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Goodness‐of‐fit Tests Based on the Kernel Density Estimator

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  • RICARDO CAO
  • GÁBOR LUGOSI

Abstract

. Given an i.i.d. sample drawn from a density f on the real line, the problem of testing whether f is in a given class of densities is considered. Testing procedures constructed on the basis of minimizing the L1‐distance between a kernel density estimate and any density in the hypothesized class are investigated. General non‐asymptotic bounds are derived for the power of the test. It is shown that the concentration of the data‐dependent smoothing factor and the ‘size’ of the hypothesized class of densities play a key role in the performance of the test. Consistency and non‐asymptotic performance bounds are established in several special cases, including testing simple hypotheses, translation/scale classes and symmetry. Simulations are also carried out to compare the behaviour of the method with the Kolmogorov‐Smirnov test and an L2 density‐based approach due to Fan [Econ. Theory10 (1994) 316].

Suggested Citation

  • Ricardo Cao & Gábor Lugosi, 2005. "Goodness‐of‐fit Tests Based on the Kernel Density Estimator," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 599-616, December.
    • Handle: RePEc:bla:scjsta:v:32:y:2005:i:4:p:599-616
    DOI: 10.1111/j.1467-9469.2005.00471.x
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    Cited by:

    1. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    2. Carlos Tenreiro, 2022. "On automatic kernel density estimate-based tests for goodness-of-fit," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 717-748, September.
    3. Roca-Pardinas, Javier & Sperlich, Stefan, 2007. "Testing the link when the index is semiparametric--a comparative study," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6565-6581, August.
    4. Pablo Martínez-Camblor & Jacobo Uña-Álvarez, 2013. "Studying the bandwidth in $$k$$ -sample smooth tests," Computational Statistics, Springer, vol. 28(2), pages 875-892, April.
    5. Pavia, Jose M., 2015. "Testing Goodness-of-Fit with the Kernel Density Estimator: GoFKernel," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 66(c01).
    6. Graciela Boente & Daniela Rodriguez & Wenceslao González Manteiga, 2014. "Goodness-of-fit Test for Directional Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 259-275, March.

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