Basic Analysis and Graphing - SAS

80 Performing Univariate Analysis Chapter 2
Statistical Details for the Distribution Platform
where, for the Johnson Su:
fy ( ) = lnY + 1 + Y 2
=
X – θ
Y = ----------- – ∞ < X < ∞
σ
where, for the Johnson Sb:
sinh
–1 Y
fY ( )
=
ln -----------
Y
1 – Y
X – θ
Y = ----------- θ < X < θ+
σ
σ
and for the Johnson Sl, where σ = ± 1 .
fY ( ) = ln( Y)
Y
=
-----------
X – θ
σ
θ < X < ∞ if σ = 1
– ∞ < X < θ if σ = – 1
Johnson Su
δ
pdf: -- 1
for ; 0 < θ,δ
σ
+ x – θ
----------
σ
2 – 1 ⁄ 2
x – θ
φ γ + δ sinh–
1 ----------
σ
– ∞ < x, θ, γ < ∞
Johnson Sb
x – θ
pdf: φ γ+
δ ln -------------------------
for θ < x < θ+σ; 0 < σ
σ – ( x – θ)
δσ
-----------------------------------------------
( x – θ) ( σ – ( x – θ)
)
Johnson Sl
pdf: ------------- δ
for θ < x if σ = 1; θ > x if σ = -1
x – θ
φ γ δ x – θ
+ ln ----------
σ
where
φ ( . )
is the standard normal pdf.
Note: The parameter confidence intervals are hidden in the default report. Parameter confidence intervals
are not very meaningful for Johnson distributions, because they are transformations to normality. To show
parameter confidence intervals, right-click in the report and select Columns > Lower 95% and Upper
95%.