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Approximate and asymptotic distributions of Chi-squared-type mixtures with applications. (English) Zbl 1117.62460
Summary: We study how to approximate a random variable $$T$$ of general chi-squared-type mixtures by a random variable of the form $$\alpha \chi_{d}^{2} + \beta$$ via matching the first three cumulants. The approximation error bounds for the density functions of the chi-squared approximation and the normal approximation are established. Applications of the results to some nonparametric goodness-of-fit tests, including those tests based on orthogonal series, smoothing splines, and local polynomial smoothers, are investigated.Two simulation studies are conducted to compare the chi-squared approximation and the normal approximation numerically. The chi-squared approximation is illustrated using a real data example for polynomial goodness-of-fit tests.

MSC:
 62-XX Statistics
KernSmooth
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