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Register a new userGaussian Universal Likelihood Ratio Testing
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Added by
Robin Dunn
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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The likelihood ratio test (LRT) based on the asymptotic chi-squared distribution of the log likelihood is one of the fundamental tools of statistical inference. A recent universal LRT approach based on sample splitting provides valid hypothesis tests and confidence sets in any setting for which we can compute the split likelihood ratio statistic (or, more generally, an upper bound on the null maximum likelihood). The universal LRT is valid in finite samples and without regularity conditions. This test empowers statisticians to construct tests in settings for which no valid hypothesis test previously existed. For the simple but fundamental case of testing the population mean of d-dimensional Gaussian data, the usual LRT itself applies and thus serves as a perfect test bed to compare against the universal LRT. This work presents the first in-depth exploration of the size, power, and relationships between several universal LRT variants. We show that a repeated subsampling approach is the best choice in terms of size and power. We observe reasonable performance even in a high-dimensional setting, where the expected squared radius of the best universal LRT confidence set is approximately 3/2 times the squared radius of the standard LRT-based set. We illustrate the benefits of the universal LRT through testing a non-convex doughnut-shaped null hypothesis, where a universal inference procedure can have higher power than a standard approach.
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The complexity underlying real-world systems implies that standard statistical hypothesis testing methods may not be adequate for these peculiar applications. Specifically, we show that the likelihood-ratio tests null-distribution needs to be modified to accommodate the complexity found in multi-edge network data. When working with independent observations, the p-values of likelihood-ratio tests are approximated using a $chi^2$ distribution. However, such an approximation should not be used when dealing with multi-edge network data. This type of data is characterized by multiple correlations and competitions that make the standard approximation unsuitable. We provide a solution to the problem by providing a better approximation of the likelihood-ratio test null-distribution through a Beta distribution. Finally, we empirically show that even for a small multi-edge network, the standard $chi^2$ approximation provides erroneous results, while the proposed Beta approximation yields the correct p-value estimation.
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The ratio between two probability density functions is an important component of various tasks, including selection bias correction, novelty detection and classification. Recently, several estimators of this ratio have been proposed. Most of these methods fail if the sample space is high-dimensional, and hence require a dimension reduction step, the result of which can be a significant loss of information. Here we propose a simple-to-implement, fully nonparametric density ratio estimator that expands the ratio in terms of the eigenfunctions of a kernel-based operator; these functions reflect the underlying geometry of the data (e.g., submanifold structure), often leading to better estimates without an explicit dimension reduction step. We show how our general framework can be extended to address another important problem, the estimation of a likelihood function in situations where that function cannot be well-approximated by an analytical form. One is often faced with this situation when performing statistical inference with data from the sciences, due the complexity of the data and of the processes that generated those data. We emphasize applications where using existing likelihood-free methods of inference would be challenging due to the high dimensionality of the sample space, but where our spectral series method yields a reasonable estimate of the likelihood function. We provide theoretical guarantees and illustrate the effectiveness of our proposed method with numerical experiments.
This study aims to evaluate the performance of power in the likelihood ratio test for changepoint detection by bootstrap sampling, and proposes a hypothesis test based on bootstrapped confidence interval lengths. Assuming i.i.d normally distributed errors, and using the bootstrap method, the changepoint sampling distribution is estimated. Furthermore, this study describes a method to estimate a data set with no changepoint to form the null sampling distribution. With the null sampling distribution, and the distribution of the estimated changepoint, critical values and power calculations can be made, over the lengths of confidence intervals.
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