ISSN 0253-2778

CN 34-1054/N

open
Free to Publish. Free to Read.
Open AccessOpen Access JUSTC Original Paper

Sequential shrinkage estimation in generalized linear models with measurement errors

  • 1.

    Dept. of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China

  • 2.

    School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China

Funds:  Supported by NNSF of China (11231010, 11471302), the Fundamental Research Funds for the Central Universities (WK2040000010).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.06.004
More Information
  • Author Bio:

    LU Haibo, male, born in 1980, master. Research field: sequential analysis and survival analysis.

  • Corresponding author: WANG Zhanfeng
  • Received Date: 08 December 2014
  • Accepted Date: 13 April 2015
  • Rev Recd Date: 13 April 2015
  • Publish Date: 30 June 2015
    Abstract

  • Abstract

    A sequential shrinkage estimation method was developed to determine a minimum sample size under which both of the variable selection and the parameter estimation with a pre-specified accuracy were achieved for the generalized linear model with measurement errors. Asymptotic properties of the proposed sequential estimation method, such as the coverage probability of the sequential confidence set and the efficiency of the minimum sample size, were studied. Simulation studies were conducted and the results show that the proposed method can save a large number of samples compared to the traditional sequential sampling method. Finally a diabetes data set was used as an example.

    Abstract

    A sequential shrinkage estimation method was developed to determine a minimum sample size under which both of the variable selection and the parameter estimation with a pre-specified accuracy were achieved for the generalized linear model with measurement errors. Asymptotic properties of the proposed sequential estimation method, such as the coverage probability of the sequential confidence set and the efficiency of the minimum sample size, were studied. Simulation studies were conducted and the results show that the proposed method can save a large number of samples compared to the traditional sequential sampling method. Finally a diabetes data set was used as an example.
    • FullText(HTML)
  • loading
    • References(19)
  • [1]
    Nelder J A, Wedderburn R W M. Generalized linear models[J]. Journal of the Royal Statistical Society, Series A, 1972, 135(3): 370-384.
    [2]
    Fahrmeir L, Kaufmann H. Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models[J]. The Annals of Statistics, 1985, 13(1): 342-368.
    [3]
    McCullagh P, Nelder J A. Generalized Linear Models[M].2nd ed. New York: Chapman & Hall, 1989.
    [4]
    Chen K, Hu I, Ying Z. Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs[J]. The Annals of Statistics, 1999, 27(4): 1 155-1 163.
    [5]
    Wang Z F, Chang Y C I. Sequential estimate for linear regression models with uncertain number of effective variables[J]. Metrika, 2013, 76:949-978.
    [6]
    Tibshirani R. Regression shrinkage and selection via the lasso[J]. Journal of the Royal Statistical Society, Series B, 1996, 58(1): 267-288.
    [7]
    Efron B, Hastie T, Johnstone I, et al. Least angle regression[J]. The Annals of Statistics, 2004, 32(2): 407-499.
    [8]
    Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties[J]. Journal of the American Statistical Association, 2001, 96(456): 1 348-1 360.
    [9]
    Chang Y C I. Sequential confidence regions of generalized linear models with adaptive designs[J]. Journal of Statistical Planning and Inference, 2001, 93: 277-293.
    [10]
    Lu H B, Wang Z F, Wu Y H. Sequential estimate for generalized linear models with uncertain number of effective variables[J]. Journal of Systems Science and Complexity, 2015, 28(2): 424-438.
    [11]
    Chang Y C I. Sequential estimation in generalized linear models when covariates are subject to errors[J]. Metrika, 2011, 73: 93-120.
    [12]
    Chow Y S, Robbins H. On the asymptotic theory of fixed-width sequential confidence intervals for the mean[J]. The Annals of Mathematical Statistics, 1965, 36(2): 457-462.
    [13]
    Chang Y C I. Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time[J]. Statistics & Probability Letters, 1999, 45(3): 237-246.
    [14]
    Siegmund D. Sequential Analysis: Tests and Confidence Intervals[M]. New York: Springer-Verlag, 1985.
    [15]
    Whitehead J. The Design and Analysis of Sequential Clinical Trials[M]. 2nd ed. New York: John Wiley & Sons, 1997.
    [16]
    Anscombe F J. Large sample theory of sequential estimation[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1952, 48(4): 600-607.
    [17]
    Woodroofe M. Nonlinear Renewal Theory in Sequential Analysis[M]. Philadelphia: Society for Industrial and Applied Mathematics, 1982.
    [18]
    Willems J, Saunders J, Hunt D, et al. Prevalence of coronary heart disease risk factors among rural blacks: A community based study[J]. Southern Medical Journal, 1997, 90(8):814-820.
    [19]
    Lai T L, Wei C Z. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems[J]. The Annals of Statistics, 1982, 10(1): 154-166.
    • Supplements(0)
    • Related Articles
    • Track Citations
  • 加载中

Catalog

    |
    Get Citation
    PDF
    XML
    [1]
    Nelder J A, Wedderburn R W M. Generalized linear models[J]. Journal of the Royal Statistical Society, Series A, 1972, 135(3): 370-384.
    [2]
    Fahrmeir L, Kaufmann H. Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models[J]. The Annals of Statistics, 1985, 13(1): 342-368.
    [3]
    McCullagh P, Nelder J A. Generalized Linear Models[M].2nd ed. New York: Chapman & Hall, 1989.
    [4]
    Chen K, Hu I, Ying Z. Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs[J]. The Annals of Statistics, 1999, 27(4): 1 155-1 163.
    [5]
    Wang Z F, Chang Y C I. Sequential estimate for linear regression models with uncertain number of effective variables[J]. Metrika, 2013, 76:949-978.
    [6]
    Tibshirani R. Regression shrinkage and selection via the lasso[J]. Journal of the Royal Statistical Society, Series B, 1996, 58(1): 267-288.
    [7]
    Efron B, Hastie T, Johnstone I, et al. Least angle regression[J]. The Annals of Statistics, 2004, 32(2): 407-499.
    [8]
    Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties[J]. Journal of the American Statistical Association, 2001, 96(456): 1 348-1 360.
    [9]
    Chang Y C I. Sequential confidence regions of generalized linear models with adaptive designs[J]. Journal of Statistical Planning and Inference, 2001, 93: 277-293.
    [10]
    Lu H B, Wang Z F, Wu Y H. Sequential estimate for generalized linear models with uncertain number of effective variables[J]. Journal of Systems Science and Complexity, 2015, 28(2): 424-438.
    [11]
    Chang Y C I. Sequential estimation in generalized linear models when covariates are subject to errors[J]. Metrika, 2011, 73: 93-120.
    [12]
    Chow Y S, Robbins H. On the asymptotic theory of fixed-width sequential confidence intervals for the mean[J]. The Annals of Mathematical Statistics, 1965, 36(2): 457-462.
    [13]
    Chang Y C I. Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time[J]. Statistics & Probability Letters, 1999, 45(3): 237-246.
    [14]
    Siegmund D. Sequential Analysis: Tests and Confidence Intervals[M]. New York: Springer-Verlag, 1985.
    [15]
    Whitehead J. The Design and Analysis of Sequential Clinical Trials[M]. 2nd ed. New York: John Wiley & Sons, 1997.
    [16]
    Anscombe F J. Large sample theory of sequential estimation[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1952, 48(4): 600-607.
    [17]
    Woodroofe M. Nonlinear Renewal Theory in Sequential Analysis[M]. Philadelphia: Society for Industrial and Applied Mathematics, 1982.
    [18]
    Willems J, Saunders J, Hunt D, et al. Prevalence of coronary heart disease risk factors among rural blacks: A community based study[J]. Southern Medical Journal, 1997, 90(8):814-820.
    [19]
    Lai T L, Wei C Z. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems[J]. The Annals of Statistics, 1982, 10(1): 154-166.
    Recommended articles
    TrendMD
    Volume 45 Issue 6 page: 449-459
    Cover

    Article Metrics

    Article views (38) PDF downloads(82)
    Proportional views

    Copyright © Editorial Office of JUSTC, All Rights Reserved. 皖ICP备05002528号

    Supported by: Beijing Renhe Information Technology Co. Ltd    

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return