Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gamma-distribution]

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

0
votes
0answers
11 views

R: using gamma distribution of the gbm package [on hold]

I wanted to use the gamma distribution of the gbm package which has not originally been developed. So, based on this topic: R: Which distribution to use with gbm for gamma distributed data? I've been ...
0
votes
0answers
40 views

PDF of human population weight distribution

I have a lot of data on human weight coupled with information about gender, age, country and platform (desktop or mobile user). Looking something like this: ...
0
votes
0answers
30 views

Hypothesis testing for the Gamma distribution - Rejection region

Suppose under $H_0$, a test statistic $T$ has a gamma distribution with paramaters $\theta$ and $k$. Suppose also that the distribution of $T$ under $H_1$ is unknown. What is the appropriate rejection ...
4
votes
2answers
133 views

Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
5
votes
2answers
120 views

Which gamma regression model to use for extrapolation?

I'm looking for a regression model which would satify these requirements: My target variable follows the exponential distribution, so to my understanding I should use gamma loss function. I have ...
5
votes
1answer
108 views

Truncated Gamma Distribution

The Gamma distribution is the conjugate prior of Poisson distribution. What about the Truncated Gamma distribution? Is it still the conjugate prior of Poisson distribution?
5
votes
2answers
156 views

Numerically/approximately integrating over independent gamma variables

Problem Statement For a problem in biology, I am testing out a joint distribution of the form: $$ X \sim Multinomial(\frac{\theta_1}{\sum \theta_i}, ...,\frac{\theta_n}{\sum{\theta_i}}) \\ \theta_i \...
0
votes
0answers
25 views

In an MCMC model, what happens if the observed data contains negative and positive observations, but a strictly positive model (Gamma) is used?

Suppose that we observe data that range from negative to positive support. Now, suppose we use a likelihood specification that is a Gamma, or some other distribution with strict positive support. I ...
0
votes
0answers
18 views

MLE For Gamma Distribution [duplicate]

The $(\alpha, \lambda)$ Gamma distribution has density function $f(x| \alpha, \lambda) = \frac{\lambda^\alpha x^{\alpha-1} e^{-\lambda x}}{\Gamma(\alpha)}$ for $x>0$ How can we obtain an MLE ...
0
votes
0answers
15 views

FITDIST: Why am I getting this shape for the gamma distribution?

I am using fitdist to fit a gamma distribution based on this link: How to draw fitted graph and actual graph of gamma distribution in one plot? My code is as follows: ...
0
votes
0answers
21 views

Poisson Gamma Distribution in R - Creating Enrollment Modeling Curve

I'm trying to create an enrollment curve for a clinical trial based on the following variables: Country start up timelines (staggered), Number of sites, Number of total subjects needed, Number of ...
2
votes
0answers
33 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 ...
1
vote
1answer
84 views

Gamma glm log link - what does predicted values mean

Does the predict function in R for gamma glm with log link predict the actual values or the mean value? There is a gamma glm model in R with log link. Using predict(model,data,type = 'response') to ...
0
votes
0answers
32 views

Shift data to use Gamma glm with “identity” link?

I am trying to fit a glm using the Gamma family and the "identity" link. As I am analyzing nutrient concentrations in coastal waters (that can not be negative and are unlikely to be zero) I assume the ...
1
vote
1answer
63 views

Gamma Distribution Sufficient Statistics

I've been asked to show the gamma distribution can be written in the form $p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$ where $T(x)$ is a sufficient statistic. .... I have ...
0
votes
1answer
55 views

Inverse of Gamma Distribution [closed]

I am using python to calculate Inverse of a CDF of gamma distribution (using scipy. stat.gamma.fit). But for probability value 1, it is coming infinite. If it is replaced from 1 to 0.99 it works but ...
3
votes
0answers
39 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 ...
0
votes
0answers
8 views

How much data is considered “sparse” for fitting a mixed (Beta Geometric) distribution with 4 shape parameters?

I'm using CamDavidsonPhillips Customer Lifetime Value library to calculate CLV, and it uses a distribution based on Peter Fader's work on the subject that fits a Gamma distribution to model customer ...
0
votes
1answer
16 views

Point mass at zero and a chi square distribution with one degree of freedom

I am unclear about the critical value of a point mass at zero and a chi square distribution with one degree of freedom. How to find this?
2
votes
1answer
27 views

Apply 3 sigma formula in gamma distribution?

Let say i have some data that follows gamma distribution, and i calculated the Mean and Standard deviation of the gamma distribution. I also know that there are some outliers(Noise) in the data i ...
2
votes
2answers
71 views

What are the assumptions of a Gamma GLM or GLMM for hypothesis testing?

What are the assumptions when doing hypothesis testing using a Gamma GLM or GLMM? Are the residuals suppose to be normally distributed and is heteroscedasticity a concern like the Gaussian (normal) ...
5
votes
1answer
120 views

Expected value of last Gamma RV in a sum

I've got a sum of $X_i \sim \text{Gamma}(k, \theta)$ i.i.d. random variables. I'm trying to find the expected value of the final $X_i$ that takes the sum above a certain value, i.e., to find the value ...
3
votes
1answer
50 views

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. ...
2
votes
0answers
72 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 ...
0
votes
0answers
25 views

Gamma family GLM fails after removal of highly influential point

I have some data where distribution seems light a rough Gamma distribution. I am therefore investigating the relationship in r using a Gamma family Generalized Linear Model. When I investigate the ...
0
votes
1answer
66 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 ...
5
votes
2answers
112 views

Independence of ratios of independent variates

If $X= x_1/(x_1+x_2)$ and $Y= (x_1+x_2)/(x_1+x_2+x_3)$ where $x_1,x_2,x_3$ independent chi-square variates with d.f $n_1,n_2,n_3$ respectively, are $X$ & $Y$ independent? I know the condition ...
0
votes
0answers
22 views

Is my data suited for an ANOVA?

I have two types of "media" (surface and sediment). I have six toxin variants "variant". All the toxins originate in the surface water, but I want to know if certain toxins are more likely to ...
0
votes
0answers
199 views

Gamma distribution as a member of exponential family

In my lecture notes I have that the distribution of a random variable $Y$ is said to be in the exponential family if it can be written as $f(y;\theta)=exp(a(y)b(\theta)+c(\theta)+d(y))$, where $a,b,c$ ...
0
votes
0answers
46 views

Gamma distribution

We consider the gamma pdf of a random variable $Y$ that is given by: $$f(y) = (s^a \Gamma (a))^{-1} y^{a-1}e^{-\frac{y}{s}},$$ where $y \geq 0$, $s$ is the scale parameter and $a$ the shape ...
1
vote
1answer
42 views

Is the convolution of independent normal and gamma also a Pearson distribution?

The answer here What is the convolution of a normal distribution with a gamma distribution? gives the pdf of convolution of normal and gamma random variables. Is there a known random variable to which ...
0
votes
1answer
97 views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
0
votes
1answer
103 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 (...
1
vote
0answers
28 views

Generate an autocorrelated Gamma sample of length N

How does one simulate an autocorrelated Gamma sample of length $N$? All I found online was about generation correlated variables and not an autocorrelated sample.
1
vote
0answers
38 views

In a gamma regression, how can i interpret coefficients?

My question is pretty simple, i have done a bayesian gamma regression with an inverse link, so: $\eta_i$=$\beta_0+\beta_1x_{i1}+\dots+\beta_px_{ip}$ < using an inverse link, mu is the ...
0
votes
0answers
67 views

hypothesis testing - gamma distribution

Let W = Y/B0 be a Random variable that has a gamma(2n,1) distribution. [Y has a gamma(2n,B) distribution and W = Y/B]. i) Suppose you want to test H0 : B ≤ B0 against H1 : B > B0 for some B0 > 0. How ...
1
vote
1answer
72 views

Robust analogues of Mean, CV and Skewness

I need to characterize the mean, CV and skewness of my observed data (it is gamma-like distributed). This data is an artifact (outliers) enriched, so I decide to use robust statistics: median, ...
1
vote
1answer
327 views

Show posterior mean can be written as a weighted average of the prior mean and MLE

Suppose $Y_1, \dots Y_n$ are exponentially distributed: $Y_i | \lambda \sim Exp(\lambda)$. Find the conjugate prior for $\lambda$, and the corresponding posterior distribution. Show that the posterior ...
1
vote
1answer
64 views

Why does not the weighted sum of gamma distribution come from weighted gamma variables?

If $Z\sim 0.3\Gamma(\alpha _1,\beta _1)+0.7\Gamma (\alpha _2,\beta_2)$, why isn't $Z=0.3X_1+0.7X_2$? $X_1\sim\Gamma(\alpha _1,\beta _1)$ and $X_2\sim\Gamma(\alpha _2,\beta _2)$?
13
votes
2answers
3k views

Do test scores really follow a normal distribution?

I've been trying to learn which distributions to use in GLMs, and I'm a little fuzzled on when to use the normal distribution. In one part of my textbook, it says that a normal distribution could be ...
1
vote
1answer
38 views

How can a sum of squared random variables distributed normally with mean $\mu$ and variance $\sigma^2$ be represented as a gamma distribution?

Firstly, I'm sorry about the formatting, I hope someone can come along and give me a hand with this. I have a sum of squares of $n$ random variables distributed normally with mean $\mu$ and variance $...
2
votes
1answer
33 views

Lower incomplete gamma function format in series representation and R [closed]

As known that the lower incomplete gamma function can be written as $\gamma(a,x) = x^{a}e^{-x}\sum_k^\infty{{x^{k}}\over a^{k+1}}.$ What is the format for $\sum_j^\infty{\gamma(v/p-j,rx^{p})} $ in ...
4
votes
1answer
144 views

Generalized incomplete gamma function

As I know there is a built in function for incomplete gamma function, incgam(x, a), in R. May I know is there a built in function for the generalized incomplete gamma function? Or how can I modify the ...
3
votes
1answer
32 views

Unexpected estimate in Gamma GLM summary output

I have a question. How on earth is it possible to have a negative estimate for a form of a nominal variable (two forms: "HM" and "LM") when it should be positive? I'm modelling a positive continuous ...
2
votes
1answer
62 views

Deriving marginal pdf from joint pdf

Problem setup: $X \sim \Gamma(\alpha,\beta)$, $f_{Y|X}(y|x)= \tau xy^{\tau-1}e^{-xy^\tau}$ for $y>0$ and $f_{Y|X}(y|x)=0$ for $y\leq0$, where $\tau\geq1$ is a constant. I am asked to derive the ...
4
votes
0answers
220 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 ...
1
vote
0answers
22 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 ...
1
vote
1answer
45 views

Gamma-gamma conjugacy for rate parameter of Gamma distribution

the question is as follows. Assume the shape $r$ is a known constant. For $x \sim$ Gamma(shape = $r$, rate = $v$), the p.d.f is: $$p(x|r,v) = x^{r-1}e^{-vx}v^r/\Gamma(r)$$ a) Show that the $\theta \...
0
votes
1answer
98 views

Survival time problem exponential with gamma prior

The survival times, in days, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival times ...
2
votes
1answer
38 views

How to evaluate $\int_0^\infty m^{x+1}e^{-2m}dm$ as $\Gamma(x+2)\frac{1}{2}^{x+2}$?

$\int_0^\infty \frac{m^{x+1}e^{-2m}}{\Gamma(x+1)\Gamma(2)}dm =\frac{\Gamma(x+2)\frac{1}{2}^{x+2}}{\Gamma(x+1)\Gamma(2)}$ How does the left side equal the right side? I understand that the gamma ...