Distribution Of Likelihood Ratio Research Articles

Estimation du risque de trisomie 21 fœtale par les marqueurs sériques maternels : impact des erreurs analytiques sur le risque calculé

Foetal Down syndrome risk computed on maternal serum markers has an accuracy which depends on the accuracy of measured concentrations. This study uses properties of mean and variance to obtain formulas for risk variance in terms of concentrations analytical errors. Statistical distributions of likelihood ratio are also examined. Using two serum markers alone and two markers with nuchal translucency is considered. L’imprécision du risque de trisomie 21 fœtale calculé par les marqueurs sériques maternels dépend de l’imprécision sur les concentrations mesurées. Utilisant les propriétés mathématiques de l’espérance et de la variance, cet article propose des formules estimant la variance du risque calculé en fonction des concentrations mesurées et de leurs erreurs analytiques. Il discute également des distributions statistiques du logarithme du rapport de vraisemblance. L’usage de deux marqueurs sériques seuls, ainsi que celui de deux marqueurs joints à la clarté nucale sont étudiés.

Asymptotic distribution of density ratios

We find limit theorems regarding asymptotic distribution of density ratios. These results are applied to analyze the goodness of classical approximations. Furthermore these theorems are used in statistical inference procedures to find asymptotic distribution of likelihood ratios.

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.

Information in tournaments under limited liability

The problem of designing tournament contracts under limited liability and alternative performance measures is considered. Under risk neutrality, only the best-performing agent receives an extra premium if the liability constraint becomes binding. Under risk aversion, more than one prize is awarded. In both situations, performance measures can be ranked if their likelihood ratio distribution functions differ by a mean-preserving spread. The latter result is applied to questions of contest design and more general forms of relative performance payment.

On the measure of the information in a statistical experiment

Setting aside experimental costs, the choice of an experiment is usually formulated in terms of the maximization of a measure of information, often presented as an optimality design criterion. However, there does not seem to be a universal agreement on what objects can qualify as a valid measure of the information in an experiment. In this article we explicitly state a minimal set of requirements that must be satisfied by all such measures. Under that framework, the measure of the information in an experiment is equivalent to the measure of the variability of its likelihood ratio statistics or which is the same, it is equivalent to the measure of the variability of its posterior to prior ratio statistics and to the measure of the variability of the distribution of the posterior distributions yielded by it. The larger that variability, the more peaked the likelihood functions and posterior distributions that tend to be yielded by the experiment, and the more informative the experiment is. By going through various measures of variability, this paper uncovers the unifying link underlying well known information measures as well as information measures that are not yet recognized as such. The measure of the information in an experiment is then related to the measure of the information in a given observation from it. In this framework, the choice of experiment based on statistical merit only, is posed as a decision problem where the reward is a likelihood ratio or posterior distribution, the utility function is convex, the utility of the reward is the information observed, and the expected utility is the information in an experiment. Finally, the information in an experiment is linked to the information and to the uncertainty in a probability distribution, and we find that the measure of the information in an experiment is not always interpretable as the uncertainty in the prior minus the expected uncertainty in the posterior.

Empirical likelihood analysis of the rank estimator for the censored accelerated failure time model

SUMMARY We use the empirical likelihood method to derive a test and thus a confidence interval based on the rank estimators of the regression coefficient in the accelerated failure time model. Standard chi-squared distributions are used to calculate the p-value and to construct the confidence interval. Simulations and examples show that the chi-squared approximation to the distribution of the log empirical likelihood ratio performs well, and has some advantages over the existing methods.

Zipf's Law in Importance of Genes for Cancer Classification Using Microarray Data

Using a measure of how differentially expressed a gene is in two biochemically/phenotypically different conditions, we can rank all genes in a microarray dataset. We have shown that the falling-off of this measure (normalized maximum likelihood in a classification model such as logistic regression) as a function of the rank is typically a power-law function. This power-law function in other similar ranked plots are known as the Zipf's law, observed in many natural and social phenomena. The presence of this power-law function prevents an intrinsic cutoff point between the “important” genes and “irrelevant” genes. We have shown that similar power-law functions are also present in permuted dataset, and provide an explanation from the well-known χ 2 distribution of likelihood ratios. We discuss the implication of this Zipf's law on gene selection in a microarray data analysis, as well as other characterizations of the ranked likelihood plots such as the rate of fall-off of the likelihood.

Sib‐parentage testing using molecular markers when parents are unknown

The formulae for computing the so-called Sib Index using codominant alleles for (1) full-sib and (2) half-sib parentage are given. Hypothesis testing is based on the distribution of conditional likelihood ratio or Bayes' factor. Thresholds for rejecting the null hypothesis and P-values were obtained in function of the number of alleles and their frequency distributions. Simulations showed that a relatively low number of marker systems (e.g. 20) are enough to accept the hypothesis of sib parentage with a reasonable power for usual significance levels, but that a higher number would be necessary if full-sib against half-sib parentage is the contrast to be carried out. The effect of sampling variation on the allele frequencies on power calculations is also analysed.

On empirical likelihood for a semiparametric mixture model

Plant and animal studies of quantitative trait loci provide data which arise from mixtures of distributions with known mixing proportions. Previous approaches to estimation involve modelling the distributions parametrically. We propose a semiparametric alternative which assumes that the log ratio of the component densities satisfies a linear model, with the baseline density unspecified. It is demonstrated that a constrained empirical likelihood has an irregularity under the null hypothesis that the two densities are equal. A factorisation of the likelihood suggests a partial empirical likelihood which permits unconstrained estimation of the parameters, and which is shown to give consistent and asymptotically normal estimators, regardless of the null. The asymptotic null distribution of the log partial likelihood ratio is chi‐squared. Theoretical calculations show that the procedure may be as efficient as the full empirical likelihood in the regular set‐up. The usefulness of the robust methodology is illustrated with a rat study of breast cancer resistance genes.

Ranking of information systems in agency models: an integral condition

Summary. We provide a condition for ranking of information systems in agency problems. The condition has a straightforward economic interpretation in terms of the sensitivity of a cumulative distribution with respect to the agent's effort. The criterion is shown to be equivalent to the mean preserving spread condition on the likelihood ratio distributions.

Minimizing response times and queue lengths in systems of parallel queues

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Minimizing response times and queue lengths in systems of parallel queues

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Minimizing response times and queue lengths in systems of parallel queues

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Calculation of roc curves in multidimensional likelihood ratio based screening with Down's syndrome as a special case.

The development of algorithms and computer programs for the analysis of screening performance in situations with multiple normally (Gaussian) distributed selection markers and a priori risks depending on a stratification of the population. The S-PLUS programming language was used to construct programs producing distributions of log likelihood ratios based on the Monte Carlo simulation. These distributions were used to construct programs for the calculation of roc curves, including a possible stratification of the population. S-PLUS programs for the analysis of screening performance are listed and described. The programs can be used without any special knowledge of S-PLUS. An example of the use of the programs is given.

Optimal Scheduling on Parallel Processors with Precedence Constraints and General Costs

We consider preemptive and nonpreemptive scheduling of partially ordered tasks on parallel processors, where the precedence relations have an intervalorder, an in-forest, or a uniform out-forest structure. Processing times of tasks are random variables with an increasing in likelihood ratio distribution in the nonpreemptive case and an exponential distribution in the preemptive case. We consider a general cost that is a function of time and of the uncompleted tasks and show that the most successors (MS) policy stochastically minimizes the cost function when it satisfies certain agreeability conditions. A consequence is that the MS policy stochastically minimizes makespan, weighted flowtime, and the weighted number of late jobs.

Testing for linkage disequilibrium in genotypic data using the Expectation-Maximization algorithm.

We generalize an approach suggested by Hill (Heredity, 33, 229-239, 1974) for testing for significant association among alleles at two loci when only genotype and not haplotype frequencies are available. The principle is to use the Expectation-Maximization (EM) algorithm to resolve double heterozygotes into haplotypes and then apply a likelihood ratio test in order to determine whether the resolutions of haplotypes are significantly nonrandom, which is equivalent to testing whether there is statistically significant linkage disequilibrium between loci. The EM algorithm in this case relies on the assumption that genotype frequencies at each locus are in Hardy-Weinberg proportions. This method can accommodate X-linked loci and samples from haplodiploid species. We use three methods for testing significance of the likelihood ratio: the empirical distribution in a large number of randomized data sets, the X2 approximation for the distribution of likelihood ratios, and the Z2 test. The performance of each method is evaluated by applying it to simulated data sets and comparing the tail probability with the tail probability from Fisher's exact test applied to the actual haplotype data. For realistic sample sizes (50-150 individuals) all three methods perform well with two or three alleles per locus, but only the empirical distribution is adequate when there are five to eight alleles per locus, as is typical of hypervariable loci such as microsatellites. The method is applied to a data set of 32 microsatellite loci in a Finnish population and the results confirm the theoretical predictions. We conclude that with highly polymorphic loci, the EM algorithm does lead to a useful test for linkage disequilibrium, but that it is necessary to find the empirical distribution of likelihood ratios in order to perform a test of significance correctly.

Efficiency of an Information System in an Agency Model

Different information systems are compared in terms of their relative efficiencies in an agency model. The mean preserving spread relation between the likelihood ratio distributions derived from the original information systems is found to be sufficient to rank information systems under quite general assumptions about the agent's utility function. Furthermore, it is shown that the mean preserving spread criterion can be applied to a broader set of information systems than Holmstrom's informativeness criterion and Blackwell's theorem

Linear regression under two separate regimes: an empirical distribution for Quandt's log likelihood ratio

In the empirical analysis of structural changes, Quandt's test for detecting a single change in a regression relationship at an unknown point of time is frequently applied. A limitation regarding the applicability of the method is that the distribution of the Quandt's log likelihood ratio is unknown. The purpose of this work is to derive an empirical distribution for Quandt's statistic which enables the researcher to approximate the level of significance when testing the stability of regression relationships over time.

The limit distributions of likelihood ratio and cumulative sum tests for a change in a binomial probability

We obtain limit theorems for likelihood ratio and cumulative sums tests. In the case of the likelihood ratio the centralising and normalising sequences go to infinity and the limit is the Gumbel (double exponential) distribution. The first and the last few observations determine the limit, which also explains why the likelihood ratio test is very powerful on the tails.

Higher-Order Approximations to the Null Distributions of Test Statistics for Nonlinear Restrictions on Regression Coefficients

Asymptotic expansions of the distributions of likelihood ratio and Lagrange multiplier test statistics for nonlinear restrictions on regression coefficients are derived under the null hypothesis. Nonlinear restrictions include, as a special case, the identifiability restrictions in the simultaneous equations models. Our analyses of simultaneous equations deal not only with single equations but also subsystems and complete systems. The asymptotic expansions we derive are informative about deviations of the real size of test from the nominal asymptotic size.