Skip to main content
SpringerLink
Log in

Analytical and Bootstrap Bias Corrections

  • Chapter
  • First Online:
An Introduction to Bartlett Correction and Bias Reduction

Part of the book series: SpringerBriefs in Statistics ((BRIEFSSTATIST))

  • 1515 Accesses

Abstract

Maximum likelihood estimators are usually biased: In finite samples, their expected value differs from the true parameter value. This is a systematic error. It typically vanishes as the sample size increases, but it can be large in small samples. Different strategies can be employed to reduce such systematic error. In this chapter, we present two analytical bias corrections. We also show how the bootstrap can be used to bias-correct estimators. Bias corrections in different statistical models are presented and discussed. In particular, we address the issue of bias-correcting covariance matrix estimators in heteroskedastic linear regressions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Bartlett, M. S. (1953). Confidence intervals II. Biometrika, 40, 306–317.

    MATH  MathSciNet  Google Scholar 

  • Botter, D. A., & Cordeiro, G. M. (1998). Improved estimators for generalized linear models with dispersion covariates. Journal of Statistical Computation and Simulation, 62, 91–104.

    Article  MATH  MathSciNet  Google Scholar 

  • Chesher, A., & Jewitt, I. (1987). The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica, 55, 1217–1222.

    Article  MATH  MathSciNet  Google Scholar 

  • Cook, D. R., Tsai, C. L., & Wei, B. C. (1986). Bias in nonlinear regression. Biometrika, 73, 615–623.

    Article  MATH  MathSciNet  Google Scholar 

  • Cordeiro, G. M., & McCullagh, P. (1991). Bias correction in generalized linear models. Journal of the Royal Statistical Society B, 53, 629–643.

    MATH  MathSciNet  Google Scholar 

  • Cordeiro, G. M., & Klein, R. (1994). Bias correction in ARMA models. Statistics and Probability Letters, 19, 169–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Cordeiro, G. M., Rocha, E. C., Rocha, J. G. C., & Cribari-Neto, F. (1997). Bias corrected maximum likelihood estimation for the beta distribution. Journal of Statistical Computation and Simulation, 58, 21–35.

    Article  MATH  Google Scholar 

  • Cordeiro, G. M., & Vasconcellos, K. L. P. (1997). Bias correction for a class of multivariate nonlinear regression models. Statistics and Probability Letters, 35, 155–164.

    Article  MATH  MathSciNet  Google Scholar 

  • Cordeiro, G. M., Ferrari, S. L. P., Uribe-Opazo, M. A., & Vasconcellos, K. L. P. (2000). Corrected maximum-likelihood estimation in a class of symmetric nonlinear regression models. Statistics and Probability Letters, 46, 317–328.

    Article  MATH  MathSciNet  Google Scholar 

  • Cox, D. R., & Snell, E. J. (1968). A general definition of residuals (with discussion). Journal of the Royal Statistical Society B, 30, 248–275.

    MATH  MathSciNet  Google Scholar 

  • Cox, D. R., & Hinkley, D. V. (1974). Theoretical Statistics. London: Chapman and Hall.

    Book  MATH  Google Scholar 

  • Cox, D. R., & Reid, N. (1987). Approximations to noncentral distributions. Canadian Journal of Statistics, 15, 105–114.

    Article  MATH  MathSciNet  Google Scholar 

  • Cribari-Neto, F., Ferrari, S. L. P., & Cordeiro, G. M. (2000). Improved heteroscedasticity-consistent covariance matrix estimators. Biometrika, 87, 907–918.

    Article  MATH  MathSciNet  Google Scholar 

  • Cribari-Neto, F., Frery, A. C., & Silva, M. F. (2002). Improved estimation of clutter properties in speckled imagery. Computational Statistics and Data Analysis, 40, 801–824.

    Article  MATH  MathSciNet  Google Scholar 

  • Cribari-Neto, F., & Vasconcellos, K. L. P. (2002). Nearly unbiased maximum likelihood estimation for the beta distribution. Journal of Statistical Computation and Simulation, 72, 107–118.

    Article  MATH  MathSciNet  Google Scholar 

  • Cribari-Neto, F., & Zeileis, A. (2010). Beta regressions in R. Journal of Statistical Software, 34(2), 1–24.

    Google Scholar 

  • Cribari-Neto, F., & Lima, M. G. A. (2011). A sequence of improved standard errors under heteroskedasticity of unknown form. Journal of Statistical Planning and Inference, 141, 3617–3627.

    Article  MATH  MathSciNet  Google Scholar 

  • Development Core Team, R. (2006). R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing.

    Google Scholar 

  • Doornik, J. A. (2009). An Object-Oriented Matrix Language Ox 6. London: Timberlake Consultants Press.

    Google Scholar 

  • Efron, B. (1979). Bootstrapping methods: Another look at the jackknife. Annals of Statistics, 7, 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  • Efron, B. (1990). More efficient bootstrap computations. Journal of the American Statistical Association, 85, 79–89.

    Article  MATH  MathSciNet  Google Scholar 

  • Ferrari, S. L. P., & Cribari-Neto, F. (2004). Beta regression for modeling rates and proportions. Journal of Applied Statistics, 31, 799–815.

    Article  MATH  MathSciNet  Google Scholar 

  • Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38.

    Article  MATH  MathSciNet  Google Scholar 

  • Giles, D. E. A. (2012). Bias reduction for the maximum likelihood estimator of the parameters in the half-logistic distribution. Communications in Statistics, Theory and Methods, 41, 212–222.

    Article  MATH  MathSciNet  Google Scholar 

  • Galea, M., Leiva, V., & Paula, G. A. (2004). Influence diagnostics in log-Birnbaum-Saunders regression models. Journal of Applied Statistics, 31, 1049–1064.

    Article  MATH  MathSciNet  Google Scholar 

  • Giles, D. E. A., & Feng, H. (2011). Reducing the bias of the maximum likelihood estimator for the Poisson regression model. Economics Bulletin, 31, 2933–2943.

    Google Scholar 

  • Giles, D. E. A., Feng, H., & Godwin, R. T. (2013). On the bias of the maximum likelihood estimator for the two-parameter Lomax distribution. Communications in Statistics, Theory and Methods, 42, 1934–1950.

    Article  MATH  MathSciNet  Google Scholar 

  • Lemonte, A. J., & Cordeiro, G. M. (2009). Birnbaum-Saunders nonlinear regression models. Computational Statistics and Data Analysis, 53, 4441–4452.

    Article  MATH  MathSciNet  Google Scholar 

  • Lemonte, A. J., & Cordeiro, G. M. (2010). Asymptotic skewness in Birnbaum-Saunders nonlinear regression models. Statistics and Probability Letters, 80, 892–898.

    Article  MATH  MathSciNet  Google Scholar 

  • Lemonte, A. J., Cribari-Neto, F., & Vasconcellos, K. L. P. (2007). Improved statistical inference for the two-parameter Birnbaum-Saunders distribution. Computational Statistics and Data Analysis, 51, 4656–4681.

    Article  MATH  MathSciNet  Google Scholar 

  • Lemonte, A. J., Simas, A. B., & Cribari-Neto, F. (2008). Bootstrap-based improved estimators for the two-parameter Birnbaum-Saunders distribution. Journal of Statistical Computation and Simulation, 78, 37–49.

    Article  MATH  MathSciNet  Google Scholar 

  • MacKinnon, J. G., & Smith, J. A. A. (1998). Approximate bias correction in econometrics. Journal of Econometrics, 85, 205–230.

    Article  MATH  MathSciNet  Google Scholar 

  • McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). London: Chapman and Hall.

    Book  MATH  Google Scholar 

  • Ospina, R., Cribari-Neto, F., & Vasconcellos, K. L. P. (2006). Improved point and interval estimation for a beta regression model. Computational Statistics and Data Analysis, 51, 960–981 (Errata: vol. 55, p. 2445, 2011).

    Google Scholar 

  • Patriota, A. G., & Lemonte, A. J. (2009). Bias correction in a multivariate regression model with general parameterization. Statistics and Probability Letters, 79, 1655–1662.

    Article  MATH  MathSciNet  Google Scholar 

  • Ratkowsky, D. A. (1983). Nonlinear Regression Modeling: A Unified Practical Approach. New York: Marcel Dekker.

    MATH  Google Scholar 

  • Rieck, J.R. (1989). Statistical analysis for the Birnbaum-Saunders fatigue life distribution. Ph.D. dissertation, Clemson University.

    Google Scholar 

  • Rieck, J. R., & Nedelman, J. R. (1991). A log-linear model for the Birnbaum-Saunders distribution. Technometrics, 33, 51–60.

    MATH  Google Scholar 

  • Simas, A. B., Barreto-Souza, W., & Rocha, A. V. (2010). Improved estimators for a general class of beta regression models. Computational Statistics and Data Analysis, 54, 348–366.

    Article  MATH  MathSciNet  Google Scholar 

  • Stosic, B., & Cordeiro, G. M. (2009). Using Maple and Mathematica to derive bias corrections for two parameter distributions. Journal of Statistical Computation and Simulation, 75, 409–423.

    MathSciNet  Google Scholar 

  • Vasconcellos, K. L. P., & Silva, S. G. (2005). Corrected estimates for Student \(t\) regression models with unknown degrees of freedom. Journal of Statistical Computation and Simulation, 79, 751–767.

    Google Scholar 

  • White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838.

    Article  MATH  MathSciNet  Google Scholar 

  • Young, D. H., & Bakir, S. T. (1987). Bias correction for a generalized log-gamma regression model. Technometrics, 29, 183–191.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Estatística, Universidade Federal de Pernambuco, Recife, PE, Brazil

    Gauss M. Cordeiro & Francisco Cribari-Neto

Authors
  1. Gauss M. Cordeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francisco Cribari-Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gauss M. Cordeiro .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Author(s)

About this chapter

Cite this chapter

Cordeiro, G.M., Cribari-Neto, F. (2014). Analytical and Bootstrap Bias Corrections. In: An Introduction to Bartlett Correction and Bias Reduction. SpringerBriefs in Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55255-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation