Nonlinear Random Coefficient Research Articles

Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal with the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

Remaining useful life prediction based on nonlinear random coefficient regression model with fusing failure time data

Remaining useful life (RUL) prediction is one of the most crucial elements in prognostics and health management (PHM). Aiming at the imperfect prior information, this paper proposes an RUL prediction method based on a nonlinear random coefficient regression (RCR) model with fusing failure time data. Firstly, some interesting natures of parameters estimation based on the nonlinear RCR model are given. Based on these natures, the failure time data can be fused as the prior information reasonably. Specifically, the fixed parameters are calculated by the field degradation data of the evaluated equipment and the prior information of random coefficient is estimated with fusing the failure time data of congeneric equipment. Then, the prior information of the random coefficient is updated online under the Bayesian framework, the probability density function (PDF) of the RUL with considering the limitation of the failure threshold is performed. Finally, two case studies are used for experimental verification. Compared with the traditional Bayesian method, the proposed method can effectively reduce the influence of imperfect prior information and improve the accuracy of RUL prediction.

When can maximal efficacy occur with repeat botulinum toxin injection in upper limb spastic paresis?

Repeated injection cycles with abobotulinumtoxinA, a botulinum toxin type A, are recommended in current clinical guidelines as a treatment option for adults with upper limb spastic paresis. However, the magnitude of the maximal therapeutic effect of repeated abobotulinumtoxinA treatment across different efficacy parameters and the number of injection cycles required to reach maximal effect remain to be elucidated. Here, we present a post hoc exploratory analysis of a randomized, double-blind, placebo-controlled trial (12–24 weeks; NCT01313299) and open-label extension study (up to 12 months; NCT0131331), in patients aged 18–80 years with hemiparesis for ≥6 months after stroke/traumatic brain injury. Two inferential methods were used to assess the changes in efficacy parameters after repeat abobotulinumtoxinA treatment cycles: Mixed Model Repeated Measures analysis and Non-Linear Random Coefficients analysis. Using the latter model, the expected maximal effect size (not placebo-controlled) and the number of treatment cycles to reach 90% of this maximal effect were estimated. Treatment responses in terms of passive and perceived parameters (i.e. modified Ashworth scale in primary target muscle group, disability assessment scale for principal target for treatment or limb position, and angle of catch at fast speed) were estimated to reach near-maximal effect in two to three cycles. Near-maximal treatment effect for active parameters (i.e. active range of motion against the resistance of extrinsic finger flexors and active function, assessed by the Modified Frenchay Scale) was estimated to be reached one to two cycles later. In contrast to most parameters, active function showed greater improvements at Week 12 (estimated maximal change from baseline-modified Frenchay Scale overall score: +0.8 (95% confidence interval, 0.6; 1.0) than at Week 4 (+0.6 [95% confidence interval, 0.4; 0.8]). Overall, the analyses suggest that repeated treatment cycles with abobotulinumtoxinA in patients chronically affected with upper limb spastic paresis allow them to relearn how to use the affected arm with now looser antagonists. Future studies should assess active parameters as primary outcome measures over repeated treatment cycles, and assess efficacy at the 12-week time-point of each cycle, as the benefits of abobotulinumtoxinA may be underestimated in the studies of insufficient duration. Abbreviated summary In this post hoc analysis of repeated abobotulinumtoxinA injection cycles in upper limb spastic paresis, Gracies et al. used statistical modelling to elucidate the maximal therapeutic effect of abobotulinumtoxinA. Notably, the number of injections required to reach this maximal effect was higher for active (e.g. active function) compared with passive (e.g. tone) parameters.

A cost-driven reliability demonstration plan based on accelerated degradation tests

Reliability demonstration tests (RDTs) have been widely adopted to verify reliability requirements of manufacturing products. In practice, due to the limited resource and tight development schedule for new products, it is preferable to determine the decision variables including the termination time and the sample size for the RDT in advance. Existing degradation models often fail to capture the nonlinear degradation characteristics of testing items with complicated degradation mechanisms. This paper proposes a reliability demonstration method using an accelerated degradation test (ADT) in the context of a nonlinear random-coefficients model. First, we present the capabilities of the proposed ADT model to degradation data. Then, the cost-effective RDT plan is derived based on two types of decision risks and reliability requirements from both producers and customers, while meeting certain testing time constraints. The proposed method is illustrated using two practical examples. Finally, sensitivity analysis is provided to evaluate the robustness of the proposed RDT plan using ADT data.

Investigation of mixed model repeated measures analyses and non-linear random coefficient models in the context of long-term efficacy data.

The longitudinal data from 2 published clinical trials in adult subjects with upper limb spasticity (a randomized placebo-controlled study [NCT01313299] and its long-term open-label extension [NCT01313312]) were combined. Their study designs involved repeat intramuscular injections of abobotulinumtoxinA (Dysport®), and efficacy endpoints were collected accordingly. With the objective of characterizing the pattern of response across cycles, Mixed Model Repeated Measures analyses and Non-Linear Random Coefficient (NLRC) analyses were performed and their results compared. The Mixed Model Repeated Measures analyses, commonly used in the context of repeated measures with missing dependent data, did not involve any parametric shape for the curve of changes over time. Based on clinical expectations, the NLRC included a negative exponential function of the number of treatment cycles, with its asymptote and rate included as random coefficients in the model. Our analysis focused on 2 specific efficacy parameters reflecting complementary aspects of efficacy in the study population. A simulation study based on a similar study design was also performed to further assess the performance of each method under different patterns of response over time. This highlighted a gain of precision with the NLRC model, and most importantly the need for its assumptions to be verified to avoid potentially biased estimates. These analyses describe a typical situation and the conditions under which non-linear mixed modeling can provide additional insights on the behavior of efficacy parameters over time. Indeed, the resulting estimates from the negative exponential NLRC can help determine the expected maximal effect and the treatment duration required to reach it.

Equivalent Accelerated Degradation Test Plans in a Nonlinear Random Coefficients Models

Design of optimal Accelerated Degradation Testing (ADT) plan has been extensively researched over several decades. In practice, due to the rapidly changing development and assessment environment, pre-established plans often fail to meet reality. Therefore, designing a test plan that is equivalent to the target plan using an different stress-loading or a testing condition is needed to allow for more flexibility. However, there exists currently little work in the development of equivalent ADT plan. In this paper, we proposes an equivalent cost-effective accelerated degradation test (ADT) plan in the context of a nonlinear random-coefficients model. The proposed model is applied to a well-known constant-stress ADT problem in the literature.

Bayesian structured additive distributional regression with an application to regional income inequality in Germany

We propose a generic Bayesian framework for inference in distributional regression models in which each parameter of a potentially complex response distribution and not only the mean is related to a structured additive predictor. The latter is composed additively of a variety of different functional effect types such as nonlinear effects, spatial effects, random coefficients, interaction surfaces or other (possibly nonstandard) basis function representations. To enforce specific properties of the functional effects such as smoothness, informative multivariate Gaussian priors are assigned to the basis function coefficients. Inference can then be based on computationally efficient Markov chain Monte Carlo simulation techniques where a generic procedure makes use of distribution-specific iteratively weighted least squares approximations to the full conditionals. The framework of distributional regression encompasses many special cases relevant for treating nonstandard response structures such as highly skewed nonnegative responses, overdispersed and zero-inflated counts or shares including the possibility for zero- and one-inflation. We discuss distributional regression along a study on determinants of labour incomes for full-time working males in Germany with a particular focus on regional differences after the German reunification. Controlling for age, education, work experience and local disparities, we estimate full conditional income distributions allowing us to study various distributional quantities such as moments, quantiles or inequality measures in a consistent manner in one joint model. Detailed guidance on practical aspects of model choice including the selection of several competing distributions for labour incomes and the consideration of different covariate effects on the income distribution complete the distributional regression analysis. We find that next to a lower expected income, full-time working men in East Germany also face a more unequal income distribution than men in the West, ceteris paribus.

A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation-maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.

Cost-effective degradation test plan for a nonlinear random-coefficients model

The determination of requisite sample size and the inspection schedule considering both testing cost and accuracy has been an important issue in the degradation test. This paper proposes a cost-effective degradation test plan in the context of a nonlinear random-coefficients model, while meeting some precision constraints for failure-time distribution. We introduce a precision measure to quantify the information losses incurred by reducing testing resources. The precision measure is incorporated into time-varying cost functions to reflect real circumstances. We apply a hybrid genetic algorithm to general cost optimization problem with reasonable constraints on the level of testing precision in order to determine a cost-effective inspection scheme. The proposed method is applied to the degradation data of plasma display panels (PDPs) following a bi-exponential degradation model. Finally, sensitivity analysis via simulation is provided to evaluate the robustness of the proposed degradation test plan.

A hierarchical modeling approach to accelerated degradation testing data analysis: A case study

AbstractIn this case study, we investigate the degradation process of light‐emitting diodes (LEDs), which is used as a light source in DNA sequencing machines. Accelerated degradation tests are applied by varying temperature and forward current, and the light outputs are measured by a computerized measuring system. A degradation path model, which connects to the LED function recommended in Mitsuo (1991), is used in describing the degradation process. We consider variations in both measurement errors and degradation paths among individual test units. It is demonstrated that the hierarchical modeling approach is flexible and powerful in modeling a complex degradation process with nonlinear function and random coefficient. After fitting the model by maximum likelihood estimation, the failure time distribution can be obtained by simulation. Copyright © 2010 John Wiley & Sons, Ltd.

A prediction model of degradation rate for membrane electrode assemblies in direct methanol fuel cells

This article proposes a new prediction model to describe the nonlinear performance degradation paths of membrane electrode assemblies (MEAs) in direct methanol fuel cell (DMFC): a bi-exponential model with random coefficients. The bi-exponential model is constructed on a mathematical basis representing second-order kinetics. Performance variation between MEAs is incorporated by random coefficients in the proposed model. A likelihood ratio test is sequentially executed to select random effects in the nonlinear random-coefficients model. Analysis results indicate that the reliability estimation can be substantially improved by using the nonlinear random-coefficients model to incorporate two heterogeneous degradation characteristics of MEA performance during continuous operation of DMFC. Confidence intervals of failure-time distributions are obtained by the parametric bootstrap method.

Estimation of the Nonlinear Random Coefficient Model when Some Random Effects Are Separable

A method is presented for marginal maximum likelihood estimation of the nonlinear random coefficient model when the response function has some linear parameters. This is done by writing the marginal distribution of the repeated measures as a conditional distribution of the response given the nonlinear random effects. The resulting distribution then requires an integral equation that is of dimension equal to the number of nonlinear terms. For nonlinear functions that have linear coefficients, the improvement in computational speed and accuracy using the new algorithm can be dramatic. An illustration of the method with repeated measures data from a learning experiment is presented.

Analysis of Nonlinear Patterns of Change with Random Coefficient Models

Nonlinear patterns of change arise frequently in the analysis of repeated measures from longitudinal studies in psychology. The main feature of nonlinear development is that change is more rapid in some periods than in others. There generally also are strong individual differences, so although there is a general similarity of patterns for different persons over time, individuals exhibit substantial heterogeneity in their particular response. To describe data of this kind, researchers have extended the random coefficient model to accommodate nonlinear trajectories of change. It can often produce a statistically satisfying account of subject-specific development. In this review we describe and illustrate the main ideas of the nonlinear random coefficient model with concrete examples.

Fitting Partially Nonlinear Random Coefficient Models as SEMs

The nonlinear random coefficient model has become increasingly popular as a method for describing individual differences in longitudinal research. Although promising, the nonlinear model it is not utilized as often as it might be because software options are still somewhat limited. In this article we show that a specialized version of the model can be fit to data using SEM software. The specialization is to a model in which the parameters that enter the function in a linear manner are random, whereas those that are nonlinear are common to all individuals. Although this kind of function is not as general as is the fully nonlinear model, it still is applicable to many different data sets. Two examples are presented to show how the models can be estimated using popular SEM computer programs.

Bias Correction in Panel Data Models with Individual Specific Parameters

In correlated random coefficient models, standard OLS and IV estimators do not estimate the average population effect. This problem can be fixed with panel data by estimating a different coefficient for each individual, and then using the sample moment of the individual coefficients to estimate the corresponding population moment of interest. These estimates, however, can be severily biased in short panels due to the incidental parameters problem. The bias arises if some of the regressors are endogenous, or if the moments to estimate are nonlinear functions of the coefficients, e.g., variances of the individual effects. This paper introduces a class of bias-corrected fixed effects estimators for these correlated random coefficient models, which do not impose restrictions on the coefficients heterogeneity. The new estimators are based on moment conditions that can be nonlinear functions in parameters and variables, encompassing both linear and nonlinear random coefficients models and allowing for the presence of endogenous regressors. The corrections are derived from large-T expansions of the finite-sample bias, and reduce the order of this bias from O(T^{-1}) to O(T^{-2}) for model parameters and other quantities of interest, such as moments of the individual-specific coefficients. The asymptotic distribution of the bias-corrected estimators are centered at the true parameter values under asymptotic sequences where n = o(T^{3}). These methods are illustrated through an analysis of earnings equations for young men allowing the effect of the union status to be different for each individual. The results suggest the presence of important heterogeneity in the union premium.

Is anoxic depolarisation associated with an ADC threshold? A Markov chain Monte Carlo analysis

A Bayesian nonlinear hierarchical random coefficients model was used in a reanalysis of a previously published longitudinal study of the extracellular direct current (DC)-potential and apparent diffusion coefficient (ADC) responses to focal ischaemia. The main purpose was to examine the data for evidence of an ADC threshold for anoxic depolarisation. A Markov chain Monte Carlo simulation approach was adopted. The Metropolis algorithm was used to generate three parallel Markov chains and thus obtain a sampled posterior probability distribution for each of the DC-potential and ADC model parameters, together with a number of derived parameters. The latter were used in a subsequent threshold analysis. The analysis provided no evidence indicating a consistent and reproducible ADC threshold for anoxic depolarisation.

Reference Points and Loss Aversion in Intertemporal Choice

Based on the insights of Loewenstein's (1988) reference point model, we specify structural model for intertemporal choice that incorporates loss aversion and reference points. We consider four scenarios: Delay of gains, delay of losses, speed-up of gains, and speed-up of losses, using six years of panel data from a Dutch representative household survey. These data contain rich information on individual time preferences and other characteristics, and employ a non-linear random coefficients model with panel data to estimate reference points of delay and speedup, the coefficient of loss aversion and the discount rate. We find that on average the reference point of delay is larger than speedup, consistent with the hypothesis of Loewenstein; the mean coefficient of loss aversion is around two, females are more loss averse than males, and high education and age make people less loss-averse; high educated and older people are also more patient. The observed relationships of these parameters can be used to better predict and understand the behavior of households for policy purposes.

Temporal relation between the ADC and DC potential responses to transient focal ischemia in the rat: a Markov chain Monte Carlo simulation analysis.

Markov chain Monte Carlo simulation was used in a reanalysis of the longitudinal data obtained by Harris et al. (J Cereb Blood Flow Metab 20:28-36) in a study of the direct current (DC) potential and apparent diffusion coefficient (ADC) responses to focal ischemia. The main purpose was to provide a formal analysis of the temporal relationship between the ADC and DC responses, to explore the possible involvement of a common latent (driving) process. A Bayesian nonlinear hierarchical random coefficients model was adopted. DC and ADC transition parameter posterior probability distributions were generated using three parallel Markov chains created using the Metropolis algorithm. Particular attention was paid to the within-subject differences between the DC and ADC time course characteristics. The results show that the DC response is biphasic, whereas the ADC exhibits monophasic behavior, and that the two DC components are each distinguishable from the ADC response in their time dependencies. The DC and ADC changes are not, therefore, driven by a common latent process. This work demonstrates a general analytical approach to the multivariate, longitudinal data-processing problem that commonly arises in stroke and other biomedical research.

Optimum percentile estimating equations for nonlinear random coefficient models

In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.

Estimation of nonlinear random coefficient models

Nonlinear random coefficient models are found to be useful in growth studies and pharmacokinetic experiments. Several methods exist in the literature for estimating the parameters of such models. The properties of the estimators are not well studied. In this paper we summarize different estimation methods and examine some of their properties. We give an example that shows that most of the commonly used estimators are inconsistent.