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Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into quotient and remainder, polynomial long division partitions a polynomial into simpler components; in this context, the dividend is the polynomial being divided, the divisor is the expression dividing it, and the result is expressed in terms of a quotient and a remainder.The division begins by arranging the dividend and divisor in standard...
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Polynomial regression with heteroscedastic measurement errors in both axes: Estimation and hypothesis testing.

Chi-Lun Cheng1, Jia-Ren Tsai2, Hans Schneeweiss3

  • 11 Institute of Statistical Science, Academia Sinica, Taiwan, Republic of China.

Statistical Methods in Medical Research
|July 11, 2018
PubMed
Summary

This study introduces new methods for polynomial regression with measurement errors in both variables. The adjusted least squares method and hypothesis tests are developed and validated with simulations and real data.

Keywords:
Adjusted least squaresequation errorheteroscedasticityhypothesis testingmeasurement error modelpolynomial model

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Area of Science:

  • Statistics
  • Econometrics
  • Biostatistics

Background:

  • Measurement errors in regression models can bias results.
  • Heteroscedasticity, where error variance changes, complicates analysis.
  • Existing methods may not handle errors in both response and regressor variables simultaneously.

Purpose of the Study:

  • To develop robust methods for point estimation in polynomial regression with heteroscedastic measurement errors.
  • To propose and evaluate hypothesis testing procedures for such models.
  • To provide practical applications using real-world data.

Main Methods:

  • Development of adjusted least squares methods and their modifications.
  • Adaptation of methods for functional, structural, and models with/without equation error.
  • Application of Wald-type and score-type tests for hypothesis testing.

Main Results:

  • The proposed adjusted least squares methods effectively handle heteroscedastic measurement errors.
  • Simulation studies demonstrate the performance of the developed estimation and testing procedures.
  • Real data applications showcase the practical utility of the statistical techniques.

Conclusions:

  • The developed methods offer reliable solutions for polynomial regression with complex error structures.
  • The study contributes to robust statistical inference in the presence of measurement errors.
  • This research provides valuable tools for researchers dealing with data affected by heteroscedastic measurement errors.