The Shannon entropy of ranked set and partially rank-ordered set samples of the same size: no ordering relationship for the uniform parent
Abstract
In partially rank-ordered set (PROS) sampling one fixes a set size S and a design parameter D={d₁,…,dₙ} splitting the ranks {1,…,S} into n consecutive blocks, and measures one unit drawn uniformly at random from each block; the measured value from block dᵣ has the mixture density f_((dᵣ))=|dᵣ|⁻¹Σ_(u ∈ dᵣ)f^((u:S)). Ranked set sampling (RSS) of size n is the case S=n. Hatefi and Jafari Jozani, who study the information content of such samples, prove m⁻¹H_S(Xᵣss) ≤ Hₙ(Xₚros) ≤ Hₙ(Xₛrs) for the balanced design and then record that they "were not able to obtain an ordering relationship among the Shannon entropy of RSS and PROS data of the same size". We settle that comparison for the Uniform(0,1) parent, and the answer is negative: both orders occur. For the balanced design PROS(2,4) one has H₂(Xₚros)=9/2-2log 2-(27)/(8)log 3<1-2log 2=H₂(Xᵣss), while for the design D=({1,2},{3},…,{7},{8,9}) at set size S=9 and sample size n=7 one has H₇(Xᵣss)<H₇(Xₚros), the gap being -(997119143)/(74118870)+(79155653)/(5764801)log 2+10log 3-2log 5-2log 7 =-0.06004185…. So no ordering relationship between the two entropies can hold. Every entropy here is exact — a rational number plus a rational combination of log 2,log 3,log 5,log 7 — and every closed form and every inequality stated as a theorem below is checked by the Lean 4 kernel over Mathlib, with no evaluation axiom and no floating-point step. Outside the formal development we report two measurements, labelled as such: an exhaustive search over every design of set size S ≤ 10 shows that no reversal occurs for S ≤ 8, so S=9 is the smallest set size at which one exists; and over all 34 balanced designs with n ≥ 2 and S=mn ≤ 16 the inequality Hₙ(Xₚros) ≤ Hₙ(Xᵣss) never fails, which locates the obstruction in the general design parameter rather than in the design family as a whole.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-09-07 03:53 UTC
File fingerprint
17a0309c9b7e9394727bbe22535c1cfb734223a4562c7c379fbc3184de97f7d1
Claim ledger
Stated results
RP1routine2026-09-03
PROS degenerates to RSS and to SRS: Hpros (List.replicate n 1) = Hrss n for EVERY n, and Hpros [S] = 0 = Hrss 1 for every S >= 1, the latter via mixDensity_full (the uniform mixture of all S order-statistic densities is the parent density)
RP2known2026-09-03
The differential entropy of the (p+1)-st order statistic of p+q+1 i.i.d. Uniform(0,1) variables, at the eight (p,q) pairs this family needs: (0,1), (0,6), (1,5), (2,4), (3,3), (2,6), (3,5), (4,4) – e.g. H(f^(3:9)) = 3107/630 - log 252 and H(f^(5:9)) = 1879/315 - log 630
RP3known2026-09-03
H₂(Xᵣss) = 1 - 2 log 2 and H₇(Xᵣss) = 21 - 4 log 2 - 4 log 3 - 3 log 5 - 7 log 7 for the Uniform(0,1) parent (numerically -0.3862943611 and -4.6167226576)
RP4routine2026-09-03
The PROS block (mixture) entropies: H = 9/4 - log 2 - (27/16) log 3 for the block 1,2 of a set of size 4 (density 2(1-x)²(1+2x)), and H = 133307284/37059435 - log(9/2) - (8388608/5764801) log 8 for the block 1,2 of a set of size 9 (density (9/2)(1-x)⁷(1+7x))
RP5routine2026-09-03
H₂(Xₚros) = 9/2 - 2 log 2 - (27/8) log 3 for the balanced design PROS(2,4), and H₇(Xₚros) = 2553615413/74118870 - (102214857/5764801) log 2 - 14 log 3 - log 5 - 5 log 7 for the design D = (1,2,3,4,5,6,7,8,9) at set size 9
RP6routine2026-09-03
PROS strictly BELOW RSS: H₂(Xₚros) < H₂(Xᵣss) for the balanced design PROS(2,4), the gap being -7/2 + (27/8) log 3 = +0.2078164743
RP7candidate2026-09-03
PROS strictly ABOVE RSS: H₇(Xᵣss) < H₇(Xₚros) for the design D = (1,2,3,4,5,6,7,8,9) at set size S = 9, the gap being -997119143/74118870 + (79155653/5764801) log 2 + 10 log 3 - 2 log 5 - 2 log 7 = -0.0600418535
RP8candidate2026-09-03
No ordering relationship: for the Uniform(0,1) parent there are PROS designs D1 and D2, each with set size strictly larger than sample size, with H(Xₚros^(D1)) < H(Xᵣss) and H(Xᵣss) < H(Xₚros^(D2)) at the respective sample sizes – so no ordering between the Shannon entropy of RSS and of PROS data of the same size can hold
RP9routine2026-09-03
Negative controls: the too-large claim 'for every non-empty design D, Hₚros(D) <= Hᵣss(|D|)' and the too-small claim 'for every non-empty design D, Hᵣss(|D|) <= Hₚros(D)' are both refuted; H₂(Xᵣss) is not 0; and neither exhibited design is the degenerate coincidence case (set size differs from sample size)
RP10measurement2026-09-03
Search extent: over EVERY PROS design at set size S <= 10 (all compositions of S into consecutive blocks, 40-digit quadrature over 32 panels) the order reverses for no S <= 8, and S = 9 with n = 7 and D = (2,1,1,1,1,1,2) is the smallest reversal. The closest non-reversal is +0.0066391870 at S = 8, n = 6, D = (2,1,1,1,1,2); the closest call anywhere is +2.7337059299e-6 at S = 9, n = 8, D = (1,1,1,1,1,1,1,2), exactly -1071238013/148237740 - (38246987/5764801) log 2 + 14 log 3 - log 5 - log 7. Extended exactly (rational arithmetic) to S <= 13 over all designs whose subsets have size at most 2
This ledger entry is reported in prose and is not bound to a Lean theorem.RP11measurement2026-09-03
Conjecture: for the source's own BALANCED design PROS(n,S) with m = S/n, the Uniform(0,1) parent satisfies Hₙ(Xₚros) <= Hₙ(Xᵣss), with equality exactly when m = 1 or n = 1. Verified at all 34 pairs (n >= 2; 50 in all) (n,m) with S = mn <= 16
This ledger entry is reported in prose and is not bound to a Lean theorem.RP12measurement2026-09-03
1.9459 < log 7 < 1.94592, a rational two-sided bound on log 7
RP13routine2026-09-03
The integral toolbox: intₐᵇ yᵏ log y dy = Gₖ(b) - Gₖ(a) for 0 <= a <= b with Gₖ(t) = tᵏ⁺¹ log t/(k+1) - tᵏ⁺¹/(k+1)² and Gₖ(0) = 0; its coefficient-indexed polynomial forms; the four-way split of -int₀¹ g log g for a density g = c xᵖ (1-x)^q (u+vx)ʳ; and the reflection symmetry x -> 1-x of the entropy
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Two designs for drawing n measured observations from a population with density f.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7