DEVIANCE Function

Returns the deviance based on a probability distribution.

Category:

Mathematical

Syntax

DEVIANCE(distribution, variable, shape-parameters<,ε> )

Required Arguments

distribution

is a character constant, variable, or expression that identifies the distribution. Valid distributions are listed in the following table:

Distribution	Argument
Bernoulli	'BERNOULLI' \| 'BERN'
Binomial	'BINOMIAL' \| 'BINO'
Gamma	'GAMMA'
Inverse Gauss (Wald)	'IGAUSS' \| 'WALD'
Normal	'NORMAL' \| 'GAUSSIAN'
Poisson	'POISSON' \| 'POIS'

variable

is a numeric constant, variable, or expression.

shape-parameter

are one or more distribution-specific numeric parameters that characterize the shape of the distribution.

Optional Argument

ε

is an optional numeric small value used for all of the distributions, except for the normal distribution.

Details

The Bernoulli Distribution

DEVIANCE('BERNOULLI', variable, p<, ε> )

Arguments

variable

is a binary numeric random variable that has the value of 1 for success and 0 for failure.

p

is a numeric probability of success with ε ≤ p ≤ 1–ε.

ε

is an optional positive numeric value that is used to bound p. Any value of p in the interval 0 ≤ p ≤ ε is replaced by ε. Any value of p in the interval 1 – ε ≤ p ≤ 1 is replaced by 1 – ε.

The DEVIANCE function returns the deviance from a Bernoulli distribution with a probability of success p, where success is defined as a random variable value of 1. The equation follows:

\begin{matrix} D E V I A N C E (^{'} B E R N^{'}, v a r i a b l e, p, ε) = {\begin{matrix} - 2 \log (1 - p) & x = 0 \\ - 2 \log (p) & x = 1 \\ . & o t h e r w i s e \end{matrix} \end{matrix}

The Binomial Distribution

DEVIANCE('BINO', variable, μ, n<, ε> )

Arguments

variable

is a numeric random variable that contains the number of successes.

Range

0 ≤ variable ≤ 1

μ

is a numeric mean parameter.

Range

nε ≤ μ ≤ n(1–ε)

n

is an integer number of Bernoulli trials parameter

Range

n ≥ 0

ε

is an optional positive numeric value that is used to bound μ. Any value of μ in the interval 0 ≤ μ ≤ nε is replaced by nε. Any value of μ in the interval n(1 – ε) ≤ μ ≤ n is replaced by n(1 – ε).

The DEVIANCE function returns the deviance from a binomial distribution, with a probability of success p, and a number of independent Bernoulli trials n. The following equation describes the DEVIANCE function for the Binomial distribution, where x is the random variable.

\begin{matrix} D E V I A N C E (' B I N O', x, μ, n) = {\begin{matrix} . & x < 0 \\ 2 (x \log (\frac{x}{μ}) + (n - x) \log (\frac{n - x}{n - μ})) & 0 \leq x \leq n \\ . & x > n \end{matrix} \end{matrix}

The Gamma Distribution

DEVIANCE('GAMMA', variable, μ <, ε> )

Arguments

variable

is a numeric random variable.

Range

variable ≥ ε

μ

is a numeric mean parameter.

Range

μ ≥ε

ε

is an optional positive numeric value that is used to bound variable and μ. Any value of variable in the interval 0 ≤ variable ≤ ε is replaced by ε. Any value of μ in the interval 0 ≤ μ ≤ ε is replaced by ε.

The DEVIANCE function returns the deviance from a gamma distribution with a mean parameter μ. The following equation describes the DEVIANCE function for the gamma distribution, where x is the random variable:

\begin{matrix} D E V I A N C E (^{'} G A M M A^{'}, x, μ) = {\begin{matrix} . & x < 0 \\ 2 (- \log (\frac{x}{μ}) + \frac{x - μ}{μ}) & x \geq ε, μ \geq ε \end{matrix} \end{matrix}

The Inverse Gauss (Wald) Distribution

DEVIANCE('IGAUSS' | 'WALD', variable, μ<, ε> )

Arguments

variable

is a numeric random variable.

Range

variable ≥ ε

μ

is a numeric mean parameter.

Range

μ ≥ε

ε

The DEVIANCE function returns the deviance from an inverse Gaussian distribution with a mean parameter μ. The following equation describes the DEVIANCE function for the inverse Gaussian distribution, where x is the random variable:

\begin{matrix} D E V I A N C E (^{'} I G A U S S^{'}, x, μ) = {\begin{matrix} . & x < 0 \\ \frac{{(x - μ)}^{2}}{μ^{2} x} & x \geq ε, μ \geq ε \end{matrix} \end{matrix}

The Normal Distribution

DEVIANCE('NORMAL' | 'GAUSSIAN', variable, μ)

Arguments

variable

is a numeric random variable.

μ

is a numeric mean parameter.

The DEVIANCE function returns the deviance from a normal distribution with a mean parameter μ. The following equation describes the DEVIANCE function for the normal distribution, where x is the random variable:

\begin{matrix} D E V I A N C E (^{'} N O R M A L^{'}, x, μ) = {(x - μ)}^{2} \end{matrix}

The Poisson Distribution

DEVIANCE('POISSON', variable, μ<, ε> )

Arguments

variable

is a numeric random variable.

Range

variable ≥ 0

μ

is a numeric mean parameter.

Range

μ ≥ε

ε

is an optional positive numeric value that is used to bound μ. Any value of μ in the interval 0 ≤ μ ≤ ε is replaced by ε.

The DEVIANCE function returns the deviance from a Poisson distribution with a mean parameter μ. The following equation describes the DEVIANCE function for the Poisson distribution, where x is the random variable:

\begin{matrix} D E V I A N C E (^{'} P O I S S O N^{'}, x, μ) = {\begin{matrix} . & x < 0 \\ 2 (x \log (\frac{x}{μ}) - (x - μ)) & x \geq 0, μ \geq ε \end{matrix} \end{matrix}