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In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. [1] Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
Although the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include: Bayesian methods, e.g. Bayesian linear regression; Percentage regression, for situations where reducing percentage errors is deemed more appropriate. [25]
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
Linear panel data models use the linear additivity of the fixed effects to difference them out and circumvent the incidental parameter problem. Even though Poisson models are inherently nonlinear, the use of the linear index and the exponential link function lead to multiplicative separability, more specifically [2] E[y it ∨ x i1...
in which the f i (X) are quantities that are functions of the variable X, in general a vector of values, while c and the w i stand for the model parameters. The term may specifically be used for: A log-linear plot or graph, which is a type of semi-log plot. Poisson regression for contingency tables, a type of generalized linear model.
Hilbe [3] notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "… the random variable y {\displaystyle y} is the count response and parameter λ {\displaystyle \lambda } (lambda) is the mean.
In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion.