User:Lchrisman/JQPN Cowlick

This page has notes by Lonnie on an oddity that I've named the "Cowlick" of the fully-bounded Johnson QPN distribution from the paper:

Hadlock & Bickel, "Johnson Quantile-Parameterized Distributions", Decision Analysis 2017

This oddity impacts the implementation of DensUncertainLMH, where I was hoping to use their algorithm but ran into this glitch. Other than this, I am enthusiastic about their paper and am hoping to use their algorithm. But to be usable as a robust built-in function in Analytica, I feel like I need to find a solution to this problem first. So this page contains my notes.

The cowlick refers to an artifact in their fully-bounded distribution in which the density goes to infinity at the upper bound when n=1 (when B < ave(L,H)), or goes to infinity at the lower bound when n=-1 (when B>ave(L,H)). This is an undesirable trait, and causes there to be what is essentially a step in the CDF at the upper bound, as if the continuous part of the CDF doesn't really go to the upper bound. It isn't a step in a strict sense (it is continuous), but it flicks up so fast at the end that it acts like a step.

The figure here is the cowlick in the illustrative example from their paper, where you can see the density jumping to infinity at the upper bound. This graph is the same as the graph that appears in their paper as Figure 7b, where you can make out this phenomena if you look carefully. Here is their graph with my arrow pointing out the start of the cowlick.

The CDF for the distribution parameterized by (-11, -10, -9.99, -9, -8) is shown next, where the cowlick causes a "step" to occur in the CDF at the upper bound. It isn't quite the smooth CDF I had hoped for.

The paper doesn't acknowledge this oddity, but it can be detected in their Figure 7b at the upper bound. The oddity does not happen with the unbounded and semi-bounded cases.

Cowlick proof

Although the cowlick was quite obvious after I had implemented the algorithm, I found it to be non-trivial to prove that it happens. Here I'll prove that the cowlick always happens when [math]\displaystyle{ B \lt (L+H)/2 }[/math]. L, B and H are the probit-transformed values for the 10th, 50th and 95th percentiles as defined in the paper. Loosely speaking, this is the case where the median is closer to the 10th percentile than to the 90th percentile.

Theorem: In the fully-bounded Hadlock & Bickel JQPN distribution, when [math]\displaystyle{ B \lt (L+H)/2 }[/math], the probability density at the upper bound [math]\displaystyle{ u }[/math] approaches infinity -- i.e.,

[math]\displaystyle{ \lim_{x\rarr u} F_B(x) = \infty }[/math]

and when [math]\displaystyle{ B\gt (L+H)/2 }[/math], the probability density at the lower bound [math]\displaystyle{ l }[/math] approaches infinity

[math]\displaystyle{ \lim_{x\rarr l} F_B(x) = \infty }[/math]

where [math]\displaystyle{ F_B(x) }[/math] is defined in Eq (8) on Page 42 of the paper.