A variety of statistics based on sample spacings has been studied in the literature for testing goodness-of-fit to parametric distributions. To test the goodness-of-fit to a nonparametric class of univariate shape-constrained densities, including widely studied classes such as k-monotone and log-concave densities, a likelihood ratio test with a working alternative density estimate based on the spacings of the observations is considered, and is shown to be asymptotically normal and distribution-free under the null, consistent under fixed alternatives, and admits bootstrap calibration. The distribution-freeness under the null comes from the fact that the asymptotic dominant term depends only on a function of the spacings of transformed outcomes that are uniformly distributed. Applications and extensions of theoretical results in the literature of shape-constrained estimation are required to show that the average log-density ratio converges to zero at a faster rate than the sample spacing term under the null, and diverges under the alternatives. Numerical studies are conducted to demonstrate that the test is applicable to various classes of shape-constrained densities and has a good balance between type-I error control under the null and power under alternative distributions.

翻译：文献中已研究了多种基于样本间距的统计量，用于检验参数分布的拟合优度。为检验单变量形状约束密度（包括广泛研究的k-单调密度和对数凹密度等类别）这一非参数类别的拟合优度，本文考虑采用基于观测值间距的备择密度估计的似然比检验。该检验在原假设下具有渐近正态性和分布无关性，在固定备择假设下具有相合性，并可采用自助法校准。原假设下的分布无关性源于其渐近主导项仅依赖于均匀分布变换结果间距的某个函数。需要应用和拓展形状约束估计文献中的理论结果以证明：在原假设下，平均对数密度比以快于样本间距项的速度收敛于零；在备择假设下，该比值发散。数值研究表明，该检验适用于各类形状约束密度，并在原假设下的I类错误控制与备择分布下的检验功效之间取得了良好平衡。