Conditional Likelihood Ratio Research Articles

Bartlett-Type Adjustment for the Conditional Likelihood Ratio Statistic of Cox and Reid

Bartlett-Type Adjustment for the Conditional Likelihood Ratio Statistic of Cox and Reid

Bartlett-type adjustment for the conditional likelihood ratio statistic of Cox and Reid

SUMMARY This paper explicitly derives a Bartlett-type adjustment for the conditional likelihood ratio statistic of Cox & Reid via that for the usual likelihood ratio statistic.

Testing for Threshold Autoregression

We consider the problem of determining whether a threshold autoregressive model fits a stationary time series significantly better than an autoregressive model does. A test statistic $\lambda$ which is equivalent to the (conditional) likelihood ratio test statistic when the noise is normally distributed is proposed. Essentially, $\lambda$ is the normalized reduction in sum of squares due to the piecewise linearity of the autoregressive function. It is shown that, under certain regularity conditions, the asymptotic null distribution of $\lambda$ is given by a functional of a central Gaussian process, i.e., with zero mean function. Contiguous alternative hypotheses are then considered. The asymptotic distribution of $\lambda$ under the contiguous alternative is shown to be given by the same functional of a noncentral Gaussian process. These results are then illustrated with a special case of the test, in which case the asymptotic distribution of $\lambda$ is related to a Brownian bridge.

Measuring Attitudes With a Threshold Model Drawing on a Traditional Scaling Concept

This paper presents a generalized Rasch model for measuring attitudes which is based on the concepts of Thurstone's method of successive intervals. The model combines the rating scale and the dispersion model proposed by Andrich and a submodel of the partial credit model proposed by Masters. An estimation pro cedure for unconditional maximum likelihood (ML) es timates is outlined. A recursion formula for the sym metric functions, which is needed for conditional ML procedures, is given. The benefits of the model are il lustrated with a study on students' interest in physics. The fit of different threshold models can be compared using conditional likelihood values and conditional likelihood ratio tests. Index terms: attitude measure ment, conditional likelihood ratio test, partial credit model, Rasch model, rating scales, successive inter vals, threshold model.

A conditional likelihood ratio test for order restrictions in exponential families

This paper considers a conditional likelihood ratio procedure for testing order restrictions on a collection of parameters indexing a set of populations from an exponential family. The class of problems dealt with here has received detailed examination in earlier work of Robertson and Wegman (1978). The conditional test amounts to a modification of the likelihood ratio procedure developed by those authors; it benefits from increased statistical power, at least for interesting classes of alternative hypotheses, and in addition, is computationally less burdensome than the likelihood ratio method. A Monte Carlo comparison of the power of the conditional and unconditional tests is carried out. An application to testing trend in a set of Poisson parameters is discussed.

Maximum likelihood estimation for doubly stochastic poisson processes with partial observations

This paper deals with a nonlinear filtering approach to the problem of intensity parameter estimation of a doubly stochastic Poisson process, driven by an unob‐ servable Markov process. We present a method for the evaluation of the conditional likelihood ratio, given the observations of one path of the point process. By the methodsof Ibragimov, Khasminskii and Kutoyants we investigate the asymptotic properties of the corresponding maximum likelihood estimator. In the ergodic case consistency, asymptotic normality, convergence of the moments and asymptotic efficiency are established.

Paternity testing. 2: Likelihood ratio tests.

It is known that the so-called paternity index is also a likelihood ratio statistic for testing that an alleged father is the true father. Unfortunately the likelihood ratio test can sometimes lead to unsatisfactory results because of its dependence on the phenotype combinations of the mother and child. A new, conditional likelihood ratio test, which we call the ancillary test, is proposed in which statistical testing is carried out conditionally on the phenotypes of the mother and alleged father. An exact procedure is available which can be executed with reasonable facility on a microcomputer.

Parameter estimation for point processes with partial observations: A filtering approach

The estimation of the parameters of a doubly stochastic Poisson process driven by a Markov chain is considered. An ML-estimator will be defined by the conditional likelihood ratio, given the observations of one path of the point process. The filler equation for point processes is used as a tool to solve the corresponding maximisation problem.

Geometrical theory of higher-order asymptotics of test, interval estimator and conditional inference

Higher order asymptotic powers of one-sided and two-sided tests, both conditional and unconditional, are evaluated for a one-parameter curved exponential family of distributions by using differential-geometrical notions. Higher order asymptotic powers and the expected size of one-sided and two-sided interval estimators, both conditional and unconditional, are also obtained. The tests and interval estimators, which are third-order most powerful at any arbitrary specified one point are explicitly designed with the help of the approximate ancillary statistic. The characteristics of the conditional inference are elucidated, by proving the equivalence up to the third order of the conditional inference and likelihood ratio inference. Geometrical notions such as curvatures and angles play a fundamental role in the present theory.

Specifically objective stochastic latency mechanisms

A two parameter additive model for psychological response latencies f vi ( t) = ( θ v + ϵ i ) exp(−( θ v + ϵ i ) t) is suggested. Conditional inference procedures are derived for the psychometric latency model. If the model is considered as a theory for speed test construction the unrestricted comparability of measurements obtained under “speed” and “power” conditions is an attractive feature of the model. An extension of the parametric structure gives the “linear exponential model” f vi ( t) = ( θ v + Σ ι m a iι η ι ) exp(−( θ v + Σ ι m a iι η ι ) t), where the time constants of the latencies are conceived as linear combinations of “basic parameters” corresponding to the experimental conditions of the latency situations. The linear exponential model includes as special cases some well known series latency mechanisms. Some generalizations of the distributional assumptions are discussed, especially the Weibull distributions. The estimation of parameters by conditional inference procedures and the evaluation of the fit of the model by conditional likelihood ratio tests is illustrated by means of the numerical data from a free recall experiment.

An alternative approach to nonlinear filtering

The structure of a nonlinear filter with observation process having continuous and discontinuous components is considered. The approach is based on the so-called “Bayes” formula for conditional expectations. “Fubini” type theorems for stochastic integrals are given and used to obtain the representations of an optimal estimate and of the conditional likelihood ratio. A linear unnormalized filtering equation for controlled system process is derived.

A goodness of fit test for the rasch model

The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties estimated from different scoregroups and over-all estimates. Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is formed. It is shown that—2 times the logarithm of this ratio isx 2-distributed when the Rasch model is true. The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves, but unequal discriminating items from a scholastic aptitude test.

The Asymptotic Distribution of Conditional Likelihood Ratio Tests

Abstract A conditional likelihood ratio test for testing a hypothesis concerning structural parameters in the presence of infinitely many incidental parameters is suggested. It is shown that the usual X2 approximation to the log-likelihood ratio fails to work in many situations involving incidental parameters. In contrast it is shown that the X2 approximation can be used in large samples under fairly general assumptions if we use conditional likelihood ratio tests instead. The relationship between the theory of UMPU-test and conditional likelihood ratio tests is discussed, and some examples are given to show that the conditional likelihood ratio approach covers more cases than the UMPU approach.

The Asymptotic Distribution of Conditional Likelihood Ratio Tests

Abstract A conditional likelihood ratio test for testing a hypothesis concerning structural parameters in the presence of infinitely many incidental parameters is suggested. It is shown that the usual X2 approximation to the log-likelihood ratio fails to work in many situations involving incidental parameters. In contrast it is shown that the X2 approximation can be used in large samples under fairly general assumptions if we use conditional likelihood ratio tests instead. The relationship between the theory of UMPU-test and conditional likelihood ratio tests is discussed, and some examples are given to show that the conditional likelihood ratio approach covers more cases than the UMPU approach.

Estimating and Testing Trend in a Stochastic Process of Poisson Type

Let $\{T_i: i = 1, 2, \cdots\}$ be a stochastic process of Poisson type, with $\lambda(t)$, the rate of occurrence of the events, depending on time. We may interpret $T_i$ as the time of occurrence of the $i$th event. In Section 2, starting with the joint density function of $T_1, \cdots, T_n$, the maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda (t) \leqq M$ being a non-decreasing function of time ($M$ some positive number) is found. In Section 3, starting with the conditional joint density of $T_1, \cdots, T_n$ given there are $n$ events in $(0, t^\ast\rbrack$, the conditional maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda(t)$ being a non-decreasing function of time is found. In Section 4, the conditional likelihood ratio test of the hypothesis that $\lambda(t)$ is constant against the alternate hypothesis that $\lambda(t)$ is not constant but is nondecreasing is found, and a limiting distribution is found which may be used to approximate the probability of a type I error for large sample size. Theorem 2.1 (Brunk-van Eeden), I believe is important in its own right. It is contained in the works of Brunk and van Eeden, although it is not explicitly stated. This theorem can be used as a basis for tests of hypotheses for constant parameters against increasing parameters or for increasing parameters against all other alternatives.