PriMera Scientific Medicine and Public Health (ISSN: 2833-5627)

Review Article

Volume 6 Issue 6

Generalized Regression Models with an Increasing Number of Unknown Parameters

Asaf Hajiyev* and Jiuping Xu

June 04, 2025

DOI : 10.56831/PSMPH-06-218

View PDF

Abstract

The paper considers the general form of regression models with an increasing number of unknown parameters and different and unknown random error variances. Such models are typical for applications and allow us to solve some practical problems that face difficulties. Linear and nonlinear regression models can be considered as a partial form of this model. The iterated process for calculating the least square estimators for generalized regression models is constructed. The approach for estimating the elements of the covariance matrix of the deviation vector is suggested. Using these results, the method of constructing a confident band for unknown functions in regression models is suggested.

Keywords: generalized regression model; increasing number of unknown parameters; least square estimator; unknown variances of random errors; design matrix; Fisher’s matrix; iterated process

References

  1. Aldrich J. “Fisher and Regression”. Statist. Sci 20.4 (2005): 401-417.
  2. Demidenko E. “Mixed Models: Theory and Applications with R”. Wiley Series in Probability and Statistics (1981).
  3. Galton F. “Typical laws of heredity”. Nature 15 (1877): 492-495, 512- 514, 532-533.
  4. Hajiyev AH. “Regression Models with Increasing Numbers of Unknown Parameters”. In: International Encyclopedia. Statist. Sciences, Springer (2011): 1208-1211.
  5. Hajiyev AH. “Nonlinear Regression Models with Increasing Numbers of Unknown Parameters”. Russian Acad. Sci., Mathem 79.3 (2009): 339-341.
  6. Hajiyev AH. “Linear Regression Models with increasing numbers of unknown Parameters”. Russian Acad.Sci., Doklady, Mathematics 399.3 (2004): 1-5.
  7. Huber PJ. “Robust regression: Asymptotics, conjectures and Monte Carlo”. The Annals of Statistics 1 (1973): 799-821.
  8. Legendre A.-M On Least Squares. Translated from the French by Professor Henry A Ruger and Professor Helen M Walker, Teachers College, Columbia University, New York City (1805).
  9. Prohorov YuV and Rozanov YuV. “Probability theory. Basic concepts-limit theorems-random processes”. Springer (1969).
  10. Rao CR. “Linear Statistical Inference and its Applications”. John Wiley Series in Probability and Statistics (1973).
  11. Seber GF. “Linear Regression Analysis”. NY, Wiley Ser. Probab. and Statistics (2003).
  12. Sheinin O. “Gauss and the method least squares”. Jahrbucher fur Nationalokonomie und statistic 219.3+4 (1999): 458-467.
  13. Shiryaev AN. Probability. Springer-Verlag, New York-Berlin- Heidelberg (2013): 580.
  14. Sun Q, Zhou WX and Fan J. “Adaptive Huber Regression”. Am Stat. Assoc 115.529 (2020): 254-265.
  15. Stanton JM. “Galton, Pearson, and the Peas: A brief history of linear regression for statistics instructors”. Journal of Statist. Education 9.3 (2001).
  16. Wand MP. “Comparison of Regression Spline Smoothing Procedures”. Computational Statistics 15 (2000): 443-462.
  17. Wang L, Wu Y and Li R. “Quantile regression for analyzing heterogeneity in ultra-high dimension”. Journal of the American Statistical Association 107 (2012): 214-222.
  18. Wu CF. “Jacknife, bootstrap, and other resampling methods in regression analysis”. Ann. Statist 14.4 (1986): 1261-1350.
  19. Wu CFJ and Hamada MS. “Experiment. Planning, Analysis and Optimization”. Wiley, Ser. Probability and Statistics (2009).
  20. Yao F, Muller H-G and Wang J-L. “Functional linear regression analysis for longitudinal data”. Ann. Statist 33.6 (2005): 2873-2903.
  21. Yule GU and Pearson K. “On theory of consistence of logical class- frequences, and its geometrical representation”. "Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (1997): 287-299.
  22. Zaks Sh. “Theory of statistical inference”. John Wiley Sons (1972).
  23. Zapf A, Wiessner C and König IR. “Regression Analysis and Their Particularities in Observational Studies”. Dtsch Arztebl Int 121.4 (2024): 128-134.