Questions tagged [poisson-distribution]
A discrete distribution defined on the non-negative integers that has the property that the mean is equal to the variance.
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Very different scale parameter estimates in Poisson regression
The background: I'm analysing survival data using a Poisson model. I've splitted the data on 2 time-scales (attained age and calendar year). Attained age is modelled using flexible parametric ...
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Poisson regression for binary data
I've been trying to read up on Poisson regression models, and it looks like it is possible to estimate such a model with a binary outcome. This has come up before on this site here (and somewhat here ...
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Bayesian inference on mean of statistic from population
Suppose that a collection of time intervals $t_i$ have occurred, for $i=1,...,n$. These should be considered as samples from a population governed by some distribution. During these time intervals, ...
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Conditional probability update for correlated Poisson variables
Some background:
I am trying to estimate the number of failures in two related machine populations. I model machine failures in a year as a correlated Poisson process as such:
$Y_0,\ Y_1$ and $Y_2$ ...
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Credit Risk and Concentration
I am working with a UK credit-union and we are looking to build a model to assess our credit risk and changes to this over time. We have a number of loans to borrowers who each have a credit rating (...
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Rule of thumb - number of predictors - Poisson regression rates
I am interested in estimating a Poisson regression for mortality rates, with number of deaths as the dependent variable and log(population size) as the offset. I have 50 observations (states).
I am ...
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How to denoise a "Poissonous" time series
I have $N$ time series each of which can be modeled as $$y_{kt}=Ax_{kt}+b+\varepsilon_{kt}\quad(1\le k\le N,1\le t\le T),$$ where $x_{kt}\sim\text{Pois}(\lambda\Delta t)$ and $\varepsilon_{kt}\sim N(0,...
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Dependent variable with Poisson distribution in lavaan in R: how to deal with it?
I am planning to create a SEM model with lavaan.
Most variables are normal, however, my dependent variable consists in a count of times and it has a Poisson distribution. Is there any Poisson ...
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Nonnegative Matrix Factorization as Maximum Likelihood
Elements of Statistical Learning has this on such NMF loss function (section 14.6 Non-negative Matrix Factorization):
The matrices $\mathbf{W}$ and $\mathbf{H}$ are found by maximizing $$ L(\mathbf{...
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Variance of quotient of Poisson random variable and sum of the Poisson sample
Let
$$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$.
What is the variance of $$\frac{...
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Why is RMSE inappropriate for Poisson distribution?
I want to build models to predict the number of car claims which follow a Poisson distribution (mean = 0.05).
One model is a Poisson regression and another is a RandomForest and I want to compare the ...
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Interpreting Random Effects for Poisson GLMM
There seem to be a few answers for normally distributed models, but after some searching I could only come across this page for Poisson mixed models. I want to be certain I am interpreting the random ...
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Processes behind statistical distribution laws: a compendium?
The simple processes that "explain" the binomial, Gaussian or Poisson distribution are relatively well-known. Johnson or shot noises may be known in restricted area of science. Sometimes, a ...
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Spatial Autoregressive Poisson model in R
I am estimating a gravity model of migration on cross-sectional data. The Moran I statistic indicates a positive and significant spatial autocorrelation in the residuals of the non-spatial model, and ...
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Multivariate generalization of Poisson-Gamma model?
I actually assumed it would be easy to find a multivariate version of the Poisson distribution, but couldn't find any concrete solution (in terms of a well cited publication). It seems that ...
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Distribution of ratio of Poisson distributed random variables
I am currently reading a paper and puzzling about a certain statement. I have a ratio
$\frac{\hat\alpha_{T+1}}{\hat\alpha_{T}}=\frac{H_{T-4}+...+H_{T}}{H_{T-3}+...+H_{T+1}}\left(\frac{D_{T-3}+...+...
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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,...
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Dependent thinning Poisson process
If $N_1$ and $N_2$ are independent Poisson processes then the superposition is a Poisson process. Is it possible to construct two dependent Poisson processes such that the superposition is a Poisson ...
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Are Poisson distributions with low mean heavy-tailed?
It is very apparent to me how using the normal distribution to estimate the probability of large, Poisson-distributed events may lead to significant underestimates of the probability of these events, ...
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What is the best way to deal with over-dispersion in a poisson GLMM?
I am currently in the process of trying to complete a poisson GLMM analysis with two fixed (with an interaction) and two random effects using the glmer() function of the lme4 package. Using the ...
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Is it possible to show that this estimator has minimum variance?
Doing some exercises I stumbled upon this tricky one:
Suppose we have an independent random sample $(X_1, ... , X_n)$ with $X_i \sim Poisson(\lambda)$. Define $\theta = e^{-\lambda}$.
Let $$ \...
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Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are "no events"
I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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two independent Poisson Arrivals
I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$
Consider two ...
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Bounds on probability that a random variable is neither the maximum nor the minimum of a set of random variables
Suppose I have $n$ independent random variables $X_1,\dotsc,X_n$ which are Poisson distributed with $X_i \sim Poi(\lambda_i)$. Without loss of generality, an additional condition $\lambda_1\le\...
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Linear regression when the conditional distribution is Poisson
Suppose $Y$ is discrete and only takes on non-negative integers and
that the conditional distribution of $Y$ given $X=x$ is Poisson, that
is, $$P(Y=y|X=x) = \frac{\exp(-x'\beta) (x'\beta)^y}{y!}$...
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Poisson Deviance - How to Estimate the Optimal Nodes for a Decision Tree
The following is the formulation for how the GBM package in R calculates the loss function and terminal node estimates for gradient boosting with decision trees. My question is generally how are the ...
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MLE estimation of Autoregressive Conditional Poisson model
The density of an Autoregressive Conditional Poisson ACP(p,q) model is defined as
$$ f(x | \lambda_{t}) = \frac{\lambda_{t}^{x}\exp[-\lambda_{t}]}{x!},$$
where
$$\lambda_{t} = \omega + \sum_{j = 1}...
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What do the outputs of my zero inflated poisson model mean? (counts of fish)
I am looking a counts of fish within different size classes (0-10, 10-20 etc) between 3 different reef sites, 3 different depths and 2 different survey methods. However, naturally on all observations ...
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What statistical test do I need?
Say I have $N$ light bulbs. Whevener one breaks down, I immediately fix it.
$k_0$ of these $N$ light bulbs do not break down during this year,
$k_1$ break down (and get fixed) once, and
$k_2$ ...
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Specifying the priors for multivariate MCMCglmm mixed model in R (Poisson distribution)
I am trying to build a model using MCMCglmm. Ideally, I would use a negative binomial distribution for my response; however, this is not an option in MCMCglmm. I don't know of any open-source ...
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Count data in a Structural Equation Model
I want to fit a structural equation model (SEM) on using the lavaan package in R.
Some of the most important variables in my ...
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Sufficient statistics of posterior (with Poisson data)
Suppose that, for year $t$, the data $y$ is Poisson with mean $a + bt$. Assume also a uniform prior on $(a,b)$. If we have $n$ years of data then I think the posterior for $(a,b)$ will be
\begin{...
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Confidence interval for the ratio of Poisson to normal variables
In an experiment, I'm counting failures on an semiconductor device for a given fluence (i.e. amount) of neutron radiation. The failures occur according to a Poisson law.
The fluence is provided by ...
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What is the distribution of independent binomial variables conditional upon the sum?
Suppose that we have independent binomial variates with differing sizes and probabilities $X_i \sim \operatorname{Binomial}(n_i,p_i)$, and $Z = \sum_iX_i$ is the sum. I understand that $Z$ is ...
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Finding the limiting distribution
I was given an exercise to do that sounded something like this:
The Arizona football team scored $45$ goals in $19$ games in the 2007/08 season. If $y_i$ denotes the number of goals scored in the $i$-...
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Probability of collisions in queues of Poisson process
I have a process whereby objects of width $W$ land on a gene at rate $F$ (per second, poisson process, lets assume), and then start to move along at constant speed $V$. I'm trying to work out the ...
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Queueing Theory: How to estimate steady-state queue length for single queue, N servers?
I have a real-life situation that can be solved using Queueing Theory.
Scenario:
There is a single Queue and N Servers.
When a server becomes free, the Task at the front of the queue gets serviced.
...
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What is the difference between normal approximation and poisson approximation of binomial distribution?
what is the difference between Poisson distribution as an approximation of Binomial distribution and Normal (Gaussian) distribution as an approximation of Binomial distribution? Both are ...
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Numerical moments of a multivariate Poisson Log-normal posterior
I have a log-density of the form:
$$P(\mathbf{x}) \propto \exp\left( - \mathbf{b}^{\top} e^{ \mathbf{x} } - \frac{1}{2}\mathbf{x}^{\top}A\mathbf{x} \right)$$
where $A$ is a symmetric positive definite ...
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Deriving a gamma distribution from a Poisson distribution
At the instant $t = 0$ a certain radioactive focus starts emitting particles. The infinitesimal probability that the focus emits a particle in the differential interval is $\lambda dt$. Let $N_t$ also ...
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$Y_1, Y_2, ... $ are iid Poisson($\lambda$)
We observe $Y_1, Y_2, ..., Y_T$ such that $T$ is the first $t\geq 1$ for which $Y_t>0$. Define $Y=Y_T$
(a) Find MLE $\hat{\lambda}$.
(b) What is the relative bias, $[E(\hat{\lambda})- \lambda]/\...
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The convolution of a Poisson Distribution and Scaled Poisson Distribution
I am trying to do a likelihood analysis on a variable, $Z$, which is defined as
$(1)$ $Z = X - cY$
where $X$ and $Y$ are both independent Poisson distributions with rate parameters $\lambda_{x}, \...
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Convergence rate about a limit concerning the Poisson CDF
The CDF of a Poisson distribution with rate parameter $\lambda$ is
$$
P(n;\lambda)=\sum_{k=0}^n \frac{\lambda^ke^{-\lambda}}{k!}.
$$
As $n$ goes to infinity, the CDF would certainly approach 1.
Now, ...
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Finding the UMVUE of $e^{3\lambda}$ in Poi($\lambda$)
Let $ X = (X_1, ... , X_n)$ iid variables coming from Poisson
distribution with mean $\lambda$.
Find the UMVUE of $e^{3\lambda}$.
I tried understanding the solution below (in the possible duplicate ...
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How to control the error rate when selecting the mode of a sample as the mode of the population?
I just looked out of my window and saw 4 white, 2 black, 1 green, 1 silver, 1 light blue, 1 lemon and 1 orange-white-striped car. Based on this, how confident should I be that white is the most common ...
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When the number occurences in a time interval are not Poisson distributed?
The lectures statistics I followed also presented the Poisson distribution. We were taught that the number of events occurring in a time interval, that this statistic follows a Poisson distribution.
$
...
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Mean of non-central Poisson deviance distribution is lower than that of the associated central distribution
We have observed some strange behaviour regarding distributions of Poisson deviance statistics.
In short, we find for certain Poisson parameters, that the non-central deviances are smaller than the ...
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Can we mix conclusions from Poisson and Quasi-Poisson?
Currently I'm working with ecological studies, where my response is a count variable. I need to estimate several models, each one represents a city. Afterwards I aggregate them to obtain meta-analysis ...
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Priors and nested random effects in MCMCglmm?
I am trying to construct a zero inflation Poisson GLMM using MCMCglmm(). I am new to Bayesian Statistics and this function and I am struggling to understand a couple of things.
For my data I am ...
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Sample size calculation for correlated count data
I am wondering how I can simulate Poisson data that is correlated. Let's say, I collect data at two time points. At both time points, the data is Poisson distributed, at time point 1 with $\lambda=4$, ...
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