Using Maximum Weighted Likelihood to Derive Lehmer and Hölder Families of Means

In this letter, we establish the links between the Lehmer and Hölder families of means and maximum weighted likelihood estimator. Considering the regular one-parameter exponential family of probability density functions, we show that the maximum weighted likelihood of the parameter is a generalized weighted mean from which Lehmer and Hölder families of means are derived. Some of the outcomes obtained provide a probabilistic interpretation of these two families and could therefore broaden their uses in various applications.