Inference under a new exponential-exponential loss capturing specified penalties for over- and under-estimation

Asymmetric loss functions have gained enormous importance over the years, with particular relevance to situations where over- and under-estimation of the parameter of interest are considered not of equal consequence. In particular, the linear-exponential (LINEX) loss has been studied and used quite extensively in classical and Bayesian inference. While LINEX loss nicely captures whether over- or under-estimation has a more serious impact, it falls short of incorporating any prior knowledge about the relative penalty for over- vis-à-vis that for under-estimation. Thus, if such prior knowledge is available as happens in many practical situations, notably in finance, medicine and reliability theory, among others, then there is a pressing need for devising a loss function that accounts for this information and hence is more realistic than the LINEX loss. More specifically, suppose the ground realities in a given situation demand that over-estimation needs to be penalized k times the penalty of under-estimation, where k is known. Clearly, over-estimation gets more penalized than under-estimation if k > 1 and it is the other way round if k