Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions

A unified approach for allele frequency estimation, SNP detection and association studies based on pooled sequencing data using EM algorithms

“…Note that the null hypothesis f = 0 is on the boundary of the region of the parameters of interest. Therefore, the asymptotic distribution of Λ is $\frac{1}{4}{I}_0+\frac{1}{2}{\chi }_1^2+\frac{1}{4}{\chi }_2^2$ when the number of pools is large according to [18], where I 0 is the point mass at 0 and ${\chi }_i^2$, i = 1, 2 are the chi-square distributions with i degrees of freedom. When the number of pools is relatively small, simulation approaches for the null distribution of Λ are needed to obtain the asymptotic distribution.…”

“…Because the null hypothesis f i = 0 is on the boundary of parameter region f i > 0, the statistic Λ i has an asymptotic distribution $\frac{1}{2}{I}_{0}MathClass-bin+\frac{1}{2}{\chi }_1^2$ when the number of pools is large according to [18]. We refer to the above method for SNP identification as EM-SNP.…”

Applying Linear Mixed Effects Models with Crossed Random Effects to Psycholinguistic Data: Multilevel Specification and Model Selection.

“…For example, one would not use this approach for an LMER to select the random structure of the model because in this context, model selection is used to test whether the variance (or covariance) terms are equal to zero. Self and Liang (1987) refer to this type of hypothesis as a nonstandard testing condition because the parameters of the null hypothesis fall at the boundary of the parameter space. Based on Self and Liang's (1987) results, Lee (1994, 1995) show that the asymptotic null distribution for the LRT statistics for testing a hypothesis of this type often consist of a mixture of chisquare distributions rather than the classical single chisquared distribution.…”

“…Self and Liang (1987) refer to this type of hypothesis as a nonstandard testing condition because the parameters of the null hypothesis fall at the boundary of the parameter space. Based on Self and Liang's (1987) results, Lee (1994, 1995) show that the asymptotic null distribution for the LRT statistics for testing a hypothesis of this type often consist of a mixture of chisquare distributions rather than the classical single chisquared distribution. Under more general conditions, for example, comparing models with k and k+k' (k'>1) random effects, the null distribution is a mixture of random variables, and the weights for each component can only be calculated analytically in a number of special cases (Raubertas, Lee, & Nordheim, 1986;Shapiro, 1988).…”

Island tameness and the repeatability of flight initiation distance in a large herbivore

“…For hypotheses involving parameters on the boundary of parameter space, such as variances, the theoretical asymptotic distribution of the likelihood ratio is a mixture of 2 variates, where the mixing probabilities are 0.5, one with 0 degrees of freedom and the other with 1 degree of freedom (Self and Liang 1987;Gilmour et al 2009). The p values from 2 tests with 1 degree of freedom were therefore divided by 2.…”