scholarly journals Universal inference

2020 ◽  
Vol 117 (29) ◽  
pp. 16880-16890 ◽  
Author(s):  
Larry Wasserman ◽  
Aaditya Ramdas ◽  
Sivaraman Balakrishnan
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Model Misspecification ◽  
Null Distribution ◽  
Regularity Conditions ◽  
Ratio Test ◽  
Finite Sample ◽  
Confidence Sets ◽  
Nonparametric Models

We propose a general method for constructing confidence sets and hypothesis tests that have finite-sample guarantees without regularity conditions. We refer to such procedures as “universal.” The method is very simple and is based on a modified version of the usual likelihood-ratio statistic that we call “the split likelihood-ratio test” (split LRT) statistic. The (limiting) null distribution of the classical likelihood-ratio statistic is often intractable when used to test composite null hypotheses in irregular statistical models. Our method is especially appealing for statistical inference in these complex setups. The method we suggest works for any parametric model and also for some nonparametric models, as long as computing a maximum-likelihood estimator (MLE) is feasible under the null. Canonical examples arise in mixture modeling and shape-constrained inference, for which constructing tests and confidence sets has been notoriously difficult. We also develop various extensions of our basic methods. We show that in settings when computing the MLE is hard, for the purpose of constructing valid tests and intervals, it is sufficient to upper bound the maximum likelihood. We investigate some conditions under which our methods yield valid inferences under model misspecification. Further, the split LRT can be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid P values and confidence sequences. Finally, when combined with the method of sieves, it can be used to perform model selection with nested model classes.

On the Asymptotic Non-Null Distribution of Randomly Stopped Log-Likelihood Ratio Statistic

1992 ◽  
Vol 42 (3-4) ◽  
pp. 255-260 ◽  
Author(s):  
A.K. Basu ◽  
Debasis Bhattacharya
Keyword(s):  
Likelihood Ratio ◽  
Alternative Hypothesis ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Regularity Conditions ◽  
Normal Distributions ◽  
Log Likelihood ◽  
Log Likelihood Ratio ◽  
Mixture Of Normal Distributions ◽  
Loglikelihood Ratio

Non-null asymptotic distribution of randomly stopped loglikelihood ratio statistic for general dependent process has been obtained. We observed that under certain regularity conditions the limiting distribution of randomly stopped log-likelihood ratio statistic under alternative hypothesis is again a mixture of Normal distributions. Some possible applications of the result has also been pointed out in this note.

Statistical inference for Markov processes when the model is incorrect

1979 ◽  
Vol 11 (04) ◽  
pp. 737-749
Author(s):  
Robert V. Foutz ◽  
R. C. Srivastava
Keyword(s):  
Maximum Likelihood ◽  
Statistical Inference ◽  
Likelihood Ratio ◽  
Markov Processes ◽  
Transition Probability ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Transition Density ◽  
Likelihood Method ◽  
Incorrect Model

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λ n are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λ n are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λ n is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λ n when the statistic is constructed from an incorrect model.

Testing and Estimation in the Block Compound Symmetry Problem

1982 ◽  
Vol 7 (1) ◽  
pp. 3-18 ◽  
Author(s):  
Ted H. Szatrowski
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Covariance Matrices ◽  
Maximum Likelihood Estimates ◽  
Testing Problem ◽  
Compound Symmetry ◽  
Estimation Problems ◽  
Symmetry Problem

Known results for testing and estimation problems for patterned means and covariance matrices with explicit linear maximum likelihood estimates are applied to the block compound symmetry problem. New results given include the constants for G.P.E. Box’s approximate null distribution of the likelihood ratio statistic. These techniques are applied to the analysis of an educational testing problem.

scholarly journals PEPA test: fast and powerful differential analysis from relative quantitative proteomics data using shared peptides

Biostatistics ◽  
2018 ◽  
Vol 20 (4) ◽  
pp. 632-647
Author(s):  
Laurent Jacob ◽  
Florence Combes ◽  
Thomas Burger
Keyword(s):  
Likelihood Ratio ◽  
Linear Time ◽  
Hypothesis Test ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Enzymatic Digestion ◽  
Ratio Test ◽  
Differential Analysis ◽  
Proteomics Data ◽  
Differential Abundance

Summary We propose a new hypothesis test for the differential abundance of proteins in mass-spectrometry based relative quantification. An important feature of this type of high-throughput analyses is that it involves an enzymatic digestion of the sample proteins into peptides prior to identification and quantification. Due to numerous homology sequences, different proteins can lead to peptides with identical amino acid chains, so that their parent protein is ambiguous. These so-called shared peptides make the protein-level statistical analysis a challenge and are often not accounted for. In this article, we use a linear model describing peptide–protein relationships to build a likelihood ratio test of differential abundance for proteins. We show that the likelihood ratio statistic can be computed in linear time with the number of peptides. We also provide the asymptotic null distribution of a regularized version of our statistic. Experiments on both real and simulated datasets show that our procedures outperforms state-of-the-art methods. The procedures are available via the pepa.test function of the DAPAR Bioconductor R package.

scholarly journals PEPA test: fast and powerful differential analysis from relative quantitative proteomics data using shared peptides

2017 ◽  
Author(s):  
Laurent Jacob ◽  
Florence Combes ◽  
Thomas Burger
Keyword(s):  
Likelihood Ratio ◽  
Linear Time ◽  
Hypothesis Test ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Enzymatic Digestion ◽  
Ratio Test ◽  
Differential Analysis ◽  
Proteomics Data ◽  
Differential Abundance

AbstractWe propose a new hypothesis test for the differential abundance of proteins in mass-spectrometry based relative quantification. An important feature of this type of high-throughput analyses is that it involves an enzymatic digestion of the sample proteins into peptides prior to identification and quantification. Due to numerous homology sequences, different proteins can lead to peptides with identical amino acid chains, so that their parent protein is ambiguous. These so-called shared peptides make the protein-level statistical analysis a challenge, so that they are often not accounted for. In this article, we use a linear model describing peptide-protein relationships to build a likelihood ratio test of differential abundance for proteins. We show that the likelihood ratio statistic can be computed in linear time with the number of peptides. We also provide the asymptotic null distribution of a regularized version of our statistic. Experiments on both real and simulated datasets show that our procedures outperforms state-of-the-art methods. The procedures are available via the pepa.test function of the DAPAR Bioconductor R package.

Statistical inference for Markov processes when the model is incorrect

1979 ◽  
Vol 11 (4) ◽  
pp. 737-749 ◽  
Author(s):  
Robert V. Foutz ◽  
R. C. Srivastava
Keyword(s):  
Maximum Likelihood ◽  
Statistical Inference ◽  
Likelihood Ratio ◽  
Markov Processes ◽  
Transition Probability ◽  
Likelihood Ratio Statistic ◽  
Null Distribution ◽  
Transition Density ◽  
Likelihood Method ◽  
Incorrect Model

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.

scholarly journals Monitoring Persistence Change in Heavy-Tailed Observations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 936
Author(s):  
Dan Wang
Keyword(s):  
Kernel Method ◽  
Alternative Hypothesis ◽  
Null Distribution ◽  
Real Data ◽  
Ratio Test ◽  
Finite Sample ◽  
Test Statistic ◽  
Bootstrap Approximation ◽  
Heavy Tailed ◽  
Better Than

In this paper, a ratio test based on bootstrap approximation is proposed to detect the persistence change in heavy-tailed observations. This paper focuses on the symmetry testing problems of I(1)-to-I(0) and I(0)-to-I(1). On the basis of residual CUSUM, the test statistic is constructed in a ratio form. I prove the null distribution of the test statistic. The consistency under alternative hypothesis is also discussed. However, the null distribution of the test statistic contains an unknown tail index. To address this challenge, I present a bootstrap approximation method for determining the rejection region of this test. Simulation studies of artificial data are conducted to assess the finite sample performance, which shows that our method is better than the kernel method in all listed cases. The analysis of real data also demonstrates the excellent performance of this method.

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