On the estimation of mixing distributions: NPMLE and NPMPLE

Lei Liu, Purdue University


If we need to compute the NPMLE of a mixing distribution, which had been proved to be discrete with a finite support set in literature, some efficient and stable algorithms are desired. We propose several new algorithms, which are called partially projected gradient algorithms. These new algorithms can fully take advantage of the information provided by gradient functions. The numerical examples show they outperform the popular VEM algorithm and Wu's algorithms. If we know the mixing distribution is continuous and intend to estimate the mixing density, then the NPMLE is not appropriate because it is a discrete distribution. The estimation of mixing densities is an ill-posed indirect estimation problem. In this dissertation, we apply the maximum penalized likelihood principle for estimating mixing densities, and derive a functional EM algorithm to compute the NPMPLE of mixing densities. In M-steps of our new algorithm, we solve some ordinary differential equations with boundary conditions numerically. The simulations show the algorithm works better than the popular EMS algorithm and the kernel method in deconvolution. We also do some theoretical analysis of the NPMPLE in the dissertation.




Zhu, Purdue University.

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