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Questions tagged [gamma-distribution]

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

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11
votes
0answers
687 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 ...
9
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0answers
2k 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 ...
6
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0answers
169 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{\...
5
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476 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 $H_1:\alpha>1$?
5
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621 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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346 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 ...
4
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0answers
113 views

Showing independence between two functions of a set of random variables

I've been working on the following problem and I'm confused about how to get started: Let $X_1, X_2,\dots, X_n$ denote i.i.d. real valued random variables, each absolutely continuous with an ...
4
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0answers
78 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. ...
4
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0answers
461 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,...
4
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0answers
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: \...
4
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0answers
155 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 ...
4
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0answers
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 ...
3
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0answers
50 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
votes
1answer
61 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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0answers
479 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
votes
1answer
58 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, ...
3
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0answers
145 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) &...
3
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0answers
109 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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0answers
416 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: ...
3
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0answers
197 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\...
3
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219 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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0answers
415 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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0answers
297 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{-\...
3
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0answers
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 $$...
3
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0answers
926 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 ...
3
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0answers
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 ...
3
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0answers
965 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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1answer
65 views

How Gamma generalized linear model with zero dependent variable value is derived?

I understand that Gamma distribution generates only positive values. And this is reflected in R gamma family glm function which ...
2
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0answers
43 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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0answers
35 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 ...
2
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0answers
85 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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0answers
111 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
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 ...
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,...
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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0answers
71 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
votes
0answers
193 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
79 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
194 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.
2
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0answers
60 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....
2
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0answers
125 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
69 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 ...
2
votes
0answers
149 views

Detect the correct distribution from a small sample size by using fitdistrplus in R

The simplest version of the issue that I am looking for help is: How to detect the correct distribution from a small sample size in R by using fitdistrplus A simpler version: I am generating some ...
2
votes
0answers
56 views

Expectation of ratio between product of gaussian r.v.'s and generalized gamma r.v

Given \begin{equation}\label{eq:definition_of_z} \begin{split} \textbf{Z} = \left[\begin{array}{cccc} {z}_{11} & {z}_{12} & \cdots & {z}_{1P} \\ {z}_{21} & {z}_{22} & \cdots & {...
2
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0answers
256 views

How to check the correctness of calculations with a gamma distribution?

I’m reading Ponomareva, Roman, and Date (2015) and trying to generate vector $P$ of the $2Ns + 3$ probability weights: $$P =\{\underbrace{p_1, p_2, \ldots, p_s, p_1, p_2, \ldots, p_s,p_1, p_2, \ldots, ...
2
votes
0answers
308 views

Log link vs logging response variable in GAM model

hope someone might be able to help: My aim is to build a value predictor of an individual’s pension pot using customer data (based in the UK). It is to be used for reporting purposes and ...
2
votes
0answers
93 views

How to compare shape of gamma distributions to detect population change

Sorry if this question is poorly composed due to lack of stats knowledge. Any advice to point me in the right direction would be greatly appreciated: I'm hoping to detect if a dataset originated from ...
2
votes
0answers
87 views

Generate Gamma distributed values with upper bound

I need to generate N random numbers from a Gamma distribution, but with an upper bound Pmax using Matlab. Right now, I see two ...
2
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0answers
141 views

Calculating the likehood from the coeficients of logistical regression

I am doing a logistical regression and need to calculate the likehood from the null model and from each feature model and to after that get the p-value.The problem states: a) Create a model that ...
2
votes
0answers
70 views

Transformation of sum of Gammas into Chi-squared with a Casella Bergian twist

My question is very similar to the ones asked before (I have looked at all of them on Cross-Validated) but it is more about house-keeping and making sure it matches precisely transformation theorems ...