The distribution of the maximum likelihood-ratio statistic used for testing a normal sample for three outliers

2012 ◽  
Vol 56 (12) ◽  
pp. 61-70
Author(s):  
L. K. Shiryaeva
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Normal Sample ◽  
Distribution Of The Maximum

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.

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.

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 Maximum Likelihood Analysis of Rare Binary Traits Under Different Modes of Inheritance

Genetics ◽  
1996 ◽  
Vol 143 (4) ◽  
pp. 1819-1829 ◽  
Author(s):  
G Thaller ◽  
L Dempfle ◽  
I Hoeschele
Keyword(s):  
Maximum Likelihood ◽  
Mixed Model ◽  
Major Gene ◽  
Likelihood Ratio Statistic ◽  
Information Criterion ◽  
Genetic Models ◽  
Parameter Estimates ◽  
Gene Effects ◽  
Binary Traits ◽  
Deterministic Algorithms

Abstract Maximum likelihood methodology was applied to determine the mode of inheritance of rare binary traits with data structures typical for swine populations. The genetic models considered included a monogenic, a digenic, a polygenic, and three mixed polygenic and major gene models. The main emphasis was on the detection of major genes acting on a polygenic background. Deterministic algorithms were employed to integrate and maximize likelihoods. A simulation study was conducted to evaluate model selection and parameter estimation. Three designs were simulated that differed in the number of sires/number of dams within sires (10/10, 30/30, 100/30). Major gene effects of at least one SD of the liability were detected with satisfactory power under the mixed model of inheritance, except for the smallest design. Parameter estimates were empirically unbiased with acceptable standard errors, except for the smallest design, and allowed to distinguish clearly between the genetic models. Distributions of the likelihood ratio statistic were evaluated empirically, because asymptotic theory did not hold. For each simulation model, the Average Information Criterion was computed for all models of analysis. The model with the smallest value was chosen as the best model and was equal to the true model in almost every case studied.

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