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Estimating functions in indirect inference

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  • Knut Heggland
  • Arnoldo Frigessi

Abstract

Summary. There are models for which the evaluation of the likelihood is infeasible in practice. For these models the Metropolis–Hastings acceptance probability cannot be easily computed. This is the case, for instance, when only departure times from a G/G/1 queue are observed and inference on the arrival and service distributions are required. Indirect inference is a method to estimate a parameter θ in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of θ. First an auxiliary model is chosen whose parameter β can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate . The parameter β is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter θ. Finally, the parameter θ which leads to the best match to is chosen as the indirect inference estimate. We analyse which properties an auxiliary model should have to give satisfactory indirect inference. We look at the situation where the data are summarized in a vector statistic T, and the auxiliary model is chosen so that inference on β is drawn from T only. Under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of T and componentwise inversely proportional to the square of the derivative, with respect to θ, of the expected value of T. We discuss how these results can be used in selecting good estimating functions. We apply our findings to the queuing problem.

Suggested Citation

  • Knut Heggland & Arnoldo Frigessi, 2004. "Estimating functions in indirect inference," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 447-462, May.
  • Handle: RePEc:bla:jorssb:v:66:y:2004:i:2:p:447-462
    DOI: 10.1111/j.1369-7412.2003.05341.x
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    References listed on IDEAS

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    1. Monica Billio & Alain Monfort & Christian P, Robert, 1998. "The Simulated Likelihood Ratio (SLR) Method," Working Papers 98-21, Center for Research in Economics and Statistics.
    2. Jean-David Fermanian & Bernard Salanié, 2001. "A Nonparametric Simulated Maximum Likelihood Estimation Method," Working Papers 2001-13, Center for Research in Economics and Statistics.
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    Citations

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    Cited by:

    1. Forneron, Jean-Jacques & Ng, Serena, 2018. "The ABC of simulation estimation with auxiliary statistics," Journal of Econometrics, Elsevier, vol. 205(1), pages 112-139.
    2. Lombardi, Marco J. & Calzolari, Giorgio, 2009. "Indirect estimation of [alpha]-stable stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2298-2308, April.
    3. Aushev, Alexander & Pesonen, Henri & Heinonen, Markus & Corander, Jukka & Kaski, Samuel, 2022. "Likelihood-free inference with deep Gaussian processes," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    4. Laurent-Emmanuel Calvet & Veronika Czellar, 2011. "State-Observation Sampling and the Econometrics of Learning Models," Working Papers hal-00625500, HAL.
    5. Frazier, David T. & Oka, Tatsushi & Zhu, Dan, 2019. "Indirect inference with a non-smooth criterion function," Journal of Econometrics, Elsevier, vol. 212(2), pages 623-645.
    6. Stéphane Guerrier & Samuel Orso & Maria-Pia Victoria-Feser, 2018. "Parametric Inference for Index Functionals," Econometrics, MDPI, vol. 6(2), pages 1-11, April.
    7. Xu, Yihuan & Iglewicz, Boris & Chervoneva, Inna, 2014. "Robust estimation of the parameters of g-and-h distributions, with applications to outlier detection," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 66-80.
    8. Anthony Ebert & Ritabrata Dutta & Kerrie Mengersen & Antonietta Mira & Fabrizio Ruggeri & Paul Wu, 2021. "Likelihood‐free parameter estimation for dynamic queueing networks: Case study of passenger flow in an international airport terminal," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(3), pages 770-792, June.
    9. C. C. Drovandi & A. N. Pettitt, 2011. "Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation," Biometrics, The International Biometric Society, vol. 67(1), pages 225-233, March.
    10. Raknerud, Arvid & Skare, Øivind, 2012. "Indirect inference methods for stochastic volatility models based on non-Gaussian Ornstein–Uhlenbeck processes," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3260-3275.
    11. Di Iorio, Francesca & Calzolari, Giorgio, 2006. "Discontinuities in indirect estimation: An application to EAR models," Computational Statistics & Data Analysis, Elsevier, vol. 50(8), pages 2124-2136, April.
    12. D.T. Frazier & G.M. Martin & C.P. Robert & J. Rousseau, 2016. "Asymptotic Properties of Approximate Bayesian Computation," Monash Econometrics and Business Statistics Working Papers 18/16, Monash University, Department of Econometrics and Business Statistics.
    13. Anna Gottard & Giorgio Calzolari, 2014. "Alternative estimating procedures for multiple membership logit models with mixed effects: indirect inference and data cloning," Econometrics Working Papers Archive 2014_07, Universita' degli Studi di Firenze, Dipartimento di Statistica, Informatica, Applicazioni "G. Parenti".
    14. Bertl Johanna & Ewing Gregory & Kosiol Carolin & Futschik Andreas, 2017. "Approximate maximum likelihood estimation for population genetic inference," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 16(5-6), pages 291-312, December.
    15. Stefano Cabras & María Castellanos & Erlis Ruli, 2014. "A Quasi likelihood approximation of posterior distributions for likelihood-intractable complex models," METRON, Springer;Sapienza Università di Roma, vol. 72(2), pages 153-167, August.
    16. Gael M. Martin & Brendan P.M. McCabe & Worapree Maneesoonthorn & Christian P. Robert, 2014. "Approximate Bayesian Computation in State Space Models," Monash Econometrics and Business Statistics Working Papers 20/14, Monash University, Department of Econometrics and Business Statistics.
    17. Calvet, Laurent E. & Czellar, Veronika, 2015. "Through the looking glass: Indirect inference via simple equilibria," Journal of Econometrics, Elsevier, vol. 185(2), pages 343-358.
    18. Efstathios Panayi & Gareth Peters, 2015. "Stochastic simulation framework for the Limit Order Book using liquidity motivated agents," Papers 1501.02447, arXiv.org, revised Jan 2015.
    19. Efstathios Panayi & Gareth W. Peters, 2015. "Stochastic simulation framework for the limit order book using liquidity-motivated agents," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-52.
    20. Gael M. Martin & Brendan P.M. McCabe & David T. Frazier & Worapree Maneesoonthorn & Christian P. Robert, 2016. "Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models," Monash Econometrics and Business Statistics Working Papers 09/16, Monash University, Department of Econometrics and Business Statistics.
    21. David T. Frazier & Gael M. Martin & Christian P. Robert, 2015. "On Consistency of Approximate Bayesian Computation," Monash Econometrics and Business Statistics Working Papers 19/15, Monash University, Department of Econometrics and Business Statistics.
    22. Thomas A. Dean & Sumeetpal S. Singh & Ajay Jasra & Gareth W. Peters, 2014. "Parameter Estimation for Hidden Markov Models with Intractable Likelihoods," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 970-987, December.
    23. Drovandi, Christopher C. & Pettitt, Anthony N., 2011. "Likelihood-free Bayesian estimation of multivariate quantile distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(9), pages 2541-2556, September.

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