Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

Atin Gayen, M. Ashok Kumar

TL;DR

The paper extends classical sufficiency to estimation problems based on Jones et al. and Basu et al. α-likelihoods, identifying power-law extensions of exponential families (including Student distributions) as the fixed-dimension sufficiency class. It develops generalized factorization and minimal sufficiency, establishes Rao-Blackwell-type theorems for these likelihoods, and derives Cramér-Rao-type bounds for estimators in power-law families. The results provide a unified framework for robust inference with probabilistic models beyond ML, yielding sharper variance bounds and best-unbiased estimators for deformed models. Applications to Student distributions illustrate improved estimator efficiency under generalized sufficiency, with implications for robust statistical practice and information-theoretic connections.

Abstract

This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among distance-based robust inference methods. We first characterize the probability distributions that always have a fixed number of sufficient statistics (independent of sample size) with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as a special case. We then extend the notion of minimal sufficient statistics and compute it for these power-law families. Finally, we establish a Rao-Blackwell-type theorem for finding the best estimators for a power-law family. This helps us establish Cramér-Rao-type lower bounds for power-law families.

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

TL;DR

The paper extends classical sufficiency to estimation problems based on Jones et al. and Basu et al. α-likelihoods, identifying power-law extensions of exponential families (including Student distributions) as the fixed-dimension sufficiency class. It develops generalized factorization and minimal sufficiency, establishes Rao-Blackwell-type theorems for these likelihoods, and derives Cramér-Rao-type bounds for estimators in power-law families. The results provide a unified framework for robust inference with probabilistic models beyond ML, yielding sharper variance bounds and best-unbiased estimators for deformed models. Applications to Student distributions illustrate improved estimator efficiency under generalized sufficiency, with implications for robust statistical practice and information-theoretic connections.

Abstract

This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among distance-based robust inference methods. We first characterize the probability distributions that always have a fixed number of sufficient statistics (independent of sample size) with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as a special case. We then extend the notion of minimal sufficient statistics and compute it for these power-law families. Finally, we establish a Rao-Blackwell-type theorem for finding the best estimators for a power-law family. This helps us establish Cramér-Rao-type lower bounds for power-law families.
Paper Structure (10 sections, 18 theorems, 75 equations)

This paper contains 10 sections, 18 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. A Generalized notion of the principle of sufficiency
  3. Generalized Fisher-Darmois-Koopman-Pitman Theorem
  4. Generalized Fisher-Darmois-Koopman-Pitman theorem for Basu et al. likelihood function
  5. Minimal sufficiency with respect to Basu et al. and Jones et al. likelihood functions
  6. Rao-Blackwell theorem for Jones et al. estimation
  7. Example:
  8. Efficiency of Rao-Blackwell estimators of power-law family
  9. Analogous results for Basu et al. likelihood function
  10. Summary and concluding remarks

Key Result

Proposition 2

Let $\Pi=\{p_\theta:\theta\in\Theta\}$ be a family of probability distributions. Suppose the underlying problem is to estimate $\theta$ by maximizing some generalized likelihood function $L_G$. Then a set of statistics $T$ is a sufficient statistic for $\theta$ with respect to $L_G$ if and only if t for all sample points ${x}_1^n$ and for all $\theta\in \Theta$.

Theorems & Definitions (34)

  • Definition 1
  • Proposition 2: Generalized Factorization Theorem
  • Proposition 3
  • Remark 1
  • Definition 4
  • Proposition 5
  • Definition 6
  • Remark 2
  • Theorem 7
  • Theorem 8
  • ...and 24 more