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.