Two-piece Normal Distribution

[Jameel Ahmed]

The overall variance and skewness of Two-piece Normal (TPN) is given by:

𝑉(𝑥) = 𝜎₁ 𝜎₂ + (1 − 𝑘²)(𝜎₂ − 𝜎₁)²

𝛾(𝑥) = 𝑘(𝜎₂ − 𝜎₁)[(2𝑘² − 1)(𝜎₂ − 𝜎₁)^2 + 𝜎₁𝜎₂]

where 𝑘 =\sqrt(2/pi)

For given values of 𝑉(𝑥) and 𝛾(𝑥), two variances (𝜎₁ and 𝜎₂) could be estimated. How can I set up this nonlinear simultaneous equations problem in STATA? Any help?

Note: I have tried methods suggested here (https://www.stata.com/support/faqs/p...ear-equations/), but am constantly getting error.

Regards,

[Scott Merryman]

Jameel Ahmed

Like this?

clear*
scalar _sigma= 2
scalar _skew=  .3
 program nlfaq
        syntax varlist(min=1 max=1) [if], at(name)

        tempname A B 
        scalar `A' = `at'[1, 1]
        scalar `B' = `at'[1, 2]

        tempvar yh
        gen double `yh' = (1 - 2/_pi) * (`B' - `A')^2 + `B'*`A' + 1 - scalar(_sigma)  in 1
        replace `yh' =  sqrt(2/_pi)*(`B' - `A') *(( (4/_pi) -1) * (`B' - `A')^2 + `B'*`A') - scalar(_skew)  in 2
        replace `varlist' = `yh'

 end

 set obs 2
 gen double y = 0 
 replace y = 1 in 1
 set trace off
 nl faq @ y, parameters(A B ) initial(A 1 B 1)

[Jameel Ahmed]

Scott Merryman .. Thanks. Much appreciated