Basic Analysis and Graphing - SAS

78 Performing Univariate Analysis Chapter 2
Statistical Details for the Distribution Platform
Gamma
Beta
The Gamma fitting option estimates the gamma distribution parameters, α >0 and σ > 0. The parameter
α, called alpha in the fitted gamma report, describes shape or curvature. The parameter σ, called sigma, is
the scale parameter of the distribution. A third parameter, θ, called the Threshold, is the lower endpoint
parameter. It is set to zero by default, unless there are negative values. You can also set its value by using the
Fix Parameters option. See “Fit Distribution Options” on page 58.
1
pdf: ------------------- for ; 0 < α,σ
Γα ( )σ α ( x – θ) α – 1 exp( –( x – θ)
⁄ σ)
θ ≤ x
E(x) = ασ + θ
Var(x) = ασ 2
• The standard gamma distribution has σ = 1. Sigma is called the scale parameter because values other
than 1 stretch or compress the distribution along the x-axis.
2
• The Chi-square χ ( ν)
distribution occurs when σ =2, α = ν/2, and θ =0.
• The exponential distribution is the family of gamma curves that occur when α =1 and θ =0.
The standard gamma density function is strictly decreasing when α ≤ 1 . When α > 1 , the density function
begins at zero, increases to a maximum, and then decreases.
The standard beta distribution is useful for modeling the behavior of random variables that are constrained
to fall in the interval 0,1. For example, proportions always fall between 0 and 1. The Beta fitting option
estimates two shape parameters, α >0 and β > 0. There are also θ and σ, which are used to define the lower
threshold as θ, and the upper threshold as θ + σ. The beta distribution has values only for the interval
defined by θ ≤ x ≤ ( θ + σ)
. The θ is estimated as the minimum value, and σ is estimated as the range. The
standard beta distribution occurs when θ = 0 and σ =1.
Set parameters to fixed values by using the Fix Parameters option. The upper threshold must be greater
than or equal to the maximum data value, and the lower threshold must be less than or equal to the
minimum data value. For details about the Fix Parameters option, see “Fit Distribution Options” on
page 58.
pdf: ------------------------------------------ 1
for ; 0 < σ,α,β
B( α,
β)σ α+
β – 1 ( x – θ ) α – 1 ( θ + σ – x ) β – 1 θ ≤ x ≤ θ + σ
E(x) = θ + σ------------
α
α + β
σ
Var(x) = ------------------------------------------------
2 αβ
( α + β) 2 ( α+ β + 1)
where
B ( . )
is the Beta function.