Exponential dispersion model

With dispersion

ϕ

and known weight

w

, an exponential dispersion density has exponent

\frac{w}{ϕ} {y θ - b (θ)} + c (y, ϕ / w) .

Its mean is

b^{'} (θ)

and its variance is

(ϕ / w) b^{''} (θ)

.

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Exponential dispersion family of order one ( $p (y; θ, ϕ) = exp {(y θ - b (θ)) / ϕ + c (y, ϕ)}$ )

An order-one exponential dispersion family has a scalar natural parameter and canonical sufficient statistic

y

; its density or mass function has the displayed form on parameter-independent support. Differentiating its normalization gives

E Y = b^{'} (θ)

and

Var (Y) = ϕ b^{''} (θ)

. Known observation weights

w

replace

ϕ

by

ϕ / w

. Its variance function is

V (μ) = b^{''} ((b^{'})^{- 1} (μ))

, and its canonical link function is

(b^{'})^{- 1}

.

Dispersion parameter ( $ϕ$ )

The dispersion parameter scales the variance of an exponential dispersion family while leaving its mean function unchanged.

Underdispersion

Underdispersion means less variation than prescribed by a reference statistical model, corresponding to a working dispersion parameter below one. It is not, by itself, evidence that the reference mean function is improved.

Overdispersion

Overdispersion occurs when observed conditional variation exceeds that prescribed by a statistical model. For count data it commonly means variance greater than the mean assumed by a Poisson model.

Variance function ( $V (μ)$ )

The variance function writes an exponential dispersion family's variance as

Var (Y) = ϕ V (μ) / w

.

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Exponential dispersion model by

Wikipedia Bot  1

The Exponential Dispersion Model (EDM) is a class of statistical models used to represent a wide range of probability distributions. These models are particularly useful in the context of generalized linear models (GLMs). The EDM framework generalizes the idea of exponential families of distributions and is characterized by a specific functional form for the distribution of the response variable.

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