Questions tagged [gamma-distribution]
A non-negative continuous probability distribution indexed by two strictly positive parameters.
0
votes
0answers
10 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
106 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
18 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
30 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
34 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
20 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
50 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
$$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
0
votes
1answer
23 views
Understanding this expression of the multivariate t-distribution
I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows
$$
G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\
Z = N (...
1
vote
0answers
21 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
22 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
37 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
42 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, ...
0
votes
0answers
107 views
Characteristic function for a Chi-squared distribution
I would like to directly derive the probability density function (PDF) for a Chi-squared distribution with $k$ degrees of freedom using characteristic functions.
If $X_{1}, X_{2}, \dots, X_{k}$ are ...
-1
votes
0answers
32 views
Pearson differential equation distribution: gamma distribution derivation
How do I do 30a? It becomes extremely complicated if I solve the diff eq. with all the constants involved...
1
vote
1answer
104 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
28 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
37 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
30 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
107 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
26 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
59 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 ...
3
votes
0answers
140 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
33 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 $\theta \...
0
votes
0answers
55 views
What is Gamma model?
Recently I saw a person's resume online and it is said he used ARMA and Gamma model to analyze time-series pattern of bond market volatility.I know what ARMA model and Gamma distribution is but not ...
0
votes
1answer
61 views
Survival time problem exponential with gamma prior
The survival times, in days, of patients diagnosed with a severe form of a terminal illness
are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival
times ...
0
votes
0answers
20 views
Could continuous concentration data (ppb) ever be modeled with negative binomial?
We have data on compound concentrations (unit: ppb, continuous variable). There are many 0s. In fact, ~70% of the observations are equal to 0 (either true 0s or below the limit of quantification).
...
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 ...
2
votes
0answers
30 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 ...
10
votes
2answers
313 views
What is the expected value of the logarithm of Gamma distribution?
If the expected value of $\mathsf{Gamma}(\alpha, \beta)$ is $\frac{\alpha}{\beta}$, what is the expected value of $\log(\mathsf{Gamma}(\alpha, \beta))$? Can it be calculated analytically?
The ...
0
votes
0answers
70 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
1answer
93 views
Is the canonical parameter (and therefore the canonical link function) for a Gamma not unique?
Consider $Y_1, \dots, Y_n$ independent from the Gamma distribution. For $y > 0$:
$$\begin{align}
f(y \mid \alpha, \beta) &= \dfrac{1}{\beta^{\alpha}\Gamma(\alpha)}y^{\alpha-1}e^{-y/\beta} \\
&...
2
votes
0answers
19 views
finding process corresponding to laplace transform
I have a positive stochastic process $X(t)$ with Laplace transforms
$$
\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 ...
1
vote
1answer
203 views
Gamma Conjugate Prior & Poisson Process
I am analyzing daily data transaction data.
I am assuming that
The number of transactions in every day of length t has the Poisson distribution with mean λt
The number of transactions in evert ...
1
vote
0answers
18 views
glmer: distribution law for number of events / time
Sorry if this is a basic question but I can't find an answer that is clear enough, so I prefer to ask.
I want to model a number of events (number of gaze) that depends on the behaviour that one ...
3
votes
2answers
82 views
sum of $N$ gamma distributions with $N$ being a poisson distribution
I have an event having poisson distribution with time intervals of one minute. Every event has accomplishment time with gamma distribution.
I $N$ number of events start in $t$ minutes, the what will ...
1
vote
1answer
41 views
Exact Confidence Interval for Poisson using Gamma-Poisson Relationship
I'm reading Casella-Berger's Statistical Inference and trying to follow along in example 9.2.15, which constructs an exact confidence interval for a Poisson rate. In this example, the authors solve ...
0
votes
1answer
123 views
clarifying exponential-gamma conjugate prior
I'm referring to page 22 of this white paper.
On page 22, it says the following: given that
$s_i \sim \text{Exp}(\theta), i = 1,..,c$
$\theta \sim\text{Gamma}(k, \Theta)$,
Then the posterior ...
0
votes
0answers
25 views
What is the motivation for the formula for the Gamma distribution? [duplicate]
In my statistics class, it was proven that the sum of independent and identically distributed random variables distributed according to an exponential distribution follows a gamma distribution using ...
3
votes
1answer
65 views
Probability distribution for independent time to event
I am looking at waiting times between two events from multiple patients, so I'm looking at a gamma distribution. Turns out, the model is plotting out an exponential distribution, which if I was to ...
0
votes
0answers
12 views
Scale mixture of normals; t-distribution; proving marginal distribution equality [duplicate]
This is a review-type homework question that I'm very much stuck on.
Basically, these are the assumptions:
$\theta|\lambda = N(\mu,\frac{\sigma^2}{\lambda})$
$\lambda = \rm{Gamma}(\nu/2,\nu/2)$
$\...
0
votes
0answers
34 views
How to implement bayesian ab testing to gamma distributed data using pymc3
I found a simple implementation for Bernoulli distributed data of the bayesian ab tesing using pymc3 in the Probabilistic Programming and Bayesian Methods for Hackers book: http://nbviewer.jupyter.org/...
2
votes
2answers
277 views
Why is regression with Gradient Tree Boosting sometimes impacted by normalization (or scaling)?
I read that normalization is not required when using gradient tree boosting (see e.g. https://stackoverflow.com/q/43359169/1551810 and https://github.com/dmlc/xgboost/issues/357).
And I think I ...
2
votes
1answer
91 views
Mean of truncated gamma distribution using threshold
Given a gamma distribution with PDF:
$f(x;\alpha;\beta) = \frac{x^{\alpha-1} e^{-\frac{x}{\beta}}}{\Gamma(\alpha) \beta^\alpha}$
with a shape parameter $\alpha > 0$, a scale parameter $\beta > ...
1
vote
1answer
500 views
Interpretation of coefficients from glm Gamma
I am attempting to fit a model to a dataset with frequency (Hz) is the dependent variable. Using a generalized linear model based on a gamma distribution seems appropriate since the values of the ...
2
votes
0answers
39 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,...
1
vote
1answer
46 views
Strange behavior of Gamma prior in a setting with binomial likelihood
I am trying to use Bayes theorem to estimate the probability of a binary event. To give a (simplified) example:
Let's say our a priori guess of the probability of the event per trial is 0.3 (and the ...
1
vote
0answers
181 views
How shape parameters are connected with mean, variance, skewness and kurtosis of generalized gamma distribution
I am writing a code in python that can generate probability distribution with given mean (m), variance (v), skewness (s) and kurtosis (k). In scipy library of python, there is a function named ...
5
votes
2answers
67 views
Did I mess up the Poisson-Gamma relationship?
Let $X_1, X_2$ be independent, exponentially distributed random variables with mean 2. So $X_1+X_2=Z$ is gamma distributed with $\alpha=2$ and $\beta=1/2$.
I am trying to solve the probability that $...
