Affine Power-Hazard Distributions on a Finite Support: Construction, Shape Theory, and Likelihood Inference

Finite-supportfirst-success models arise when an event must occur by a prescribed terminal trial. The Alê distribution uses the linear hazard hx=x/n, which does not permit independent adjustment of the baseline success propensity and hazard curvature for fixed n. We introduce the affine power-hazard (APH) distribution on {1,…,n}, defined by hx=α+(1−α)(x/n)γ, where n≥2, 0≤α<1, and γ>0. The Alê distribution is recovered at (α,γ)=(0,1). We derive the main distributional functions, a stochastic first-success representation, stochastic orderings, and limiting properties. The hazard is increasing and is concave, linear, or convex according to 0<γ<1, γ=1, or γ>1. The probability mass function is unimodal for 0<γ≤1, whereas bimodality may occur for γ>1. For n≥3, the continuous parameters are structurally identifiable, although identification of γ becomes weak in the large-power regime, where the model approaches a right-censored geometric distribution. We develop likelihood-based inference for known and unknown endpoints and examine finite-sample estimation and sensitivity to endpoint-search bounds. Two applications illustrate both the usefulness of an externally specified finite endpoint and the inferential limitations arising near the geometric limiting regime.