An Equal-Probability Partition of the Sample Space: A Non-parametric Inference from Finite Samples

• URL: http://arxiv.org/abs/2507.21712v1
• Date: Tue, 29 Jul 2025 11:39:13 GMT
• Title: An Equal-Probability Partition of the Sample Space: A Non-parametric Inference from Finite Samples
• Authors: Urban Eriksson,
• Abstract summary: This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it.<n>The $N$ sorted sample points partition the real line into $N+1$ segments, each carrying an expected probability mass of exactly $1/(N+1)$.<n>I compare this partition-based framework to the conventional ECDF and discuss its implications for robust non-parametric inference.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper investigates what can be inferred about an arbitrary continuous probability distribution from a finite sample of $N$ observations drawn from it. The central finding is that the $N$ sorted sample points partition the real line into $N+1$ segments, each carrying an expected probability mass of exactly $1/(N+1)$. This non-parametric result, which follows from fundamental properties of order statistics, holds regardless of the underlying distribution's shape. This equal-probability partition yields a discrete entropy of $\log_2(N+1)$ bits, which quantifies the information gained from the sample and contrasts with Shannon's results for continuous variables. I compare this partition-based framework to the conventional ECDF and discuss its implications for robust non-parametric inference, particularly in density and tail estimation.

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