Random Parameter Estimation Research Articles

An Approximate MSE Expression for Maximum Likelihood and Other Implicitly Defined Estimators of Non-Random Parameters

An approximate mean square error (MSE) expression for the performance analysis of implicitly defined estimators of non-random parameters is proposed. An implicitly defined estimator (IDE) declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well known examples of this class. In this paper, an exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is given. It is shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and misspecified CRLB in the large sample size regime for ML and misspecified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. In addition, extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems.

Accident prediction modelling for expressways: a review

Abstract Expressways are the need for fast movement of goods and human being for long distances in current scenario for every country, but at the same time these facilities have the highest severity rate among all the road categories. Various researchers around the world have tried to find out the effect of traffic volume, road geometry and environmental factors on frequency of accident on expressway by using different accident prediction modeling methods during last three decades. The purpose of this review paper is to find out the appropriate modeling method which can be used to predict accident frequency on the expressways in the developing country like India where traffic conditions, vehicular characteristics, and the driver behaviour are very different from the developing countries and to find out the research areas where the findings are inconclusive. Literature review suggests that among various models used so far, correlated random parameter model found to be the most advanced model to simultaneously account for both the heterogeneous effects of explanatory factors across the road segments and the cross correlations among the random parameter estimates. Findings related to variables like percentage of heavy vehicles, vertical gradient and lane width are not conclusive. However, safety effect of variables like speed limit, roadway lighting, pavement type and fog are less studied on expressway. The effect of allowing two wheelers on the expressways safety has not been studied yet, and the accidents related to fatigue or drowsiness also need examination.

ВОЗМОЖНОСТИ ПРИМЕНЕНИЯ БАЙЕСОВСКОГО ПОДХОДА В ИССЛЕДОВАНИИ СЛОЖНЫХ СОЦИАЛЬНО-ЭКОНОМИЧЕСКИХ СИСТЕМ

When studying complex socio-economic systems, there is a high degree of uncertainty, which leads to the need to introduce the concept of subjective probability and a vague set into the model. The concept of subjective probability implies personal beliefs and the degree of personal faith of an individual, which control the probability of a possible occurrence of some event. Under conditions of uncertainty, one of the consistent methods of mathematical reasoning when working with numerical information is the Bayesian approach, in which the randomness of parameters is interpreted as a property of the real world and that objects are subject to continuous unforeseen changes. According to this approach, estimates of random parameters are not random, based on this, the search for these estimates is carried out.

A note on out-of-sample prediction, marginal effects computations, and temporal testing with random parameters crash-injury severity models

Random parameters logit models have become an increasingly popular method to investigate crash-injury severities in recent years. However, there remain potential elements of the approach that need clarification including out-of-sample prediction, the calculation of marginal effects, and temporal instability testing. In this study, four models are considered for comparison: a fixed parameters multinomial logit model; a random parameters logit model; a random parameters logit model with heterogeneity in means; and a random parameters logit model with heterogeneity in means and variances. A full simulation of random parameters is undertaken for out-of-sample injury-severity predictions, and the prediction accuracy of the estimated models was assessed. Results indicate, not surprisingly, that the random parameters logit model with heterogeneity in the means and variances outperformed other models in predictive performance. Following this, two alternative methods for computing marginal effects are considered: one using Monte Carlo simulation and the other using individual estimates of random parameters. The empirical results indicate that both methods produced defensible results since the full distributions of random parameters are considered. Finally, two testing alternatives for temporal instability are evaluated: a global test across all time periods being considered, and a pairwise time-period to time-period comparison. It is shown that the pairwise comparison can provide more detailed insights into possible temporal variability.

Analyzing Prospective Owners’ Choice Decision towards Plug-in Hybrid Electric Vehicles in Urban India: A Stated Preference Discrete Choice Experiment

Plug-in Hybrid Electric Vehicles (PHEVs) can help decarbonize road transport in urban India. To accelerate the diffusion of PHEVs, investigation of commuter preferences towards the attributes of PHEVs is necessary. Therefore, the present study analyzes prospective owners’ choice decisions towards PHEVs in a typical Indian context. A stated preference survey was designed to collect responses from the current owners of conventional vehicles (CVs) in Delhi, India, and Mixed Logit (ML) models were developed to estimate commuters’ Willingness To Pay (WTP) for a set of key PHEV-specific attributes. The decomposition effect of prospective owners’ sociodemographic characteristics and trip characteristics on the mean estimates of random parameters was investigated by developing ML models with heterogeneity. Subsequently, the influence of improvement of each PHEV-specific attribute on prospective owners’ choice probability was investigated by calculating marginal effects. Among the various PHEV-specific attributes considered in the present study, high WTPs are observed for decrease in battery recharging time, reduction in tailpipe emission and increase in electric range. Therefore, an added emphasis on these attributes by vehicle manufacturers is likely to enhance the attractiveness of PHEVs to Indian commuters. The results also highlight the importance of government subsidy for promoting PHEVs in the Indian market. Prospective owners’ income, availability of home-based parking space, and average daily trip length are found to significantly influence the choice decision of Indian commuters towards PHEVs.

Spatial tourism supply: the case of ASEAN-5 countries

This article is the applied econometric analyses in estimating the spatial stochastic frontier (SSF) model. The heart of the modelling inference is Bayesian statistics with Gibbs sampling, which can provide flexible parametric distributions for capturing abnormal distributional observations. Yearly panel data between 2008 and 2018 regarding tourism supplies such as numbers of hotels, service rooms, and labour forces in ASEAN-5 countries (Vietnam, Indonesia, Malaysian, Singapore, and Thailand) is observed. The empirical results are divided into two major parts. The first section is the understanding of spatial effects which causally relate to the way for expanding tourism supplies is explained by the model validation using deviance information criterion (DIC). The panel spatial-error stochastic frontier (random-effect model) contains the minimum DIC which implies the parameters of locational effects is the minor, but it should be still considered for stimulating tourism supply sides in these five countries. The second section is the estimation of random parameters are processed to monitor technical efficiencies. Spatial effects are a specific impact on the group of connected borderline countries. On the other hand, this impact is dependently reduced when a country contains a remote area condition. Ultimately, this paper is the alternative to better analyse supply-chain management in the tourism sector without the restrictions of P-value, asymptotic struggling, and multicollinear problems. For further researches, subjective statistics (Bayesian) is the main issue that applies for modern econometrics.

Do high visibility crosswalks improve pedestrian safety? A correlated grouped random parameters approach using naturalistic driving study data

In this study, the effectiveness of High-Visibility Crosswalks (HVCs) in improving pedestrian safety at urban settings is assessed using SHRP2 (Second Strategic Highway Research Program) Naturalistic Driving Study (NDS) data. Various HVCs located at different positions on the roadway segment (mid-block vs end-of-block) and featuring different HVC marking designs (continental, bar-pair, and ladder) were selected for the assessment. As no pedestrian-vehicle crashes or conflicts were identified from the forward-facing videos and time series information of the SHRP2 Naturalistic Driving Study data, crash surrogate measures (i.e., speed; acceleration; throttle pedal actuation; and brake application) were employed to identify and analyze modifications in driving behavior at or near the HVCs.The surrogate measures were statistically modeled using a correlated grouped random parameters estimation framework. This can account for panel effects arising from multiple traversals undertaken by each participant, for the effect of unobserved characteristics, as well as for their unobserved correlations, which constitute possible misspecification issues of statistical modeling. The results of the analysis showed that the presence of HVC modifies driving behavior, thus reducing the risk of motor vehicle – pedestrian conflicts. Apart from the presence of HVC, the HVC type (ladder, continental or bar-pair), the HVC location (mid-block or end-of-block) and various driver, roadway and trip characteristics were found to affect the vehicle speed, acceleration, throttle pedal actuation, and brake application.

Assessing the impact of traffic signal performance on crash frequency for signalized intersections along urban arterials: A random parameter modeling approach

The recent development of Automated Traffic Signal Performance Measures (ATSPMs), has provided new opportunities and insights into traffic signal operations. As agencies begin to make decisions regarding investment in infrastructure and operation systems, it is imperative to understand the impacts these systems may have on safety. Past research has thoroughly investigated the impact of geometry and signal timing parameters on the safety of intersections, but little is understood on the relationship between improved signal performance and safety. This study uses vehicle trajectory data to create performance metrics for 121 signalized intersections on ten corridors near Columbus, Ohio. These metrics are used to understand the relationship between signal performance and safety. Two performance metrics, percent arrivals on green (POG) and level of travel time reliability (LOTTR), were used along with other volume and geometric data to model the total crash frequency on signalized mainline approaches. The crash data were modeled using a random parameters negative binomial approach. In consideration of potential unobserved heterogeneity between intersections, a correlated random parameters specification was tested alongside the traditional uncorrelated random parameters and fixed parameters model. Based on goodness of fit measures, the correlated random parameter model was chosen to interpret results because this model explains the complex cross-correlation among the estimates of random parameters. The elasticity values revealed a one percent increase in percent arrivals on green is associated with a reduction in total crashes by 1.12 %. The results of this study show the investment in signal operations and optimization result in an improvement in safety at signalized intersections. Further research should be explored to expand this study to additional intersections over a larger time period.

Remaining Useful Life Prediction Under Imperfect Prior Degradation Information

The remaining useful life (RUL) prediction is the core of equipment maintenance and decision-making. Accurate RUL prediction can make effective maintenance before the failure occurs to reduce the probability of equipment failure. However, in industrial practice, we often face the situation that prior information is insufficient or inaccurate, which could influence our prediction of RUL and reduce the prediction accuracy severely. To solve this problem, we study the issue of RUL prediction under imperfect prior degradation data by reasonably fusing the prior information and the field degradation data. Firstly, based on the linear Wiener degradation process, we prove a conclusion that the random parameter estimation results by the joint Bayesian algorithm and the expectation-maximization (EM) algorithm are the same as that by the maximum likelihood estimation (MLE) method. It shows that the joint estimation method completely overcomes the influence of prior information, and the more iterations, the smaller proportion of prior information. Secondly, we also prove that the random parameter joint updating method for the nonlinear Wiener degradation process has the same characteristics. Then, a heuristic algorithm that reasonably fuses the prior information and the field degradation data is proposed, which controls the number of iterations of the joint updating algorithm based on the credibility of the prior information. It is also applied to the nonlinear degradation process by not iterating the nonlinear coefficients. Finally, the correctness of the conclusion of this paper is verified by the simulated degradation data, and the effectiveness of the proposed heuristic updating method is verified by the actual lithium battery degradation data.

Noise Benefits in Combined Nonlinear Bayesian Estimators

This paper investigates the benefits of intentionally adding noise to a Bayesian estimator, which comprises a linear combination of a number of individual Bayesian estimators that are perturbed by mutually independent noise sources and multiplied by a set of adjustable weighting coefficients. We prove that the Bayes risk for the mean square error (MSE) criterion is minimized when the same optimum weighting coefficients are assigned to the identical estimators in the combiner. This property leads to a simplified analysis of the noise benefit to the MSE of the combined Bayesian estimator even when the number of individual estimators tends to infinity. It is shown that, for a sufficiently large number of individual estimators, the MSE of the designed Bayesian estimator approaches a plateau for a wide range of added noise levels. This robust feature facilitates the incorporation of the added noise into the design of Bayesian estimators without tuning the noise level. For an easily implementable Bayesian estimator composed of quantizers, the benefit of the symmetric scale-family noise is demonstrated, and the optimal noise probability density function is approximated by solving a constrained nonlinear optimization problem. We further extend this potential Bayesian estimator to the nonlinear filter design. Finally, examples of the noise benefits in random parameter estimation and nonlinear filtering support the theoretical analyses.

Analyzing road crash frequencies with uncorrelated and correlated random-parameters count models: An empirical assessment of multilane highways

Recent literature on highway safety research has focused on methodological advances to minimize misspecifications and the potential for erroneous estimates and invalid statistical inferences. To further these efforts, this study carries out an empirical assessment of uncorrelated and correlated random-parameters count models for analyzing road crash frequencies on multilane highways considering two crash severities; injury and no-injury. The empirical results indicate that the relative statistical performance of these models is comparable; however, the correlated random-parameters approach accounts for both the heterogeneous effects of explanatory factors across the road segments and the cross-correlations among the random-parameter estimates. As noted in the results, statistically significant correlation effects among the random parameters confirm the adequacy of this approach. The safety models for multilane highways presented in this study can be useful in (i) the detection of critical risk factors on these road types, (ii) the assessment of crash reduction due to improvements in pavement condition and retrofitting of roadway geometric features and, (iii) the prediction of crash frequency while comparing different design alternatives. As such, the outcomes of this study may assist design engineers and highway agencies in designing new or calibrating existing multilane highways from a safety standpoint.

Non-decreasing threshold variances in mixed generalized ordered response models: A negative correlations approach to variance reduction

Mixed Generalized Ordered Response (MGOR) models, that allow random heterogeneity in thresholds, are widely used to model ordered outcomes such as accident injury severity. This study highlights a potential limitation of these models, as applied in most empirical research, that the variances of the random thresholds are implicitly assumed to be in a non-decreasing order. This restriction is unnecessary and can lead to difficulty in estimation of random parameters in higher order thresholds. In this study, we investigate the use of negative correlations between random parameters as a variance reduction technique to relax the property of non-decreasing variances of thresholds in MGOR models. To this end, a simulation-based approach was used (where multiple datasets were simulated assuming a known negative correlation structure between the true parameters), and two models were estimated on each dataset – one allowing correlations between random parameters and the other not allowing such correlations. Allowing negative correlations helped relax the non-decreasing variance property of MGOR models. However, maximum simulated likelihood estimation of parameters on data with correlations occasionally encountered model convergence and parameter identification issues. Comparison of the models that did converge suggests that ignoring correlations leads to an estimation of fewer random parameters in the higher order thresholds and results in bias and/or loss of precision for a few parameter estimates. However, ignoring correlations leads to an adjustment of other parameter estimates such that overall likelihood values, predicted percentage shares, and the marginal effects are similar to those from the models with correlations.

Bayesian Estimation in the Presence of Deterministic Nuisance Parameters—Part I: Performance Bounds

How accurately can one estimate a random parameter subject to unknown deterministic nuisance parameters? The hybrid Cramer-Rao bound (HCRB) provides an answer to this question for a restricted class of estimators. The HCRB is the most popular performance bound on the mean-square-error (MSE) for random parameter estimation problems which involve deterministic parameters. The HCRB is useful when one is interested in both the random and the deterministic parameters and in the coupling between their estimation errors. This bound refers to locally weak-sense unbiased estimators with respect to (w.r.t.) the deterministic parameters. However, if these parameters are nuisance, it is unnecessary to restrict their estimation as unbiased. This paper is the first of a two-part study of Bayesian parameter estimation in the presence of deterministic nuisance parameters. It begins with a study on order relations between existing Cramer-Rao (CR)-type bounds of mean-unbiased Bayesian estimators. Then, a new CR-type bound is developed with no assumption of unbiasedness on the nuisance parameters. Alternatively, Lehmann’s concept of unbiasedness is employed rather than conventional mean-unbiasedness. It is imposed on a risk that measures the distance between the estimator and the minimum MSE (MMSE) estimator which assumes perfect knowledge of the nuisance parameters. In the succeeding paper, asymptotic performances of some Bayesian estimators with maximum likelihood based estimates for the nuisance parameters are investigated. The proposed risk-unbiased bound (RUB) is proved to be asymptotically achieved by the MMSE estimator with maximum likelihood estimates for the nuisance parameters, while the existing CR-type bounds are not necessarily achievable.

Optimal quantisation for random parameter estimation

In this study, the optimal quantiser design for random parameter estimation is investigated. The objective is to find a quantiser to minimise the variance of the estimation error by the minimum mean-square estimation. The main results are presented for the cases of high and low resolutions, respectively. For high resolution, multi-dimensional quantisation is considered and a quantitative relationship between the quantisation density and the probability density function is presented. For low-resolution case, an indirect method is developed for one-dimensional optimal quantisation by exploiting the results of high resolution case. The measurement space is first evenly divided into a number of small intervals, then the quantisation is approximately represented by the grouping of the small intervals. At last, a dynamic programming-based method is presented for the optimal grouping.

Stand basal area model for Cunninghamia lanceolata (Lamb.) Hook. plantations based on a multilevel nonlinear mixed-effect model across south-eastern China

Based on a multilevel nonlinear mixed-effect model approach, a stand basal area model was developed for Cunninghamia lanceolata (Lamb.) Hook. plantations belonging to the National Forest Inventory in China. The database consists of 583 plots embracing 18 different blocks within three seed source sites in this study. Of the plots, 80% were randomly selected for model fitting and 20% were carried out for model validation. The modified Chapman–Richards and Schumacher models were evaluated to find a basic model. The explanatory variables included stand dominant height, stand age and total number of stems per hectare. After selection of the basic model, the fitting and predictive ability of a multilevel nonlinear mixed-effect model was analysed. Site-, block- and plot-level random-effects terms were assessed for their contributions to improve model prediction over the ordinary least squares (OLS) method widely used in forest management. In addition, within-plot variance–covariance structure was taken into account owing to the repeated measurements and hierarchical structure of the data set. When evaluating the predictive accuracy of the final model, the first measurement was used for estimation of random parameters. The Chapman–Richards model was finally selected for the basic model based on model-fitting statistics, and both the fitting model and validation data with site-, block- and plot-level random effects showed a substantial improvement compared with the OLS method. After taking into account a reasonable variance–covariance structure, the model performed better than the model developed using only random effects.

Adjusting for differential misclassification in multilevel models: the relationship between child exposure to smoke and cognitive development

Misclassification is found in many of the variables used in social sciences and, in practice, tends to be ignored in statistical analyses, and this can lead to biased results. This paper shows how to correct for differential misclassification in multilevel models and illustrates the extent to which this changes fixed and random parameter estimates. Reliability studies on self-reported behaviour of pregnant women suggest that there may be differential misclassification related to smoking and, thus, to child exposure to smoke. Models are applied to the Millennium Cohort Study data. The response variable is the child cognitive development assessed by the British Ability Scales at 3 years of age and explanatory variables are child exposure to smoke and family income. The proposed method allows a correction for misclassification when the specificity and sensitivity are known, and the assessment of potential biases occurring in the multilevel model parameter estimates if a validation data sample is not available, which is often the case.

Estimating Nonparametric Random Utility Models with an Application to the Value of Time in Heterogeneous Populations

The estimation of random parameters by means of mixed logit models is now current practice for the analysis of transportation behaviour. One of the most straightforward applications is the derivation of willingness-to-pay distribution over a heterogeneous population, an important element for dynamic tolling strategies on congested networks. In numerous practical cases, the underlying discrete choice models involve parametric distributions that are a priori specified and whose parameters are estimated. This approach can however lead to many problems for realistic interpretation, such as negative value of time, etc. In this paper, we propose to capture the randomness present in the model by using a new nonparametric estimation method, based on the approximation of inverse cumulative distribution functions. This technique is applied to simulated data, and the ability to recover both parametric and nonparametric random vectors is tested. The nonparametric mixed logit model is also used on real data derived from a stated preference survey conducted in the region of Brussels (Belgium). The model presents multiple choices and is estimated on repeated observations. The obtained results provide a more realistic interpretation of the observed behaviours.

Parameters Estimation in Modelling of Gas-Gap in RBMK Type Reactor Using Bayesian Approach

This study presents developed algorithm for assessment and updating estimates of parameters in the mathematical models of non-stationary processes (for instance, system ageing model, dynamic system models and so on) with respect of prior information and new obtained observations. Proposed algorithm for updating estimates of random parameters is based on modified application of Bayesian approach (BA). Developed algorithm was applied for Ignalina NPP Unit 2 RBMK-1500 reactor's closure of the gas-gap between the pressure tubes and the graphite bore probabilistic analysis.

Local influence in multilevel models

AbstractThe authors study the local influence of observations in multilevel regression models. To this end, they perturb simultaneously the variances, responses and design matrix. To measure the local change caused by these perturbations, they use generalized Cook statistics for the fixed and random parameter estimates. Closed form local influence measures also allow them to assess the joint influence of various observations. They suggest a simple computation method and illustrate their results using two examples.

On estimating the scale parameter of the selected gamma population under the scale invariant squared error loss function

Let X 1 and X 2 be two independent random variables representing the populations Π 1 and Π 2 , respectively, and suppose that the random variable X i has a gamma distribution with shape parameter p, same for both the populations, and unknown scale parameter θ i , i = 1 , 2 . Define, M = 1 , if X 1 > X 2 , M = 2 , if X 2 > X 1 and J = 3 - M . We consider the component wise estimation of random parameters θ M and θ J , under the scale invariant squared error loss functions L 1 ( θ ̲ , δ 1 ) = ( δ 1 / θ M - 1 ) 2 and L 2 ( θ ̲ , δ 2 ) = ( δ 2 / θ J - 1 ) 2 , respectively. Sufficient conditions for the inadmissibility of equivariant estimators of θ M and θ J are derived. As a consequence, various natural estimators are shown to be inadmissible and better estimators are obtained.