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On the Existence of UMVU Estimators for Bernoulli Experiments in the Non-Identically Distributed Case with Applications to the Randomized Response Method and the Unrelated Question Model

  • A. Dannwerth
  • D. Plachky
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 114)

Abstract

It is shown that there does not exist any UMVU estimator of the type \( {d^*}\left( {\Sigma _{j = 1}^n{X_j}} \right) \) based on independent random variables X 1,...,X n, where X j has a Bernoulli distribution with corresponding probability aj p +bj ∈ [0, 1] for p ∈ [0,1],aj≠ 0, j= 1,...n and where \( {d^*}:\left\{ {0, \ldots,n} \right\} \to \mathbb{R} \) is one-to-one, if the condition aj= a k and b j=b k or aj=-a kandb j =1-b k is not valid for all \( \left( {j,k} \right) \in {\left\{ {1, \ldots,n} \right\}^2} \) In particular, it is proved that the model parametera j b j j = 1,...,n, in connection with the unrelated question model must be kept fixed, if there should exist some UMVU estimator forpof this type. Furthermore, it is shown that the existence of some non-constant UMVU estimator forpaccording to the randomized response method implies that the randomization parameter \( {p_j} \in {\raise0.7ex\hbox{${\left[ {0,1} \right]}$} \!\mathord{\left/ {\vphantom {{\left[ {0,1} \right]} {\left\{ {\frac{1}{2}} \right\}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\left\{ {\frac{1}{2}} \right\}}$}},j = 1, \ldots,n \) must satisfy the condition p j=pk orp j + pk=1 for all \( \left( {j,k} \right) \in {\left\{ {1, \ldots,n} \right\}^2} \)

Keywords

Bernoulli Distribution Unbiased Estimator Randomization Parameter Randomization Procedure Covariance Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. Chaudhuri, A. and R. Mukerjee (1988): Randomized Response. M. Dekker, New YorkMATHGoogle Scholar
  2. Lehmann, E. L. (1983): Theory of Point Estimation. Wiley, New YorkMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • A. Dannwerth
    • 1
  • D. Plachky
    • 1
  1. 1.Institute of Mathematical StatisticsUniversity of MünsterMünsterGermany

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