Likelihood Inference for the Marshall–Olkin Bivariate Weibull Model with Modified Singularity Under Exact and Rounded Data

The Marshall–Olkin bivariate Weibull model with modified singularity assigns positive probability to an unknown ray x=μy. We study inference for its location under exact and finite-resolution observation. With exact observations, repeated common-shock ratios permit support recovery, and conditional profiling reduces the rate-parameter optimization. With grouped observations, a rectangle likelihood retains continuous and singular mass, and a breakpoint-aware search accommodates the piecewise-smooth likelihood. In simulations with 1000 replications per scenario, the ratio of mean RMSE across four recording resolutions, comparing midpoint substitution with interval-aware estimation of μ, was approximately 29 and 39 for common-shock probabilities 0.30 and 0.50. At probability 0.10 and the coarsest resolution, 24.4% of replications met an operational weak-information criterion. A direct comparison also shows lower rate-parameter RMSE for interval-aware estimation at moderate and large singular masses, whereas shape-parameter accuracy is similar and not uniformly better. Hessian-based intervals were unavailable in many replications and their conditional coverage could be below nominal. An exploratory computational illustration with two river-response episodes demonstrates the calculations and sensitivity without supporting substantive hydrological conclusions. The study clarifies how finite recording resolution changes inference on a moving singular support and limits local uncertainty summaries.