An examination of the mean and quantiles from a relational system with fixed just unnoticeable difference representation.

摘要

Given a set A, and preferences defined on A x A it may be possible to find a function f:A → R such that (f(x) > f(c) + 1 iff x is preferred to c). Such functions are called fixed just unnoticeable difference (jud) representations; the values of these functions are called ratings.; For fixed c ∈ A and arbitrary x ∈ A we determine supf∈F {lcub}f(x) - f(c){rcub} = MxJ(x ≺ c) + mxJ(x ≻ ≈ c) and inff∈F {lcub}f(x) - f(c){rcub} = MxJ(x ≻ c) + mxJ(x ≺ c) - mxJ(x ≈ c). Here x ≻ c implies x is rated above c; x ≈ c implies x and c can be rated equally; x ≺ c implies x is rated below c, Mx is the length of the longest preference path between x and c, subject to direction; mx is the length of the shortest indifference path between x and c, subject to direction; F denotes the set of fixed jud representations; J is the indicator function. These two extrema are determined by constructing sequences in F that converge to each. Additionally, it is determined that any value between these extrema is a possible rating difference of x and c.; Let gamma ∈ (0,1) and X be the result of a random process on A with probability measure P. We examine the validity of many statements involving the position of c relative to the gammath quantile. The most interesting result is: under the conditions P(X ≺ ≈ c) ≥ gamma, P(X ≻ ≈ c) ≥ 1 - gamma and either (P(X ≺ c) + P(c) gamma or P(X ≻ c) + P(c) 1 - gamma), the statement "c is rated at the gammath quantile" is true for some, but not all, fixed jud representations. The analysis of every statement regarding the position of c relative to the gammath quantile follows from the results cited in the previous paragraph.; Our most significant result is that when it is assumed there is no b ∈ A with P(b) = 1 and both E( inff∈F {lcub}f(X) - f(c){rcub}) and E( supf∈F {lcub}f(X) - f(c){rcub}) exist then the statement "c is rated above the mean" is always true iff E( supf∈F (f(X) - f(c){rcub}) ≤ 0, always false iff E( inff∈F {lcub}f(X) - f(c){rcub}) ≥ 0, and true for some, but not all, f ∈ F iff E( inff∈F {lcub}f(X) - f(c){rcub}) 0 E( supf∈F {lcub}f(X) - f(c){rcub}). This result follows when it is established that inff∈F {lcub}E(f(X) - f(c)){rcub} = E( inff∈F (f(X) - f(c){rcub}) and supf∈F {lcub}E(f(X) - f(c)){rcub} = E( supf∈F {lcub}f(X) - f(c){rcub}).; Also provided is an algorithm that determines inff∈F {lcub}f(X) - f(c){rcub}, supf∈F {lcub}f(X) - f(c){rcub}, E( inff∈F {lcub}f(X) - f(c){rcub}) and E( supf∈F {lcub}f(X) - f(c){rcub}) when A is finite, an analysis of our results in comparison to Stevens' appropriateness criteria, and the theoretical considerations of defining a probability measure on A.