Questions tagged [gamma-distribution]

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

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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 ...
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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 \...
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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}...
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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).
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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 { ...
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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, ...
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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 ...
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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 ...
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Gamma distribution what is scale and rate

I have question regarding the gamma distribution when using ...
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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 ...
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Z = X1/(X1+X2) where X1 and X2 are gamma distributed

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 ...
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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 ...
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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 ...
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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)....
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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 ...
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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 (...
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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 ...
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1answer
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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 ...
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1answer
59 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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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)}...
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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 ...
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1answer
130 views

GLMM hurdle model for continuous data -Truncated negative binomial family in glmmTMB?

I am running a hurdle model using the glmmTMB function. My dependent variable is continuous and >= 0. I was looking for a function that would allow me to model the binary response in a logistic ...
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Gamma Regression as the Last Layer of the Neural Network

My current task involves predicting data that follows a Gamma distribution. To avoid confusion of notations, in the following discussion, the p.d.f will be $$\mathbb{P}(y|\alpha, \beta)=\frac{\beta^\...
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Validation metric for gamma regression

I have a regression task to predict the loan default amount in the case of a default (which is always positive). I am using Gamma regression for this in LightGBM. ...
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Posterior mean of $\mu$ in Bayesian Hierarchical model (Poisson-Gamma)

Chapter 7 of Jim Albert's book considers the case of using a hierarchical model, to estimate heart-transplant mortality rates ($\lambda_i$) from 94 hospitals, each with it's own exposure (# of ...
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Parameterization of gamma distribution in XGBoost 'reg:gamma'

I am looking into gamma regression using XGBoost in R. I am trying to understand the negative log-likelihood used in estimating the model. Github has the following ...
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How to scale gamma distribution to longer interval?

I'm trying to gain intuition for how to scale the parameters of the Gamma distribution when the desired variance is not known. I'll make up an example: Let $\lambda$ be the number of 100-degree days ...
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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 \...
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2answers
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best regression model for a percentage of function variable

I am trying to choose the best regression model for a dependent variable 'percentage of normal shoulder function'. This variable is non normal and clustered at the high end of the possible range, ie ...
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Use gamma distribution or log-transform with OLS?

I have blood levels of a chemical as the response or dependent variable. The minimum can be 0 and it has distribution as shown in figure below: I believe this is a gamma distribution. I have to ...
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GLMs with skewed distributions - why use mean and not mode?

There's something that's a bit troubling for me. The unit deviance in GLM is defined as $2[t(y,y) - t(y,\mu)]$, when $t(y,\mu) = y\theta(\mu) - b(\theta(\mu))$ (theta being the natural parameter). For ...
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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 ...
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How to use scipy stats gamma pdf to update the posterior distribution?

I'm trying to "get my bearings" performing bayesian analysis, specifically I'm exploring the Gamma-Poisson conjugate prior. The definition of the PDF is below If the prior takes the form of ...
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1answer
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Robust SE clustered GLM Gamma Log Link to match GEE Robust SE

How do I get the robust standard errors/sandwich variance estimators for GLM using a Gamma family with a log-link to match the robust standard errors from the GEE output? ...
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F test and T test in GLM gamma Log link

We get both t-test and F-test for the estimation of GLM Gamma with a log link model. F-test: T-test: From my knowledge for ANOVA, I can report the main effect of League to use the F test(F4,53699=20,...
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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?
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Generalized Gamma Distribution in Lifetime Models

It seems to me that a version of generalized gamma distribution is often used in lifetime models. For example, survival analysis uses this distribution. Is there some intuition as to why this ...
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Box-Cox vs GLM for skewed non homoskedastic data

I need to do a regression on a variable Y that is skewed right (non normal) and heteroskedastic and therefore violates two assumptions of the normal linear model. The data is non negative (some 0 ...
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1answer
107 views

QQ Plot for GLM Regression Model in R with Family NOT Gaussian

I apologize if this has been asked and answered elsewhere -- I've tried to find the answer and could not. In R, if you use the normal plot() command on a ...
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1answer
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Identifying the best distribution to this data?

I'm trying to fit an appropriate distribution to a data with 216 values and estimate parameters. From Cullen and Frey graph, it looks like lognormal could be a good fit. From q-q plot, Weibull seems ...
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Conditional expectation of a Gamma Distribution

$X$ a Gamma Distributed random variable. Calculate $E[X\mid X \in [a,b]]$, where $a>0$, $b>0$. Is there a closed form solution for this, and if so, how can I calculate it?
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Citation for Gamma Probability Density Function?

Wondering if anyone has a reference for the Gamma PDF? I cannot find any leads. Thanks
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How to compute categorical probability given a continuous distribution?

I have a continuous target column that consists of IQ scores for kids of age 10. My main purpose is to forecast the probability that a kid is genius given 3 covariates. If a kid has IQ over 160 she/he ...
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fitting multiplicative non linear model in R

I have samples per month over several years (assuming stationary behavior). My data contains an independent variable: hours (representing available hours per month). The model to be fitted should ...

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