A one-parameter decreasing-hazard lifetime model with exact inference on a bounded-below support

We introduce the modified exponential root distribution (MERD), a one-parameter lifetime model on $ [1, \infty) $ defined as the square of a unit-shifted exponential variate. Being a monotone transform of an exponential variable, MERD admits closed-form distribution, survival, hazard (strictly decreasing), cumulative and reversed hazard, quantile, and mean-residual-life functions, along with closed-form moments, quantile-based shape measures, order statistics, entropies, and stress-strength reliability via a single root substitution. Since the transformed data are exactly exponential, the pivot $ 2\lambda S\sim\chi^2_{2n} $ gives exact confidence intervals for the rate parameter at any sample size, unlike most comparably simple lifetime models. We derive closed-form maximum-likelihood and conjugate-gamma Bayesian estimators and evaluate their bias, mean squared error (MSE), and coverage via simulation. The MERD exhibits competitive performance with respect to one- and two-parameter models on the parsimony-penalized information criteria when evaluated on empirical datasets of biomedical survival and actuarial science, while also providing exact inference. We emphasize that these findings are dataset-specific and do not constitute a claim of universal superiority over all competing models; rather, they illustrate the practical utility of the MERD as a tractable and exactly-inferable option for bounded decreasing-hazard data. To address the fixed lower bound and scale, we develop an identifiable three-parameter location-scale extension, show its threshold is estimated by the sample minimum, and examine how far the exact-inference properties persist once threshold and scale are estimated.