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Fundamental Concepts in Parametric Inference

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Book cover A Parametric Approach to Nonparametric Statistics

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Abstract

In this chapter we review some terminology and basic concepts in probability and classical statistical inference which provide the notation and fundamental background to be used throughout this book. In the section on probability we describe some basic notions and list some common distributions along with their mean, variance, skewness, and kurtosis. We also describe various modes of convergence and end with central limit theorems. In the section on statistical inference, we begin with the subjects of estimation and hypothesis testing and proceed with the notions of contiguity and composite likelihood.

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References

  • Albert, J. (2008). Bayesian Computation with R. Springer, second edition.

    Google Scholar 

  • Billingsley, P. (2012). Probability and Measure. John Wiley and Sons, anniversary edition.

    Google Scholar 

  • Box, George, E. and Tiao, George, C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley Publishing Company.

    Google Scholar 

  • Casella, G. and Berger, R. L. (2002). Statistical Inference. Duxbury Press., second edition.

    Google Scholar 

  • Casella, G. and George, E. I. (1992). Explaining the Gibbs sampler. The American Statistician, 46:167–174.

    MathSciNet  Google Scholar 

  • Cox, D. and Hinkley, D. (1974). Theoretical Statistics. Chapman Hall, London.

    Book  Google Scholar 

  • Ferguson, T. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York and London.

    MATH  Google Scholar 

  • Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741.

    Article  Google Scholar 

  • Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57:97–109.

    Article  MathSciNet  Google Scholar 

  • Hájek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, New York.

    MATH  Google Scholar 

  • Jarque, C. and Bera, A. (1987). A test of normality of observations and regression residuals. International Statistical Review, 55(2):163–172.

    Article  MathSciNet  Google Scholar 

  • Kendall, M. and Stuart, A. (1979). The Advanced Theory of Statistics, volume 2. Griffin, London, fourth edition.

    Google Scholar 

  • Lehmann, E. (1975). Nonparametrics: Statistical Methods Based on Ranks. McGraw-Hill, New York.

    MATH  Google Scholar 

  • Liang, F., Liu, C., and Carroll, J. D. (2010). Advanced Markov Chain Monte Carlo Methods. John Wiley & Sons.

    Book  Google Scholar 

  • Lindsay, B. G. and Qu, A. (2003). Inference functions and quadratic score tests. Statist. Sci., 18(3):394–410.

    Article  MathSciNet  Google Scholar 

  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21:1087–1092.

    Article  Google Scholar 

  • Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer, New York, 2nd edition.

    Book  Google Scholar 

  • Serfling, Robert, J. (2009). Approximating Theorems of Mathematical Statistics. John Wiley and Sons.

    Google Scholar 

  • Stein, C. (1956). Efficient nonparametric testing and estimation. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pages 187–195, Berkeley, Calif. University of California Press.

    Google Scholar 

  • Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion and rejoinder). Annals of Statistics, 22(4):1701–1762.

    Article  MathSciNet  Google Scholar 

  • van der Vaart, A. (2007). Asymptotic Statistics. Cambridge University Press.

    Google Scholar 

  • Varin, C., Reid, N., and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21:5–42.

    MathSciNet  MATH  Google Scholar 

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Alvo, M., Yu, P.L.H. (2018). Fundamental Concepts in Parametric Inference. In: A Parametric Approach to Nonparametric Statistics. Springer Series in the Data Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-94153-0_2

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