Normal Likelihood Function Research Articles

Conditional minimum volume ellipsoid with application to multiclass discrimination

In this paper, we present a new formulation for constructing an n-dimensional ellipsoid by generalizing the computation of the minimum volume covering ellipsoid. The proposed ellipsoid construction is associated with a user-defined parameter β?[0,1), and formulated as a convex optimization based on the CVaR minimization technique proposed by Rockafellar and Uryasev (J. Bank. Finance 26: 1443---1471, 2002). An interior point algorithm for the solution is developed by modifying the DRN algorithm of Sun and Freund (Oper. Res. 52(5):690---706, 2004) for the minimum volume covering ellipsoid. By exploiting the solution structure, the associated parametric computation can be performed in an efficient manner. Also, the maximization of the normal likelihood function can be characterized in the context of the proposed ellipsoid construction, and the likelihood maximization can be generalized with parameter β. Motivated by this fact, the new ellipsoid construction is examined through a multiclass discrimination problem. Numerical results are given, showing the nice computational efficiency of the interior point algorithm and the capability of the proposed generalization.

ESTIMATING DENSITY DEPENDENCE, PROCESS NOISE, AND OBSERVATION ERROR

We describe a discrete-time, stochastic population model with density dependence, environmental-type process noise, and lognormal observation or sampling error. The model, a stochastic version of the Gompertz model, can be transformed into a linear Gaussian state-space model (Kalman filter) for convenient fitting to time series data. The model has a multivariate normal likelihood function and is simple enough for a variety of uses ranging from theoretical study of parameter estimation issues to routine data analyses in population monitoring. A special case of the model is the discrete-time, stochastic exponential growth model (density independence) with environmental-type process error and lognormal observation error. We describe two methods for estimating parameters in the Gompertz state-space model, and we compare the statistical qualities of the methods with computer simulations. The methods are maximum likelihood based on observations and restricted maximum likelihood based on first differences. Both offer adequate statistical properties. Because the likelihood function is identical to a repeated-measures analysis of variance model with a random time effect, parameter estimates can be calculated using PROC MIXED of SAS. We use the model to analyze a data set from the Breeding Bird Survey. The fitted model suggests that over 70% of the noise in the population's growth rate is due to observation error. The model describes the autocovariance properties of the data especially well. While observation error and process noise variance parameters can both be estimated from one time series, multimodal likelihood functions can and do occur. For data arising from the model, the statistically consistent parameter estimates do not necessarily correspond to the global maximum in the likelihood function. Maximization, simulation, and bootstrapping programs must accommodate the phenomenon of multimodal likelihood functions to produce statistically valid results.

Robust Inference for the Correlation Coefficient—A Parametric Method

ABSTRACT This article introduces a parametric robust way of making valid inferences for the correlation coefficient. More specifically, it will be demonstrated that the bivariate normal likelihood function can be made asymptotically valid for practically all underlying bivariate continuous distributions. The adjustment to the bivariate normal model that achieves the robustness property will be presented. Simulation studies will be performed to demonstrate the finite sample performance of the novel robust procedure.

Comparing Two Population Means and Variances: A Parametric Robust Way

This article introduces a parametric robust way of comparing two population means and two population variances. With large samples the comparison of two means, under model misspecification, is lesser a problem, for, the validity of inference is protected by the central limit theorem. However, the assumption of normality is generally required, so that the inference for the ratio of two variances can be carried out by the familiar F statistic. A parametric robust approach that is insensitive to the distributional assumption will be proposed here. More specifically, it will be demonstrated that the normal likelihood function can be adjusted for asymptotically valid inferences for all underlying distributions with finite fourth moments. The normal likelihood function, on the other hand, is itself robust for the comparison of two means so that no adjustment is needed.

Score test for mapping quantitative‐trait loci with sibships of arbitrary size when the dominance effect is not negligible

In the linakge analysis of quantitative traits, an additive model that assumes no dominance effect is often adopted. Intuitively, when the no-dominance-effect assumption does not hold, such a practice does not make efficient use of the data, and its power to detect linkage can be improved. Here we introduce a score statistic for detecting quantitative trait loci when the dominance effect is not neglible or the dominance effect is a concern. This statistic is derived from a normal likelihood function for sibships of arbitrary size. In the derivation, the inherent genetic constraints on model parameters are fully taken into consideration. This score statistic is asymptotically equivalent to the corresponding likelihood ratio statistic, but it is much easier to compute. The asymptotic distribution of this statistic is derived, which is a mixture of chi(0) (2), chi(1) (2), and chi(2) (2). Weights for distribution components are functions of the informativeness of the marker data. The type I error rate and the power of the proposed statistic in finite sample are evaluated via simulations.

On Repeated Measures Analysis with Misspecified Covariance Structure

Summary In recent years various sophisticated methods have been developed for the analysis of repeated measures, or longitudinal data. The more traditional approach, based on a normal likelihood function, has been shown to be unsatisfactory, in the sense of yielding asymptotically biased estimates when the covariance structure is misspecified. More recent methodology, based on generalized linear models and quasi-likelihood estimation, has gained widespread acceptance as ‘generalized estimating equations’. However, this also has theoretical problems. In this paper a suggestion is made for improving the asymptotic behaviour of estimators by using the older approach, implemented via Gaussian estimation. The resulting estimating equations include the quasi-score function as one component, so the methodology proposed can be viewed as a combination of Gaussian estimation and generalized estimating equations which has a firmer asymptotic basis than either alone has.

An estimate of the covariance between variables which are not jointly observed

Situations sometimes arise in which variables collected in a study are not jointly observed. This typically occurs because of study design. An example is an equating study where distinct groups of subjects are administered different sections of a test. In the normal maximum likelihood function to estimate the covariance matrix among all variables, elements corresponding to those that are not jointly observed are unidentified. If a factor analysis model holds for the variables, however, then all sections of the matrix can be accurately estimated, using the fact that the covariances are a function of the factor loadings. Standard errors of the estimated covariances can be obtained by the delta method. In addition to estimating the covariance matrix in this design, the method can be applied to other problems such as regression factor analysis. Two examples are presented to illustrate the method.

Foundations of the theory of evidence: Resolving conflict among schemata

Schematic conflict occurs when evidence is interpreted in different ways (for example, by different people, who have learned to approach the given evidence with different schemata). Such conflicts are resolved either by weighting some schemata more heavily than others, or by finding common-ground inferences for several schemata, or by a combination of these two processes. Belief functions, interpreted as representations of evidence strength, provide a natural model for weighting schemata, and can be utilized in several distinct ways to compute common-ground inferences. In two examples, different computations seem to be required for reasonable common-ground inference. In the first, competing scientific theories produce distinct, logically independent inferences based on the same data. In this example, the simple product of the competing belief functions is a plausible evaluation of common ground. In the second example (sensitivity analysis), the conflict is among alternative statistical assumptions. Here, a product of belief functions will not do, but the upper envelope of normalized likelihood functions provides a reasonable definition of common ground. Different inference contexts thus seem to require different methods of conflict resolution. A class of such methods is described, and one characteristic property of this class is proved.

Interval estimation using the likelihood function

The general properties of two commonly-used methods of interval estimation for population parameters in physics are examined. Both of these methods employ the likelihood function: (i) Obtaining an interval by finding the points where the likelihood decreases from its maximum by some specified ratio; (ii) Obtaining an interval by finding points corresponding to some specified fraction of the total integral of the likelihood function. In particular, the conditions for which these methods give a confidence interval are illuminated, following an elaboration on the definition of a confidence interval. The first method, in its general form, gives a confidence interval when the parameter is a function of a location parameter. The second method gives a confidence interval when the parameter is a location parameter. A potential pitfall of performing a likelihood analysis without understanding the underlying probability distribution is discussed using an example with a normal likelihood function.

Hidden Markov modeling using a dominant state sequence with application to speech recognition

Approximate maximum likelihood (ML) hidden Markov modeling using the most likely state sequence (MLSS) is examined and compared with the exact ML approach that considers all possible state sequences. It is shown that for any hidden Markov model (HMM), the difference between the approximate and the exact normalized likelihood functions cannot exceed the logarithm of the number of states divided by the dimension of the output vectors (frame length), which is negligible for typically used values of vector dimension (128–256) and number of states (2–30). Furthermore, for Gaussian HMMs and a given observation sequence, the MLSS is typically the sequence of nearest neighbor states in the Itakura-Saito sense, and the posterior probability of any state sequence which departs from the MLSS in a single time instant decays exponentially with the frame length. Hence, for a sufficiently large frame length the exact and approximate ML approaches provide similar model estimates and likelihood values. The results and their implications on speech recognition are demonstrated in a set of experiments.

Maximum likelihood hidden Markov modeling using a dominant sequence of states

Approximate maximum likelihood (ML) hidden Markov modeling using the most likely state sequence (MLSS) is examined and compared with the exact ML approach that considers all possible state sequences. It is shown that for any hidden Markov model (HMM), the difference between the approximate and the exact normalized likelihood functions cannot exceed the logarithm of the number of states divided by the dimension of the output vectors (frame length). Furthermore, for Gaussian HMMs and a given observation sequence, the MLSS is typically the sequence of nearest neighbor states in the Itakura-Saito sense, and the posterior probability of any state sequence which departs from the MLSS in a single time instant, decays exponentially with the frame length. Hence, for a sufficiently large frame length the exact and approximate ML approach provide similar model estimates and likelihood values. >

Likelihood as approximate pivotal distribution

SUMMARY For a single-parameter curved exponential family, it is shown that the conditional distri- bution of the maximum likelihood estimator is the normalized likelihood function, except for a relative error proportional to inverse sample size. Recently it has been shown (Efron & Hinkley, 1978; Peers, 1978) that there is a conditional normal approximation for the distribution of the maximum likelihood estimator in the one- parameter case. This conditional normal approximation uses an observed characteristic of the likelihood as an approximate variance. In principle the approximation is unnecessary for the special case of estimating a location parameter, because Fisher (1934) derived the exact, fully-informative conditional distribution of the maximum likelihood estimator. That distri- bution is the normalized likelihood function. In this paper we present theoretical evidence that Fisher's special case result is a good approximation in general. Specifically, let lik (0) denote the likelihood for a sample of size n from a curved exponential family distribution indexed by a single parameter 0 and let 6 denote the maximum likelihood estimator. Then we find that the probability density of the pivot #- 0, conditional on a natural approximate ancillary a, is

A probabilistic approach to the statistical analysis of experiments on helminth parasites

Abstract A probabilistic method, based on a Bayesian approach, is proposed for the evaluation of results obtained in experimental infections induced by microscopic eggs (or larvae) whose exact number is not directly known. The statistical analysis leads to the evaluation of the probability that the eggs (or larvae) develop “normally”; the results are represented by normalized likelihood functions (NLF) or, more generally, through final probability distributions. Any difference in the various experiments that possibly emerged may be judged on the NLF graphs. Some real and fictional cases are discussed as examples. In particular, the unreliability of the usual method that considers only the number of recovered parasites but not the number of eggs used may be easily verified.