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Likelihood ratio tests for positivity in polynomial regressions

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  • Kato, Naohiro
  • Kuriki, Satoshi

Abstract

A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone K. In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial. By considering hierarchical hypotheses including the hypothesis of positivity, we define nested likelihood ratio tests, and derive their null distributions as mixtures of chi-square distributions by using the volume-of-tubes method. The mixing probabilities are obtained by utilizing the parameterizations for the cone K and its dual provided in the framework of Tchebycheff systems for polynomials of degree at most 4. For polynomials of degree greater than 4, the upper and lower bounds for the null distributions are provided. Moreover, we propose associated simultaneous confidence bounds for polynomial regression curves. Regarding computation, we demonstrate that symmetric cone programming is useful to obtain the test statistics. As an illustrative example, we conduct data analysis on growth curves of two groups. We examine the hypothesis that the growth rate (the derivative of growth curve) of one group is always higher than the other.

Suggested Citation

  • Kato, Naohiro & Kuriki, Satoshi, 2013. "Likelihood ratio tests for positivity in polynomial regressions," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 334-346.
    • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:334-346
    DOI: 10.1016/j.jmva.2012.10.016
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    1. NESTEROV, Yu., 2000. "Squared functional systems and optimization problems," LIDAM Reprints CORE 1472, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. W. Liu & F. Bretz & A. J. Hayter & H. P. Wynn, 2009. "Assessing Nonsuperiority, Noninferiority, or Equivalence When Comparing Two Regression Models Over a Restricted Covariate Region," Biometrics, The International Biometric Society, vol. 65(4), pages 1279-1287, December.
    3. Satoshi Kuriki & Akimichi Takemura, 2000. "Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X 2 Distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(1), pages 1-14, March.
    4. Kato, Naohiro & Yamada, Takayuki & Fujikoshi, Yasunori, 2010. "High-dimensional asymptotic expansion of LR statistic for testing intraclass correlation structure and its error bound," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 101-112, January.
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    Cited by:

    1. Kuriki, Satoshi & Takemura, Akimichi & Taylor, Jonathan E., 2022. "The volume-of-tube method for Gaussian random fields with inhomogeneous variance," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. Sophia Rosen & Ori Davidov, 2017. "Ordered Regressions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 817-842, December.

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