Questions tagged [gamma-distribution]

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

Filter by
Sorted by
Tagged with
1
vote
0answers
16 views

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 ...
12
votes
1answer
220 views

Name for a distribution between exponential and gamma?

The density $$f(s)\propto \frac{s}{s+\alpha}e^{-s},\quad s > 0$$ where $\alpha \ge 0$ is a parameter, lives between the exponential ($\alpha=0$) and $\Gamma(2,1)$ ($\alpha \to \infty$) ...
2
votes
1answer
666 views

Interpretation Beta coefficient regression gamma distribution

I am currently working on a panel data model of 30 companies over 10 years where the dependent variable is a score (decimal bounded between 0 and 1, continuous) while the independent are dummies and ...
0
votes
0answers
15 views

Drawing from inverse gamma and inverse chi-square [closed]

I am drawing samples from $\frac{a}{\chi(b)}$. This should be the same as drawing a sample from $\frac{1}{\Gamma(\alpha, \beta)}$. I want to figure out what are $\alpha$ and $\beta$ in terms of $a$ ...
1
vote
3answers
1k views

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 ...
4
votes
1answer
627 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 ...
5
votes
1answer
915 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
votes
0answers
54 views

Poisson Counting Process over Variable Threshold

Let's consider a Poisson Counting Process $N(t)$ with generation rate $\lambda$ and a piece-wise constant threshold $M(t)$ that changes with time. $$ M(t)=m_i \ \ for \ \ t\in(t_{i-1},t_{i}],\ i\in \...
1
vote
1answer
22 views

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 ...
0
votes
1answer
248 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 ...
2
votes
2answers
44 views

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 ...
2
votes
0answers
35 views

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 ...
0
votes
0answers
24 views

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{...
7
votes
1answer
109 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. ...
1
vote
2answers
519 views

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 ...
1
vote
3answers
53 views

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 ...
0
votes
0answers
5 views

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 \...
0
votes
0answers
23 views

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}...
0
votes
0answers
30 views

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 { ...
1
vote
1answer
213 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 ...
0
votes
0answers
9 views

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).
2
votes
1answer
55 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 ...
5
votes
1answer
475 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?
0
votes
1answer
55 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, ...
1
vote
1answer
40 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 ...
3
votes
0answers
32 views

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 ...
0
votes
1answer
27 views

Gamma distribution what is scale and rate

I have question regarding the gamma distribution when using ...
2
votes
1answer
972 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 (...
4
votes
0answers
100 views

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)....
2
votes
0answers
23 views

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 ...
0
votes
2answers
69 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 ...
0
votes
1answer
76 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 ...
0
votes
0answers
28 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 ...
0
votes
2answers
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 ...
0
votes
1answer
58 views

Gamma-Poisson conjugate prior, posterior exploding?

I've been looking for simple code that can model ad clicks per day. Notionally, gamma-poisson would be a good conjugate prior. However, I'm finding that for slightly large daily click rate values, the ...
7
votes
2answers
162 views

Approximating the median of a $\Gamma(\alpha,1)$ distribution with $0<\alpha<1$

Is there a good approximation (or useful bounds) for the median $\nu_\alpha$ of a $\Gamma(\alpha,1)$ distribution with $0<\alpha<1$? I have only been able to find things like Berg & ...
1
vote
1answer
263 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 (...
1
vote
1answer
76 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 ...
1
vote
0answers
22 views

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 ...
3
votes
1answer
67 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 ...
0
votes
1answer
94 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 ...
22
votes
2answers
38k views

How to interpret parameters in GLM with family=Gamma

I have a question regarding parameter interpretation for a GLM with a gamma distributed dependent variable. This is what R returns for my GLM with a log-link: ...
-1
votes
1answer
44 views

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 ...
1
vote
0answers
12 views

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 ...
0
votes
0answers
16 views

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 ...
2
votes
1answer
55 views

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 ...
4
votes
1answer
122 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 ...
2
votes
0answers
68 views

AIC doesn't agree with model checking [duplicate]

I have two glm, one with a gaussian distribution and identity link and one with gamma family and log link. The predictors are the same, the only thing that change is the response that in the gaussian ...
3
votes
1answer
45 views

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 ...
5
votes
1answer
64 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)}...

1
2 3 4 5
14