A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes

Summary Some important aspects of the asymptotic likelihood theory for stochastic processes is reviewed, with particular attention to martingale properties and to information and higher order likelihood quantities. Interrelations between those quantities are discussed. Furthermore, results are given on the asymptotic distribution of the score function and the maximum likelihood estimator when standardized by the various information matrices. Model robust procedures for constructing confidence regions are considered, and a robust measure of information is discussed in some detail. Finally, extensions of the ideas to estimating equations are considered. Throughout the theory is illustrated by examples.

[1]  R. Liptser.  A strong law of large numbers for local martingales , 1980 .

[2]  Grace L. Yang,et al.  Asymptotics In Statistics , 1990 .

[3]  Markov stopping sets and stochastic integrals. Application in sequential estimation for a random diffusion field , 1989 .

[4]  T. W. Anderson.  On Asymptotic Distributions of Estimates of Parameters of Stochastic Difference Equations , 1959 .

[5]  E. Wong,et al.  Likelihood ratios and transformation of probability associated with two-parameter Wiener processes , 1977 .

[6]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[7]  O. Barndorff-Nielsen.  Information And Exponential Families , 1970 .

[8]  S. Silvey.  A Note on Maximum‐Likelihood in the Case of Dependent Random Variables , 1961 .

[9]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .

[10]  C. Heyde.  On combining quasi-likelihood estimating functions , 1987 .

[11]  P. I. Nelson,et al.  Quasi-likelihood estimation for semimartingales , 1986 .

[12]  Andrew Chesher,et al.  Testing for Neglected Heterogeneity , 1984 .

[13]  J. Aitchison,et al.  Maximum-Likelihood Estimation of Parameters Subject to Restraints , 1958 .

[14]  T. Sweeting.  Uniform Asymptotic Normality of the Maximum Likelihood Estimator , 1980 .

[15]  M. Sørensen.  On quasi likelihood for semimartingales , 1990 .

[16]  Ishwar V. Basawa,et al.  Asymptotic optimal inference for non-ergodic models , 1983 .

[17]  R. Döhler.  Dominierbarkeit und suffizienz in der sequentialanalyse , 1981 .

[18]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[19]  C. Heyde,et al.  On Efficiency and Exponential Families in Stochastic Process Estimation , 1975 .

[20]  P. Jeganathan,et al.  Asymptotic theory of extimation when the limit of the log - likelihood ratios is mixed normal , 1980 .

[21]  Niels Keiding.  Maximum Likelihood Estimation in the Birth-and-Death Process , 1975 .

[22]  Paul D. Feigin,et al.  STABLE CONVERGENCE OF SEMIMARTINGALES , 1985 .

[23]  R. Höpfner.  Null recurrent birth-and-death processes, limits of certain martingales, and local asymptotic mixed normality , 1990 .

[24]  Yosihiko Ogata,et al.  Inference for earthquake models: A self-correcting model , 1984 .

[25]  Y. Ogata.  Maximum likelihood estimates of incorrect Markov models for time series and the derivation of AIC , 1980, Journal of Applied Probability.

[26]  Rolando Rebolledo,et al.  Central limit theorems for local martingales , 1980 .

[27]  Nils Lid Hjort,et al.  On inference in parametric survival data models , 1992 .

[28]  P. Feigin.  Conditional Exponential Families and a Representation Theorem for Asympotic Inference , 1981 .

[29]  John B. Walsh,et al.  Stochastic integrals in the plane , 1975 .

[30]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[31]  O. Barndorff-Nielsen.  Parametric statistical models and likelihood , 1988 .

[32]  David Aldous,et al.  On Mixing and Stability of Limit Theorems , 1978 .

[33]  Niels Keiding,et al.  Estimation in the birth process , 1974 .

[34]  Some comments concerning a curious singularity , 1979 .

[35]  John S. White.  THE LIMITING DISTRIBUTION OF THE SERIAL CORRELATION COEFFICIENT IN THE EXPLOSIVE CASE , 1958 .

[36]  Ganapati P. Patil,et al.  Statistical Distributions in Scientific Work , 1981 .

[37]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[38]  R. Morton,et al.  Efficiency of estimating equations and the use of pivots , 1981 .

[39]  On Maximum Likelihood Estimation in Randomly Stopped Diffusion-Type Processes , 1983 .

[40]  Estimation for Diffusion Processes under Misspecified Models. , 1984 .

[41]  Y. Kutoyants,et al.  Parameter estimation for stochastic processes , 1984 .

[42]  Patrick Billingsley,et al.  Statistical inference for Markov processes , 1961 .

[43]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Statistics of random processes , 1977 .

[44]  R. Royall.  Model robust confidence intervals using maximum likelihood estimators , 1986 .

[45]  C. C. Heyde,et al.  Remarks on efficiency in estimation for branching processes , 1975 .

[46]  K. Murali Rao,et al.  On Decomposition Theorems of Meyer. , 1969 .

[47]  Inge S. Helland,et al.  Central Limit Theorems for Martingales with Discrete or Continuous Time , 1982 .

[48]  Alan F. Karr,et al.  Point Processes and Their Statistical Inference , 1991 .

[49]  Paul D. Feigin,et al.  Maximum likelihood estimation for continuous-time stochastic processes , 1976, Advances in Applied Probability.

[50]  Michael Sørensen,et al.  Exponential families of stochastic processes and Lévy processes , 1994 .

[51]  D. Lépingle.  Sur le comportement asymptotique des martingales locales , 1978 .