A note on the likelihood-ratio statistic under model misspecification

1998 ◽  
Vol 26 (1) ◽  
pp. 161-168 ◽  
Author(s):  
K. Viraswami ◽  
N. Reid
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Model Misspecification

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.

scholarly journals On the Correction of the Asymptotic Distribution of the Likelihood Ratio Statistic If Nuisance Parameters Are Estimated Based on an External Source

2014 ◽  
Vol 10 (2) ◽  
Author(s):  
Marianne Jonker ◽  
Aad Van der Vaart
Keyword(s):  
Confidence Interval ◽  
Asymptotic Distribution ◽  
Confidence Intervals ◽  
Likelihood Ratio ◽  
Statistical Models ◽  
Likelihood Ratio Statistic ◽  
External Source ◽  
Nuisance Parameters ◽  
Testing Hypotheses

AbstractIn practice, nuisance parameters in statistical models are often replaced by estimates based on an external source, for instance if estimates were published before or a second dataset is available. Next these estimates are assumed to be known when the parameter of interest is estimated, a hypothesis is tested or confidence intervals are constructed. By this assumption, the level of the test is, in general, higher than supposed and the coverage of the confidence interval is too low. In this article, we derive the asymptotic distribution of the likelihood ratio statistic if the nuisance parameters are estimated based on a dataset that is independent of the data used for estimating the parameter of interest. This distribution can be used for correctly testing hypotheses and constructing confidence intervals. Four theoretical and practical examples are given as illustration.

scholarly journals On sequential versions of the generalized likelihood ratio test

1979 ◽  
Vol 86 (1) ◽  
pp. 85-90 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Likelihood Ratio ◽  
Null Hypothesis ◽  
Exponential Family ◽  
Likelihood Ratio Statistic ◽  
Generalized Likelihood Ratio Test ◽  
Ratio Test ◽  
Uhlenbeck Process ◽  
Large Sample ◽  
Sequential Tests ◽  
Composite Hypotheses

AbstractIt is shown that the Wilks large sample likelihood ratio statistic λn, for testing between composite hypotheses Θ0 ⊂ Θ1 on the basis of a sample of size n, behaves as n varies like a diffusion process related to an equilibrium Ornstein-Uhlenbeck process, whenever the null hypothesis is true. This fact is used to construct large sample sequential tests based on λn, which are the same whatever the underlying distributions. In particular, the underlying distributions need not belong to an exponential family.

The Validation of a Test to Measure Knowledge about the Fat Content of Food Products

1996 ◽  
Vol 10 (4) ◽  
pp. 331-339 ◽  
Author(s):  
I.H.M. Steenhuis ◽  
J. Brug ◽  
P. Van Assema ◽  
Tj. Imbos
Keyword(s):  
Likelihood Ratio ◽  
Rasch Model ◽  
Likelihood Ratio Statistic ◽  
Food Products ◽  
Nutrition Knowledge ◽  
Fat Content ◽  
Retest Reliability ◽  
Log Likelihood ◽  
Test Retest Reliability ◽  
The Rasch Model

The objective of the study was to develop and validate a 2l-item nutrition knowledge test to measure people's knowledge of the fat content of food-products. After pretesting and provisional development, the test was administered twice to study test-retest reliability. Furthermore, various sub-populations with expected differences in nutrition knowledge completed the test in order to study the construct validity of the questionnaire. The subpopulations consisted of lay-people (N=81), students of human nutrition and dietetics (N=108), and professional experts (N=79) on human nutrition. The internal consistency and uni-dimensionality of the test were determined by calculating the KR-20 statistic and the log-likelihood ratio statistic for the Rasch model. Pearson's correlation and gross misclassification between T1 and T2 were calculated to assess the test-retest reliability. Analysis of variance was used to test for differences in mean knowledge scores between subpopulations. Test-retest reliability was found to be sufficient (R=.85). The internal consistency was moderate (KR20=.68). According to the Rasch model, two items had to be removed from the test before the log-likelihood ratio statistic of the Rasch model indicated that knowledge about the fat content of food products as assessed by the questionnaire is a uni-dimensional construct. The differences in mean knowledge scores between the subpopulations were significant (p < .01) and in the expected direction (experts > students > lay people). It can be concluded that the test is a reliable and valid instrument to measure knowledge about total fat content in food products and that the Rasch model is a comprehensive method to indicate the reliability of nutritional knowledge tests.

Is the ‘improved likelihood ratio statistic’ really improved in the discrete case?

Biometrika ◽  
1989 ◽  
Vol 76 (4) ◽  
pp. 655-661
Author(s):  
MORTEN FRYDENBERG ◽  
JENS LEDET JENSEN
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Discrete Case
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