Non- and semi-parametric maximum likelihood estimators and the Von Mises method

Having shown in Part I of this paper that some well-known NPMLE's are compactly differentiable functions of the empirical data, and moreover solve a compactly differentiable generalized likelihood equation, we prove an efficiency theorem which shows that such an estimator asymptotically achieves the information bound in the generalized Hajek-Le Cam convolution and asymptotic local minimax theorems.

[1]  R. Fisher,et al.  On the Mathematical Foundations of Theoretical Statistics , 1922 .

[2]  J. Doob.  Stochastic processes , 1953 .

[3]  J. Kiefer,et al.  CONSISTENCY OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE PRESENCE OF INFINITELY MANY INCIDENTAL PARAMETERS , 1956 .

[4]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[5]  Fu Cheng Hsiang.  On differentiable functions , 1960 .

[6]  R. M. Dudley,et al.  Weak Convergence of Probabilities on Nonseparable Metric Spaces and Empirical Measures on Euclidean Spaces , 1966 .

[7]  O. Smolyanov,et al.  The theory of differentiation in linear topological spaces , 1967 .

[8]  B. Efron.  The two sample problem with censored data , 1967 .

[9]  O. Smolyanov,et al.  The various definitions of the derivative in linear topological spaces , 1968 .

[10]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[11]  Wayne Nelson,et al.  Hazard Plotting for Incomplete Failure Data , 1969 .

[12]  Bernard Altshuler,et al.  Theory for the measurement of competing risks in animal experiments , 1970 .

[13]  Patrick Billingsley,et al.  Weak convergence of measures - applications in probability , 1971, CBMS-NSF regional conference series in applied mathematics.

[14]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[15]  W. Vervaat.  Functional central limit theorems for processes with positive drift and their inverses , 1972 .

[16]  N. Breslow,et al.  A Large Sample Study of the Life Table and Product Limit Estimates Under Random Censorship , 1974 .

[17]  B. Turnbull.  The Empirical Distribution Function with Arbitrarily Grouped, Censored, and Truncated Data , 1976 .

[18]  O. Aalen.  Nonparametric Inference for a Family of Counting Processes , 1978 .

[19]  Søren Johansen,et al.  The product limit estimator as maximum likelihood estimator , 1978 .

[20]  Ren Johansen.  An Empirical Transition Matrix for Non-homogeneous Markov Chains Based on Censored Observations , 1978 .

[21]  James A. Reeds,et al.  Jackknifing Maximum Likelihood Estimates , 1978 .

[22]  Thomas R. Fleming,et al.  Nonparametric Estimation for Nonhomogeneous Markov Processes in the Problem of Competing Risks , 1978 .

[23]  Richard D. Gill,et al.  Nonparametric estimation based on censored observations of a Markov renewal process , 1978 .

[24]  Richard D. Gill,et al.  Testing with replacement and the product limit estimator : (preprint) , 1981 .

[25]  N. Reid.  Influence functions for censored data , 1979 .

[26]  F. W. Scholz,et al.  Towards a unified definition of maximum likelihood , 1980 .

[27]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[28]  D. Freedman,et al.  Some Asymptotic Theory for the Bootstrap , 1981 .

[29]  R. Gill,et al.  Cox's regression model for counting processes: a large sample study : (preprint) , 1982 .

[30]  R. Gill.  Large Sample Behaviour of the Product-Limit Estimator on the Whole Line , 1983 .

[31]  S. Geman,et al.  Nonparametric Maximum Likelihood Estimation by the Method of Sieves , 1982 .

[32]  G. M. Laslett,et al.  The survival curve under monotone density constraints with applications to two-dimensional line segment processes , 1982 .

[33]  L. Fernholz.  von Mises Calculus For Statistical Functionals , 1983 .

[34]  W. J. Hall,et al.  Information and Asymptotic Efficiency in Parametric-Nonparametric Models , 1983 .

[35]  S. Johansen.  An Extension of Cox's Regression Model , 1983 .

[36]  B. R. Clarke.  Uniqueness and Fréchet differentiability of functional solutions to maximum likelihood type equations , 1983 .

[37]  D. Pollard.  Convergence of stochastic processes , 1984 .

[38]  Martin Jacobsen,et al.  Maximum Likelihood Estimation in the Multiplicative Intensity Model: A Survey , 1984 .

[39]  Ørnulf Borgan,et al.  Counting process models for life history data: a review , 1984 .

[40]  K. Dzhaparidze.  On asymptotic inference about intensity parameters of a counting process , 1985 .

[41]  Differentiable statistical functionals , 1985 .

[42]  Jon A. Wellner,et al.  Semiparametric models: progress and problems , 1985 .

[43]  D. C. Taylor.  Asymptotic distribution theory for general statistical functionals , 1985 .

[44]  W. Parr.  Jackknifing Differentiable Statistical Functionals , 1985 .

[45]  J. Crowley,et al.  Correction: A Large Sample Study of Generalized Maximum Likelihood Estimators from Incomplete Data via Self-Consistency , 1985 .

[46]  W. Esty,et al.  Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators , 1985 .

[47]  D. Pollard.  New Ways to Prove Central Limit Theorems , 1985, Econometric Theory.

[48]  W. Parr.  The bootstrap: Some large sample theory and connections with robustness , 1985 .

[49]  M. Woodroofe.  Estimating a Distribution Function with Truncated Data , 1985 .

[50]  R. Dudley.  An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions , 1985 .

[51]  B. R. Clarke.  Nonsmooth analysis and Fréchet differentiability of M-functionals , 1986 .

[52]  E. Giné,et al.  Lectures on the central limit theorem for empirical processes , 1986 .

[53]  Grace L. Yang,et al.  Strong Consistency of a Nonparametric Estimator of the Survival Function with Doubly Censored Data , 1987 .

[54]  R. Beran,et al.  Convergence of stochastic empirical measures , 1987 .

[55]  J. Wellner,et al.  UNIFORMITY IN P OF SOME LIMIT THEOREMS FOR EMPIRICAL MEASURES AND PROCESSES , 1987 .

[56]  V. Dobric,et al.  The central limit theorem for stochastic processes II , 1988 .