1 . The J-QPD-S (Semi-Bounded) Distributions
Let l denote a specified finite lower bound of support, and let {x , x0.5 , x1 } denote a specified symmetric
percentile triplet for some given   (0, 0.5) . For example, {x0.1 , x0.5 , x0.9 } corresponds to the {10th, 50th,
90th} percentiles. Finally, let ϕ, Ф, and Ф-1 denote the PDF, CDF, and quantile function for the standard
normal distribution, respectively. The quantile function, CDF, and PDF for the J-QPD-S distribution
parameterized by   (l , x , x0.5 , x1 ) are given as follows:
1.1 Standard Parameterizations
Quantile Function



QS ( p)  l   exp  sinh sinh 1   1 ( p)   sinh 1 (nc ) .
Cumulative Distribution Function (CDF)
 1 


 1   x  l 
1
FS ( x)      sinh  sinh 1    log 
   sinh  nc    .
  
     



Probability Density Function (PDF)



 xl 

log 

 

 
1
 


     1
f S ( x)  
   k  nc  
    
  ( x  l )  
2
2  x l 




log

  








where,
c   1 (1   ) ,
L  log( x  l ), B  log( x0.5  l ), H  log( x1  l ),
n  sgn( L  H  2 B ),
 x  l , n  1

   x0.50  l , n  0 ,
 x  l , n  1
 1
1


H L

    sinh  cosh 1 
 ,
c
 2min( B  L, H  B)  

 1 
 min( H  B, B  L),
c 
 
k  1  (c ) 2 .
1

 
 x l 
2
2  x l 

k
log

nc



log

 



  .
 
  
    
1.2 Alternative (“Expanded”) Parameterizations
Alternatively, “expanding out” the hyperbolic functions in the expressions above gives the following
alternative forms for the J-QPD-S quantile function and CDF1:
Quantile Function
2 
 
QS ( p)  l    exp    k  1 ( p)  nc 1     1 ( p)    .

 
Cumulative Distribution Function (CDF)
 1
FS ( x)    
  


 
 x l 
2
2  x l 
   k log 
  nc   log 
   .
 
  
   
1.3 Special Case
The lognormal distributions occur as a special case of the J-QPD-S distributions. Specifically, for the
special case in which L+H–2B=0, we have n  sgn( L  H  2 B )  sgn(0)  0 , and thus:



QS ( p)  l   exp  sinh sinh 1   1 ( p)   l    exp    1 ( p)  ,
 1
FS ( x)    
  

 x  l 
  log 
,

  

  1
1
f S ( x)  
   
  ( x  l )    

 x  l 
  log 
 .

  
This corresponds to a lognormal distribution with parameters, μ = log(θ) and σ = λδ, and shifted to have
support on (l, ∞).
2 . The J-QPD-B (Bounded) Distributions
Let l and u denote specified finite lower and upper bounds of support, respectively, and let {x , x0.5 , x1 }
denote a specified symmetric percentile triplet for some given   (0, 0.5) . For example, {x0.1 , x0.5 , x0.9 }
corresponds to the {10th, 50th, 90th} percentiles. Finally, let ϕ, Ф, and Ф-1 denote the PDF, CDF, and quantile
function for the standard normal distribution, respectively. The quantile function, CDF, and PDF for the JQPD-B distribution parameterized by   (l , x , x0.5 , x1 , u ) are given as follows:
Quantile Function



QB ( p)  l  (u  l )    sinh    1 ( p)  nc  .
Cumulative Distribution Function (CDF)
 1 

 1 

 xl 
FB ( x)      sinh 1      1 
     nc  .

  

u l 

  


1
The PDF is already in expanded form.
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Probability Density Function (PDF)
f B ( x) 

 1 
1
 x  l 
  nc    sinh 1        1 

 
 u  l    
  

2

 x  l    1  x  l  
 (u  l )       1 
   

 u l  
 u l 

2
where,
c   1 (1   ) ,
 x l 
 x l 
 x l 
L   1  
, B   1  0.50
, H   1  1


,
 u l 
 u l 
 u l 
n  sgn( L  H  2 B ),
 L, n  1

 =  B, n  0
 H , n  1

1

H L

    cosh 1 
,
c
 2 min( B  L, H  B) 
H L

.
sinh(2 c)
2.1 Special Case
For the special case in which n = 0, the quantile function, CDF, and PDF for J-QPD-B are given by:

 H  L  1 
QB ( p)  l  (u  l )  B  
  ( p)  ,
 2c 


  2c  
1  x  l   
FB ( x)    
   B   
 ,
 u  l 
 H  L  
  2c  
1  x  l   
2c    
   B   

 u  l 
 H  L  
f B ( x) 
.
 1  x  l  
( H  L)(u  l )     

 u  l 

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