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End‐Point Estimation for Decreasing Densities: Asymptotic Behaviour of the Penalized Likelihood Ratio

Abstract.  We consider the problem of estimating the modal value of a decreasing density on the positive real line. This has application in several interesting phenomena arising, for example, in renewal theory, and in biased and distance samplings. We use a penalized likelihood ratio‐based approach for inference and derive the scale‐free universal large sample null distribution of the log‐likelihood ratio, using a suitably chosen penalty parameter. We present simulation results and a real data analysis to corroborate our findings, and compare the performance of the confidence sets with the existing results.