DEVIANCE Function

DEVIANCE Function

Computes the deviance

Category: Mathematical

Syntax

DEVIANCE ( distribution , variable , shape-parameter(s) <, µ >)

Arguments

distribution

• is a character string that identifies the distribution. Valid distributions are

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 random variable.

shape-parameter(s)

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

µ

• 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 <, µ >)

where

• 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 ‰ p ‰ 1- µ .

• µ

  • is an optional positive numeric value that is used to bound p . Any value of p in the interval 0 ‰ p ‰µ is replaced by µ . Any value of p in the interval 1 - µ ‰ p ‰ 1 is replaced by 1 - µ .

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 :

The Binomial Distribution

• DEVIANCE ('BINO', variable , ¼ , n <, µ >)

where

• variable

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

  • Range: ‰ variable ‰ 1

• ¼

  • is a numeric mean parameter.

  • Range: n ‰ ¼ ‰ n (1- µ )

• n

  • is an integer number of Bernoulli trials parameter

  • Range: n ‰

• µ

  • is an optional positive numeric value that is used to bound . Any value of in the interval 0 ‰ ¼ ‰ n µ is replaced by n µ . Any value of in the interval n (1 - µ ) ‰ ¼ ‰ n is replaced by n (1 - µ ).

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.

The Gamma Distribution

• DEVIANCE ('GAMMA', variable , ¼ <, µ >)

where

• variable

  • is a numeric random variable.

  • Range: variable ‰ µ

• ¼

• is a numeric mean parameter.

• Range: ¼ ‰ µ

• µ

  • is an optional positive numeric value that is used to bound variable and µ . Any value of variable in the interval 0 ‰ variable ‰ µ µ is replaced by µ . Any value of ¼ in the interval 0 ‰ ¼ ‰ µ is replaced by µ .

The DEVIANCE function returns the deviance from a gamma distribution with a mean parameter ¼ . The following equation describes the DEVIANCE function for the gamma distribution, where x is the random variable:

The Inverse Gauss (Wald) Distribution

• DEVIANCE ('IGAUSS' 'WALD', variable , ¼ <, µ >)

where

• variable

  • is a numeric random variable.

  • Range: variable ‰ µ

• ¼

  • is a numeric mean parameter.

  • Range: ¼ ‰ µ

• µ

  • is an optional positive numeric value that is used to bound variable and ¼ . Any value of variable in the interval 0 ‰ variable ‰ is replaced by µ . Any value of ¼ in the interval 0 ‰ ¼ ‰ µ is replaced by µ .

The DEVIANCE function returns the deviance from an inverse Gaussian distribution with a mean parameter ¼ . The following equation describes the DEVIANCE function for the inverse Gaussian distribution, where x is the random variable:

The Normal Distribution

• DEVIANCE ('NORMAL' 'GAUSSIAN', variable , ¼ )

where

• variable

  • is a numeric random variable.

• ¼

  • is a numeric mean parameter.

The DEVIANCE function returns the deviance from a normal distribution with a mean parameter ¼ . The following equation describes the DEVIANCE function for the normal distribution, where x is the random variable:

The Poisson Distribution

• DEVIANCE ('POISSON', variable , ¼ <, µ >)

where

• variable

  • is a numeric random variable.

  • Range: variable ‰

• ¼

  • is a numeric mean parameter.

  • Range: ¼ ‰ µ

• µ

  • is an optional positive numeric value that is used to bound ¼ . Any value of ¼ in the interval 0 ‰ ¼ ‰ µ is replaced by µ .

The DEVIANCE function returns the deviance from a Poisson distribution with a mean parameter ¼ . The following equation describes the DEVIANCE function for the Poisson distribution, where x is the random variable: