model where observations at each time are random variables drawn from a different (non-Gaussian) distribution

I have a non-linear (and not clearly linearizable) function of a few parameters that models a response over an independent variable (time): $$ f(t;\lambda_1,\lambda_2,\lambda_3). $$ The function $f$ ...
3
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477 views

Problem with Newton--Raphson Implementation of Inverse-Linked Gamma GLM in R

Recently, I've been trying to implement functions in R that use Newton--Raphson to find the MLE of parameters for various GLMs. My focus has (thus far) been on data with responses $y$ that are ...
3
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329 views

Conjugate Gamma Prior

If I have a normal distributed variable $N(\mu,\sigma^2)$ so with fixed $\mu$ the conjugate prior for $\lambda:=\frac{1}{\sigma^2}$ is given by the gamma distribution $\propto \lambda^{\alpha-1}exp{-\...
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686 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 ...
3
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1k views

How to compare models with different distributional assumptions for response variable in GLM?

Let's say I have measurements $Y$ which are all positive, and the distribution seems to be somewhat skewed. I'm modelling $Y$ in GLM framework. Now I could set my GLM using different distributional ...
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85 views

Estimating parameters of inifinite scale mixture from data

Suppose that I have an infinite scale mixture of zero-mean normal distributions, whose mixing distribution is gamma with parameters $\alpha$ and $\beta$. The data is thus distributed according to a ...
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1k views

MLE for Gamma Shifted Distribution

I need to fit a gamma distribution that is shifted to the left and truncated at zero (so that for example, my data may only come from the right tail of the full distribution, and I don't have any ...
2
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0answers
23 views

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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21 views

Estimation of generalized gamma convolutions

How can i estimate on a data sample parameters of a generalised gamma convolution ? To be more specific, if my estimation gives me only a gamma convolution and not a generalised gamma convolution i'll ...
2
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0answers
40 views

Compounding a gamma distribution with another distribution to yield a gamma

I have a gamma distributed random variable $X$, with its mean $\mu$ distributed as some other function $$ X \sim \text{Gamma}(\mu,k)\\ \mu \sim P(\theta) $$ What is the distribution $P(\theta)$ such ...
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35 views

Posterior (conjugate) prior of two parameter Gamma likelihood

This question is related to a previous question on this site. Assume some data is generated from Gamma distribution $p(x\mid\alpha,\beta) \sim \operatorname{Gamma} (\alpha,\beta)$, and both parameters ...
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72 views

Characterizing a distribution

I have a set of words which in a given year has a frequency of occurrence k. I can observe that these words follow frequencies k1, k2, k3,....kn in the following year. If I have some data in the form ...
2
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38 views

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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88 views

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 ...
2
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1answer
35 views

Bayesian Inference: Modeling checkout times at a store

I am currently learning how to use Bayesian inference. I have been making up problems (by defining some population parameters) and then trying to infer those values from samples. I recently made up a ...
2
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0answers
135 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 ...
2
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0answers
717 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
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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 ...
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0answers
53 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,...
2
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0answers
24 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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0answers
74 views

Analyzing standardized / fractional count data

In my experiment I want to figure out how the size of different planting containers, i.e. their volume, affects the number of regenerated plant shoots from root fragments (terminology here is root ...
2
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0answers
219 views

How to derive posterior distribution for non-informative Gamma prior distribution?

Lets assume I have 1000 cases of cancer occurring per year in Barcelona. How would I proceed to estimate/derive posterior distribution for this data? I know that posterior defined as: $Posterior = ...
2
votes
1answer
84 views

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 ...
2
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0answers
216 views

How can i express non central chi square random variable in terms of gamma function?

I know the relation between central chi square and gamma random variables.But i am not able to get relation between gamma and non central chi-square distribution.
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63 views

Put Together Results from GLM Gamma Models

I have a set of healthcare data in which I used GLM Gamma to model the healthcare spending. Then I am trying to put together the results in a style that is similar to this: http://onlinelibrary.wiley....
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243 views

Let $X$ have the gamma $(r, \lambda)$ distribution. Show that $P(X \ge 2E(X)) \le (2/e)^r$

Let $X$ have the gamma $(r, \lambda)$ distribution. Show that $P(X \ge 2E(X)) \le (2/e)^r$. I do not know how to approach this. I am thinking of Chernoff Bounds, but what trips me up is how to ...
2
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0answers
85 views

Convolve Gamma distribution with Triangle distribution?

I am working on the use of distributed delay applied to pharmacometric models. Specifically, the delay kernel I am interested in is the Gamma distribution, with non-integer shape. The historical ...

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