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Deriving Lehmer and Hölder means as maximum weighted likelihood estimates for the multivariate exponential family

Djemel Ziou, Issam Fakir

Abstract

The links between the mean families of Lehmer and Hölder and the weighted maximum likelihood estimator have recently been established in the case of a regular univariate exponential family. In this article, we will extend the outcomes obtained to the multivariate case. This extension provides a probabilistic interpretation of these families of means and could therefore broaden their uses in various applications.

Deriving Lehmer and Hölder means as maximum weighted likelihood estimates for the multivariate exponential family

Abstract

The links between the mean families of Lehmer and Hölder and the weighted maximum likelihood estimator have recently been established in the case of a regular univariate exponential family. In this article, we will extend the outcomes obtained to the multivariate case. This extension provides a probabilistic interpretation of these families of means and could therefore broaden their uses in various applications.
Paper Structure (10 sections, 43 equations, 3 figures)

This paper contains 10 sections, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Lehmer's and Hölder's Central tendencies
  3. Maximum likelihood estimates
  4. Case studies
  5. Lehmer's mean
  6. Hölder's mean
  7. Comparison
  8. Conclusions
  9. The MWLE
  10. Estimator uniqueness

Figures (3)

  • Figure 1: (a) Three means between $0.6$ and $2$ as function of $\alpha$. The arithmetic mean is displayed as a basis for comparison. (b) The weights $m_h(0.6)$ and $m_l(0.6)$ as a function of $\alpha$. (c) The weights $m_h(2)$ and $m_l(2)$ as a function of $\alpha$.
  • Figure 2: MWLE of $\lambda$ as the Lehmer's mean as function of $\beta$
  • Figure 3: MWLE of $\lambda$ as the Hölder's mean as function of $k$