Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of his 65th Birthday

## Convergence and consistency of Newton’s algorithm for estimating mixing distribution

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, Chapter

Ghosh, JK; Tokdar, ST

January 1, 2006

We provide a new convergence and consistency proof of Newton’s algorithm for estimating a mixing distribution under some rather strong conditions. An auxiliary result used in the proof shows that the Kullback Leibler divergence between the estimate and the true mixing distribution converges as the number of observations tends to infinity. This holds under much weaker conditions. It is pointed out that Newton’s proof of convergence, based on a representation of the algorithm as a nonhomogeneous weakly ergodic Markov chain, is incomplete. Our proof is along quite different lines. We also study various other aspects of the estimate, including its claimed superiority to the Bayes estimate based on a Dirichlet mixture.

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Ghosh, J. K., & Tokdar, S. T. (2006). Convergence and consistency of Newton’s algorithm for estimating mixing distribution. In

*Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of his 65th Birthday*(pp. 429–443). https://doi.org/10.1142/9781860948886_0019Ghosh, J. K., and S. T. Tokdar. “Convergence and consistency of Newton’s algorithm for estimating mixing distribution.” In

*Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of His 65th Birthday*, 429–43, 2006. https://doi.org/10.1142/9781860948886_0019.Ghosh JK, Tokdar ST. Convergence and consistency of Newton’s algorithm for estimating mixing distribution. In: Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of his 65th Birthday. 2006. p. 429–43.

Ghosh, J. K., and S. T. Tokdar. “Convergence and consistency of Newton’s algorithm for estimating mixing distribution.”

*Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of His 65th Birthday*, 2006, pp. 429–43.*Scopus*, doi:10.1142/9781860948886_0019.Ghosh JK, Tokdar ST. Convergence and consistency of Newton’s algorithm for estimating mixing distribution. Frontiers in Statistics: Dedicated to Peter John Bickel in Honor of his 65th Birthday. 2006. p. 429–443.