A maximum likelihood approach towards aggregating partial orders

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈A, R〉 to be n-dimensional if there are n linear orderings of A, 〈A, L1〉, 〈A, L2〉 …, 〈A, Ln〉 such that R = L1 ∩ L2 ∩ … ∩ Ln. Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈x1, y1〉 ≤Q×Q 〈x2, y2, if and only if x1 ≤ x2 and y1 ≤ y2) is 2-dimensional, since ≤Q×Q is the intersection of the two lexicographic orderings of Q × Q. In fact, as shown by Dushnik and Miller, a countable partial ordering is n-dimensional if and only if it can be embedded as a subordering of Qn.Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

The uniform content of partial and linear orders

Information from multiple informants is frequently used to assess psychopathology. We consider marginal regression models with multiple informants as discrete predictors and a time to event outcome. We fit these models to data from the Stirling County Study; specifically, the models predict mortality from self report of psychiatric disorders and also predict mortality from physician report of psychiatric disorders. Previously, Horton et al. found little relationship between self and physician reports of psychopathology, but that the relationship of self report of psychopathology with mortality was similar to that of physician report of psychopathology with mortality. Generalized estimating equations (GEE) have been used to fit marginal models with multiple informant covariates; here we develop a maximum likelihood (ML) approach and show how it relates to the GEE approach. In a simple setting using a saturated model, the ML approach can be constructed to provide estimates that match those found using GEE. We extend the ML technique to consider multiple informant predictors with missingness and compare the method to using inverse probability weighted (IPW) GEE. Our simulation study illustrates that IPW GEE loses little efficiency compared with ML in the presence of monotone missingness. Our example data has non-monotone missingness; in this case, ML offers a modest decrease in variance compared with IPW GEE, particularly for estimating covariates in the marginal models. In more general settings, e.g., categorical predictors and piecewise exponential models, the likelihood parameters from the ML technique do not have the same interpretation as the GEE. Thus, the GEE is recommended to fit marginal models for its flexibility, ease of interpretation and comparable efficiency to ML in the presence of missing data.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

In longitudinal studies, missing data are ubiquitous. This article made a comparison of three model-based techniques for handling different types of missing data (i.e., missing at random (MAR)-based maximum likelihood (ML) approach, missing not at random (MNAR)-based Diggle–Kenward (DK) selection model and MNAR-based pattern mixture (PM) model) in longitudinal studies through a Monte Carlo simulation study. Two influential factors were considered: the dropout rates (5%, 10%, 20%, and 40%) and the sample sizes (100, 300, 500, and 1000) under MAR and MNAR missingness mechanisms respectively. The results indicated that the model selection was a crucial issue when researchers were dealing with missing data in longitudinal studies because under MNAR mechanism, DK method outperformed MAR-based ML approach, but PM method performed worse than MAR-based method did. The differences of the parameter estimation among three methods became more significant as the sample size and the dropout rate increased.

본 연구는 비연속 분할함수 성장혼합모형의 접근법에 따라 계층 간 원격 결과변인 추정치에 어떠한 차이가 있는지를 확인하는 데 목적이 있다. 이를 위하여 본 연구는 3단계 ML접근법과 3단계 BCH 접근법을 적용하여 모의실험 연구를 수행하였다. 모의실험에 계층 분류정도(1.3, 2.6), 계층 혼합비율(90:10, 70:30, 50:50), 표본크기(200, 500, 1000, 2000)의 조건을 설정하였다. 분석결과, 3단계 ML접근법과 3단계 BCH접근법의 원격 결과변인에서 양호도의 차이는 미미한 것으로 나타났지만, 소표본일 때 3단계 BCH접근법이 3단계 ML접근법에 비해 상대적으로 효율성이 높은 것으로 나타났다. 본 연구는 원격 결과변인의 보조변수 접근법에 기존에 활용되지 않았던 비연속 분할함수 성장혼합모형을 적용하고, 다양한 모의실험 조건을 설정하여 모의실험 연구를 수행하였다는 점에서 의의가 있다. 본 연구가 연구자들에게 실질적인 모형 활용 방안을 제시할 수 있을 것으로 기대하며, 연구의 제한점을 바탕으로 후속연구를 제언하였다.The purpose of this study was to compare the estimates of distal outcome variable in discontinuous growth mixture model. For this purpose, this study carried out a simulation study using a three-step maximum likelihood (ML) approach and a three-step BCH approach. The manipulated conditions were the class separation (1.3, 2.6), the mixing proportions (90:10, 70:30, 50:50), and the sample size (200, 500, 1000, 2000). The differences between ML and a three-step BCH approaches were not substantial, however, the three-step BCH approach was relatively more efficient than the three-step ML approach in the conditions of relatively small sample sizes. In particular, the bias of standard error was smaller in the conditions of the relatively less mixing proportions when three-step BCH approach was used. Limitations of the current study and suggestions for future study were provided.

The construction of rank hierarchies based on agonistic interactions between two individuals (“dyads”) is an important component in the characterization of the social structure of groups. To this end, winner‐loser matrices are typically created, which collapse the outcome of dyadic interactions over time, resulting in the loss of all information contained in the temporal domain. Methods that track changes in the outcome of dyadic interactions (such as “Elo‐ratings”) are receiving increasing interest. Critically, individual ratings are not just based on the succession of wins and losses, but depend on the values of start ratings and a shift coefficient. Recent studies improved existing methods by introducing a point estimation of these auxiliary parameters on the basis of a maximum likelihood (ML) approach. For a sound assessment of the rank hierarchies generated this way, we argue that measures of uncertainty of the estimates, as well as a quantification of the robustness of the methods, are also needed. We introduce a Bayesian inference (BI) approach using “partial pooling”, which rests on the assumption that all start ratings are samples from the same distribution. We compare the outcome of the ML approach to that of the BI approach using real‐world data. In addition, we simulate different scenarios to explore in which way the Elo‐rating responds to social events (such as rank changes), and low numbers of observations. Estimates of the start ratings based on “partial pooling” are more robust than those based on ML, also in scenarios where some individuals have only few observations. Our simulations show that assumed rank differences may fall well within the “uncertain” range, and that low sampling density, unbalanced designs, and coalitionary leaps involving several individuals within the hierarchy may yield unreliable results. Our results support the view that Elo‐rating can be a powerful tool in the analysis of social behaviour, when the data meet certain criteria. Assessing the uncertainty greatly aids in the interpretation of results. We advocate running simulation approaches to test how well Elo‐ratings reflect the (simulated) true structure and how sensitive the rating is to true changes in the hierarchy.

A comparative study of short cut procedures for parameter estimation in differential equations

Uncertainty assessment of flow and contaminant transport in the vadose zone entails probability density functions (PDFs) of soil hydraulic parameters. An unconventional maximum likelihood (ML) approach was used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten α and m) for a situation common in field‐scale modeling where core samples are sparse and prior PDFs of the parameters are unknown. In this situation, the unconventional ML approach approximates the PDFs as multivariate Gaussian. This study developed a method of estimating the mean and covariance of the multivariate Gaussian PDF based on the results of least square methods that can be easily obtained in practice. The developed method was applied to and evaluated through numerical simulation of unsaturated flow and tracer transport at the proposed Yucca Mountain geologic repository. Another focus of this study was to investigate the effect of uncertainty in the water retention parameters on predictive uncertainty. By comparing the predictive uncertainty before and after incorporating random water retention parameters, it was found that the random water retention parameters had limited effects on the mean predictions of the state variables including percolation flux and tracer travel time from the potential repository to the water table. Incorporating the uncertainty in the water retention parameters, however, significantly increased the magnitude and spatial extent of predictive uncertainty of the state variables. In particular, incorporating the random water retention parameters significantly changed the 5th and 95th percentiles of the tracer travel time by tens of thousands of years.

A fundamental issue in knowledge-aided space-time adaptive processing (KA-STAP) is to determine the degree of accuracy of the a~priori knowledge and the optimal emphasis that should be placed on it. In KA-STAP, the a priori knowledge consists usually of an initial guess of the clutter covariance matrix. This can be obtained either by previous radar probings or by a map-based study. We consider a linear combination of the a~priori clutter covariance matrix with the sample covariance matrix obtained from secondary data, and derive an optimal weighting factor on the a priori knowledge by a maximum likelihood (ML) approach. The performance of the ML approach for KA-STAP is evaluated based on numerically simulated data.

Co-clustering addresses the problem of simultaneous clustering of both dimensions of a data matrix. When dealing with high dimensional sparse data, co-clustering turns out to be more beneficial than one-sided clustering even if one is interested in clustering along one dimension only. Aside from being high dimensional and sparse, some datasets, such as document-term matrices, exhibit directional characteristics, and the $$L_2$$ normalization of such data, so that it lies on the surface of a unit hypersphere, is useful. Popular co-clustering assumptions such as Gaussian or Multinomial are inadequate for this type of data. In this paper, we extend the scope of co-clustering to directional data. We present Diagonal Block Mixture of Von Mises–Fisher distributions (dbmovMFs), a co-clustering model which is well suited for directional data lying on a unit hypersphere. By setting the estimate of the model parameters under the maximum likelihood (ML) and classification ML approaches, we develop a class of EM algorithms for estimating dbmovMFs from data. Extensive experiments, on several real-world datasets, confirm the advantage of our approach and demonstrate the effectiveness of our algorithms.

Building forecasting models for tropical cyclone intensity is one of the most challenging area in tropical cyclone research. Most, if not all, of the existing models have been built using variants of Maximum Likelihood (ML) approach. The need to partition data into two sets for model development is seen to be one of the drawbacks of ML approach in the face of limited available data. This paper proposes a way to build forecasting model using a number of model selection criteria which take the penalized-likelihood approach, namely MML, MDL, CAICF, SRM. These criteria claim to have the mechanism to balance between model complexity and goodness of fit. The models selected are then compared with the benchmark models being used in operation.

This essay is intended as an expository paper on how the seemingly abstract social
\nchoice models are related to the practical problems of formulating models of economic
\npolicy. The flexibility of the social choice models in dealing with the problem of relating
\nethical propositions to propositions concerning social actions is not widely understood.
\nThe arguments found in the literature are frequently somewhat mathematical and
\nauthors seldom attempt to relate their results, or even the reasons for their research,
\nto the underlying problems and motivations as they see them. In fact, however, as will be
\nargued here, social choice models provide a natural framework for the evaluation and
\nconstruction of economic policy. Most results can be directly interpreted in terms
\nof policy consequences. Let me make clear at the outset, however, that this paper
\nis not a summary of the arguments of several authors. The interpretations given here
\nvary considerably from the interpretations offered by those who first developed some
\nof the ideas.

On surfaces of constant societal loss in a model of social choice

In this paper we develop a mathematical model of social choice that provides the basis for a computer simulation of a political decisionmaking process. Prior research has demonstrated the nonexistence of an equilibrium for multidimensional voting games, a result that is obtained both globally and locally. Our simulation, while successfully modeling these results, provides a capability for analyzing the systemic effects (voting cycle characteristics) of iterated voting games that has not before been available to researchers. The structure of our computer model also anticipates development of dynamic models of social choice and provides a basic framework for eventual stability analyses of decisionmaking behavior in simulated and actual political environments.