Exact Closed-Form Formulae for Linear and Circular Continuous Scan Statistics: $P_c(N-1;N,w)$, $P_c(3;N,w)$, and $P(3;N,w)$

Abstract

The continuous linear $P(k;N,w)$ and circular scan statistics $P_c(k;N,w)$ are fundamental tools in probability and spatial statistics, frequently used to detect clustering in uniform data. Let $X_1,X_2,\dots ,X_N$ be independently and uniformly distributed random variables on a unit interval or unit ring. The exact distribution of these scan statistics relies on the minimum window width required to capture exactly $k$ points. Furthermore, the survival function $1-P_c(k;N,w)$ directly corresponds to the geometric probability that if $N$ arcs of length $1-w$ are uniformly and randomly placed on a unit circle, every point on the circle is covered at least $N+1-k$ times. Historically, evaluating the exact cumulative distribution functions, $P(k;N,w)$ and $P_c(k;N,w)$, relies heavily on complex recursive approximations. In this paper, we bypass these traditional recursive methods to derive direct, generalized closed-form expressions for some linear and circular continuous scan statistics. Specifically, we present the exact analytical solutions for $P_c(N-1;N,w)$, $P_c(3;N,w)$, and $P(3;N,w)$ for arbitrary values of $N$ and window width $w$. These newly derived closed-form expressions not only provide exact baseline distributions for extreme spacings but also significantly simplify computational complexity compared to existing iterative approaches.

9 pages. Derives exact closed-form formulae for $P_c(N-1;N,w)$, $P_c(3;N,w)$, and $P(3;N,w)$