MMP.11:5.1 - Construct a response from its known shape

A normalized input u lies in [0,1]. The subject account supports a nondecreasing response r with r(0)=0 and r(1)=1. Its intermediate shape is unknown. Begin with these properties, rather than choosing a straight line as the only candidate.

Choose an integrable h with h(u)>=0 almost everywhere and H=integral_0^1 h(v) dv>0. Define

r(u)=integral_0^u h(v) dv / H.

The endpoints follow by substitution. For u2>=u1, the difference is the nonnegative integral of h over [u1,u2], divided by H. Thus every member is nondecreasing. These curves are absolutely continuous. Every absolutely continuous nondecreasing response with these endpoints has such a representation using its almost-everywhere derivative, but a jump response is outside this family. The needed regularity must come from the question or remain a declared restriction.

For a small calculable family, use linear segments through (0,0), (1/4,q), (1/2,1/2) and (1,1). Their slopes are 4*q, 2-4*q and 1. They are nonnegative exactly when 0<=q<=1/2. This is a construction of admissible candidates, not a conclusion from measurements alone.

Suppose observations establish only the three values at 0, 1/2 and 1. Every q in that interval agrees with them. The consequence r(1/4)<=0.6 follows for this whole family; it also follows for every nondecreasing response with the given midpoint. There is no need to identify q for that question.

Now the receiving use asks whether r(1/4)>0.3. Candidates q=0.2 and q=0.4 satisfy the same observations and give opposite answers. Another repetition at the three old input values does not distinguish these ideal candidates. An observation near the disputed input may help; its precision and cost belong to that new question. Alternatively, a supported additional shape relation could narrow the family.