Restricted Least-squares Research Articles

Analytic solution of chemical evolution models with Type Ia supernovae

Context. In recent years, a significant number of works have been focussed on finding analytic solutions for the chemical enrichment models of galactic systems, including the Milky Way. Some of these solutions, however, are not able to account for the enrichment produced by Type Ia supernovae (SNe) due to the presence of the delay time distributions (DTDs) in the models. Aims. We present a new analytic solution for the chemical evolution model of the Galaxy. This solution can be used with different prescriptions of the DTD, including the single- and double-degenerate scenarios, and allows for the inclusion of an arbitrary number of pristine gas infalls. Methods. We integrated the chemical evolution model by extending the instantaneous recycling approximation with the contribution of Type Ia SNe. This implies an extra term in the modelling that depends on the DTD. For DTDs that lead to non-analytic integrals, we describe them as a superposition of Gaussian, exponential, and 1/t functions using a restricted least-squares fitting method. Results. We obtained the exact solution for a chemical model with Type Ia SNe widely used in previous works, while managing to avoid numerical integration errors. This solution is able to reproduce the expected chemical evolution of the α and iron-peak elements in less computing time than numerical integration methods. We compare the pattern in the [Si/Fe] versus [Fe/H] plane observed by APOGEE DR17 with that predicted by the model. We find the low α sequence can be explained by a delayed gas infall. We exploit the applicability of our solution by modelling the chemical evolution of a simulated Milky Way-like galaxy from its star formation history. The implementation of our solution has been released as a PYTHON package. Conclusions. Our solution constitutes a promising tool for Galactic archaeology studies and it is able to model the observed trends in α element abundances versus [Fe/H] in the solar neighbourhood. We infer the chemical information of a simulated galaxy modelled without chemistry.

Intermittent demand, inventory obsolescence, and temporal aggregation forecasts

ABSTRACT Forecasting for intermittent demand is considered a difficult task and becomes even more challenging in the presence of obsolescence. Traditionally the problem has been dealt with modifications in the conventional parametric methods such as Croston. However, these methods are generally applied at the observed frequency, ignoring any additional information, such as trend that becomes prominent at higher levels of aggregation. We evaluate established Temporal Aggregation (TA) methods: ADIDA, Forecast Combination, and Temporal Hierarchies in the said context. We further employ restricted least-squares estimation and propose two new combination approaches tailored to decreasing demand scenarios. Finally, we test our propositions on both simulated and real datasets. Our empirical findings support the use of variable forecast combination weights to improve TA’s performance in intermittent demand items with a risk of obsolescence.

Covariate adjustment in multiarmed, possibly factorial experiments

AbstractRandomized experiments are the gold standard for causal inference and enable unbiased estimation of treatment effects. Regression adjustment provides a convenient way to incorporate covariate information for additional efficiency. This article provides a unified account of its utility for improving estimation efficiency in multiarmed experiments. We start with the commonly used additive and fully interacted models for regression adjustment in estimating average treatment effects (ATE), and clarify the trade-offs between the resulting ordinary least squares (OLS) estimators in terms of finite sample performance and asymptotic efficiency. We then move on to regression adjustment based on restricted least squares (RLS), and establish for the first time its properties for inferring ATE from the design-based perspective. The resulting inference has multiple guarantees. First, it is asymptotically efficient when the restriction is correctly specified. Second, it remains consistent as long as the restriction on the coefficients of the treatment indicators, if any, is correctly specified and separate from that on the coefficients of the treatment-covariate interactions. Third, it can have better finite sample performance than the unrestricted counterpart even when the restriction is moderately misspecified. It is thus our recommendation when the OLS fit of the fully interacted regression risks large finite sample variability in case of many covariates, many treatments, yet a moderate sample size. In addition, the newly established theory of RLS also provides a unified way of studying OLS-based inference from general regression specifications. As an illustration, we demonstrate its value for studying OLS-based regression adjustment in factorial experiments. Importantly, although we analyse inferential procedures that are motivated by OLS, we do not invoke any assumptions required by the underlying linear models.

Weighted-average least squares (WALS): Confidence and prediction intervals

We extend the results of De Luca et al. (2021) to inference for linear regression models based on weighted-average least squares (WALS), a frequentist model averaging approach with a Bayesian flavor. We concentrate on inference about a single focus parameter, interpreted as the causal effect of a policy or intervention, in the presence of a potentially large number of auxiliary parameters representing the nuisance component of the model. In our Monte Carlo simulations we compare the performance of WALS with that of several competing estimators, including the unrestricted least-squares estimator (with all auxiliary regressors) and the restricted least-squares estimator (with no auxiliary regressors), two post-selection estimators based on alternative model selection criteria (the Akaike and Bayesian information criteria), various versions of frequentist model averaging estimators (Mallows and jackknife), and one version of a popular shrinkage estimator (the adaptive LASSO). We discuss confidence intervals for the focus parameter and prediction intervals for the outcome of interest, and conclude that the WALS approach leads to superior confidence and prediction intervals, but only if we apply a bias correction.

Statistical inference for semiparametric varying-coefficient partially linear models with a diverging number of components

In applications, other than sample information, some prior information on parameters can be used to improve the estimation efficiency. In the framework of varying-coefficient partially linear models with the number of parametric and nonparametric components diverging, this paper proposes a restricted profile least-squares estimation for the parametric components after the varying coefficients are estimated by basis function approximations. This estimator is shown to be consistent and asymptotically normal under certain regularity conditions. To check the validity of the linear constraints on the parametric components, we construct a profile generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null and alternative hypotheses. Simulation studies are conducted and the Boston housing data is analyzed to illustrate the proposed method.

Sparsely restricted penalized estimators

Parameters of a linear regression model might be estimated with Ordinary Least Squares (OLS). If there are linear restrictions on model parameters, it is possible to use Restricted Least Squares (RLS). Both OLS and RLS fail when the number of predictors is large and not suitable for model selection in a sparse model. In contrast to OLS and RLS, Least Absolute Shrinkage and Selection Operator (LASSO) and Bridge estimators perform well in sparse models and can be used for model selection and estimation, simultaneously. A Restricted LASSO (RLASSO) estimator is proposed in the literature recently but it doesn’t provide sparse solutions. In this paper, we propose new sparsely restricted penalized estimators called Sparsely Restricted LASSO (SRL) and Sparsely Restricted Bridge (SRB) based on RLASSO and Restricted Bridge (RBridge). We show that SRL and SRB produces sparse solutions while satisfying restrictions of model parameters. We compare aforementioned estimators with Monte Carlo simulations. We also use prostate cancer dataset that is widely used in the literature for a numerical application. Our comparisons show that SRL and SRB outperform the remaining estimators in terms of model selection performance and mean squared error.

Immigration and the rate of population mixing: explorations with a stylized model

The integration or mixing of immigrants with non-immigrants is an important issue in many countries. There are various forms of mixing. We consider here cross-parenting, the bearing of children with one immigrant parent and one non-immigrant. Our objective is to model cross-parenting as a demographic process and investigate the rate at which such mixing could occur.We identify three populations within an overall total: non-immigrant, immigrant, and mixed. A model is constructed to track the three as they change and interact through cross-parenting. The populations evolve by simulation in accordance with a common stable projection matrix. However, as cross-parenting between immigrants and non-immigrants occurs, the progeny are transferred to the mixed population; the immigrant and non-immigrant populations are thus depleted by the transfers and the mixed population augmented in each generation. The transfers are governed by underlying preferences, but the preference pattern must be modified to recognize constraints imposed by differences in population size. A restricted least-squares procedure effects the modification so that the actual pattern is as close as possible to the preferred one. Simulations are carried out with alternative preferential patterns and rates of immigration. Of particular interest is the proportion of mixed population in the total in each generation and the final steady state.The paper develops a new framework and model to show the rate at which population mixing could occur under alternative assumptions about the immigration rate and preferences for cross-parenting.JEL Classification: J10, J15

Determination of Markov Transition Probabilities to be Used in Bridge Management from Mechanistic-Empirical Models

Many bridge management systems use Markov models to predict the future deterioration of structural elements. This information is subsequently used in the determination of optimal intervention strategies and intervention programs. The input for these Markov models often consists of the condition states of the elements and how they have changed over time. This input is used to estimate the probabilities of transition of an object from each possible condition state to each other possible condition state in one time period. A complication in using Markov models is that there are situations in which there is an inadequate amount of data to estimate the transition probabilities using traditional methods (e.g., due to the lack of recording past information so that it can be easily retrieved, or because it has been collected in an inconsistent or biased manner). In this paper, a methodology to estimate the transition probabilities is presented that uses proportional data obtained by mechanistic-empirical models of the deterioration process. A restricted least-squares optimization model is used to estimate the transition probabilities. The methodology is demonstrated by using it to estimate the transition probabilities for a reinforced concrete (RC) bridge element exposed to chloride-induced corrosion. The proportional data are generated by modeling the corrosion process using mechanistic-empirical models and Monte Carlo simulations.

Immigration and the Rate of Population Mixing: Explorations with a Stylized Model

Immigrants can mix with the population of a receiving country in various ways. We consider demographic mixing by which we mean cross-mating, and more particularly the bearing of children where one parent is of immigrant descent and the other is not – cross-parenting as we term it. We consider a hypothetical country with an initial stable population and introduce immigration. The results of cross-parenting are taken into account by identifying three separate populations within the overall total: non-immigrant population, immigrant population (immigrants and their descendants), and mixed population. We develop a stylized model to track the three populations, with interest focusing in particular on how the proportion of mixed population changes through time as it moves toward a steady state. The model has a stable projection (Leslie) matrix that holds for all three populations and moves them forward from generation to generation as each evolves in its own way. As cross-parenting occurs the resulting progeny are transferred from the other populations to the mixed population. The pattern of cross-parenting is determined in the first instance by a matrix representing preferences among the three populations and alternative preferential patterns are experimented with, ranging from complete isolation to indifference as to cross-parenting choices. However the matrix must be modified to recognize supply constraints imposed by the sizes of the available populations and a restricted least-squares procedure is employed to effect the modification while remaining as close as possible to the original preference pattern. Alternative rates of immigration are experimented with also.

The feasible generalized restricted ridge regression estimator

ABSTRACTThe presence of autocorrelation in errors and multicollinearity among the regressors have undesirable effects on the least-squares regression. There are a wide range of methods which are proposed to overcome the usefulness of the ordinary least-squares estimator or the generalized least-squares estimator, such as the Stein-rule, restricted least-squares or ridge estimator. Therefore, we introduce a new feasible generalized restricted ridge regression (FGRR) estimator to examine multicollinearity and autocorrelation problems simultaneously for the general linear regression model. We also derive some statistical properties of the FGRR estimator and comparisons have been conducted using matrix mean-square error. Moreover, a Monte Carlo simulation experiment is performed to investigate the performance of the proposed estimator over the others.

Comparisons of estimators for regression coefficient in a misspecified linear model with elliptically contoured errors

In a misspecified linear regression model with elliptically contoured errors, the exact risks of generalized least squares (GLS), restricted least squares (RLS), preliminary test (PT), Stein-rule (SR) and positive-rule shrinkage (PRS) estimators of regression coefficient are derived. Risk superiority conditions dependent on prior constraint error and the model specification error are obtained. When the model is misspecified and the error terms obey the elliptically contoured distribution, it is shown analytically that the PRS estimator dominates uniformly the SR estimator. However, the PRS and SR estimators do not dominate uniformly the GLS estimator. Furthermore, the dominance of the RLS estimator over the GLS estimator does not hold necessarily even if the model has linear constraint.

Improved Liu-type estimator in partial linear model

In this article, a Liu-type estimation is proposed for the vector-parameter in a partial linear model. This new estimator can be regarded as generalization of the restricted least-squares estimator, the restricted ridge estimator and the restricted Liu estimator. We also obtain the asymptotic distributional bias and risk of these estimators and we also discuss some properties of the new estimator. The selection of the tuning parameter in the proposed estimator is also presented. Finally, a simulation study is presented to explain the performance of the new estimator.

A proposedformula for solving the problem of multicollinearity and restricted data with A simulation

This study found out a method derived from ridge regression and restricted least-squares methods to solve the problem of multicollinearity and restricted data at the same time. A simulation study including these problems was carried out . The study used the mean squares error (MSE) to test the efficiency of the Proposedformula. The study concludes that the Proposed formula results in the lowest mean squares error compared with other methods.

The relative efficiency of the restricted estimators in linear regression models

This paper deals with the problem of multicollinearity in a multiple linear regression model with linear equality restrictions. The restricted two parameter estimator which was proposed in case of multicollinearity satisfies the restrictions. The performance of the restricted two parameter estimator over the restricted least squares (RLS) estimator and the ordinary least squares (OLS) estimator is examined under the mean square error (MSE) matrix criterion when the restrictions are correct and not correct. The necessary and sufficient conditions for the restricted ridge regression, restricted Liu and restricted shrunken estimators, which are the special cases of the restricted two parameter estimator, to have a smaller MSE matrix than the RLS and the OLS estimators are derived when the restrictions hold true and do not hold true. Theoretical results are illustrated with numerical examples based on Webster, Gunst and Mason data and Gorman and Toman data. We conduct a final demonstration of the performance of the estimators by running a Monte Carlo simulation which shows that when the variance of the error term and the correlation between the explanatory variables are large, the restricted two parameter estimator performs better than the RLS estimator and the OLS estimator under the configurations examined.

Profile Inferences on Restricted Additive Partially Linear EV Models

We consider the testing problem for the parameter and restricted estimator for the nonparametric component in the additive partially linear errors-in-variables (EV) models under additional restricted condition. We propose a profile Lagrange multiplier test statistic based on modified profile least-squares method and two-stage restricted estimator for the nonparametric component. We derive two important results. One is that, without requiring the undersmoothing of the nonparametric components, the proposed test statistic is proved asymptotically to be a standard chi-square distribution under the null hypothesis and a noncentral chi-square distribution under the alternative hypothesis. These results are the same as the results derived by Wei and Wang (2012) for their adjusted test statistic. But our method does not need an adjustment and is easier to implement especially for the unknown covariance of measurement error. The other is that asymptotic distribution of proposed two-stage restricted estimator of the nonparametric component is asymptotically normal and has an oracle property in the sense that, though the other component is unknown, the estimator performs well as if it was known. Some simulation studies are carried out to illustrate relevant performances with a finite sample. The asymptotic distribution of the restricted corrected-profile least-squares estimator, which has not been considered by Wei and Wang (2012), is also investigated.

Statistical inference on semi-parametric partial linear additive models

In the framework of partial linear additive models, we first develop a profile least-squares estimation of the parametric component based on Liang et al.’s [(2008), ‘Additive Partial Linear Models with Measurement Errors’, Biometrika, 95(3), 667–678] work. This estimator is shown to be asymptotically normal and root-n consistent without requirement of undersmoothing of the nonparametric component. Next, when some additional linear restrictions on the parametric component are available, we postulate a restricted profile least-squares estimator for the parametric component and prove the asymptotic normality of the resulting estimator. To check the validity of the linear constraints on the parametric component, we explore a generalised likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Thus, the result unveils a new Wilks type of phenomenon. Simulation studies are conducted to illustrate the proposed methods. An application to the crime rate data in Columbus (Ohio) has been carried out.

Prioritizing strategies for comprehensive liver cancer control in Asia: a conjoint analysis

BackgroundLiver cancer is a complex and burdensome disease, with Asia accounting for 75% of known cases. Comprehensive cancer control requires the use of multiple strategies, but various stakeholders may have different views as to which strategies should have the highest priority. This study identified priorities across multiple strategies for comprehensive liver cancer control (CLCC) from the perspective of liver cancer clinical, policy, and advocacy stakeholders in China, Japan, South Korea and Taiwan. Concordance of priorities was assessed across the region and across respondent roles.MethodsPriorities for CLCC were examined as part of a cross-sectional survey of liver cancer experts. Respondents completed several conjoint-analysis choice tasks to prioritize 11 strategies. In each task, respondents judged which of two competing CLCC plans, consisting of mutually exclusive and exhaustive subsets of the strategies, would have the greatest impact. The dependent variable was the chosen plan, which was then regressed on the strategies of different plans. The restricted least squares (RLS) method was utilized to compare aggregate and stratified models, and t-tests and Wald tests were used to test for significance and concordance, respectively.ResultsEighty respondents (69.6%) were eligible and completed the survey. Their primary interests were hepatitis (26%), hepatocellular carcinoma (HCC) (58%), metastatic liver cancer (10%) and transplantation (6%). The most preferred strategies were monitoring at-risk populations (p<0.001), clinician education (p<0.001), and national guidelines (p<0.001). Most priorities were concordant across sites except for three strategies: transplantation infrastructure (p=0.009) was valued lower in China, measuring social burden (p=0.037) was valued higher in Taiwan, and national guidelines (p=0.025) was valued higher in China. Priorities did not differ across stakeholder groups (p=0.438).ConclusionsPriorities for CLCC in Asia include monitoring at-risk populations, clinician education, national guidelines, multidisciplinary management, public awareness and centers of excellence. As most priorities are relatively concordant across the region, multilateral approaches to addressing comprehensive liver cancer would be beneficial. However, where priorities are discordant among sites, such as transplantation infrastructure, strategies should be tailored to local needs.

Network-based Collaborative Navigation in GPS-Denied Environment

Global Positioning System (GPS) has been used as a primary source of navigation in land and airborne applications. However, challenging environments cause GPS signal blockage or degradation, and prevent reliable and seamless positioning and navigation using GPS only. Therefore, multi-sensor based navigation systems have been developed to overcome the limitations of GPS by adding some forms of augmentation. The next step towards assured robust navigation is to combine information from multiple ground-users, to further improve the chance of obtaining reliable navigation and positioning information. Collaborative (or cooperative) navigation can improve the individual navigation solution in terms of both accuracy and coverage, and may reduce the system's design cost, as equipping all users with high performance multi-sensor positioning systems is not cost effective. Generally, ‘Collaborative Navigation’ uses inter-nodal range measurements between platforms (users) to strengthen the navigation solution. In the collaborative navigation approach, the inter-nodal distance vectors from the known or more accurate positions to the unknown locations can be established. Therefore, the collaborative navigation technique has the advantage in that errors at the user's position can be compensated by other known (or more accurate) positions of other platforms, and may result in the improvement of the navigation solutions for the entire group of users. In this paper, three statistical network-based collaborative navigation algorithms, the Restricted Least-Squares Solution (RLESS), the Stochastic Constrained Least-Squares Solution (SCLESS) and the Best Linear Minimum Partial Bias Estimation (BLIMPBE) are proposed and compared to the Kalman filter. The proposed statistical collaborative navigation algorithms for network solution show better performance than the Kalman filter.

Ridge regression methodology in partial linear models with correlated errors

In this article, a generalized restricted difference-based ridge estimator is defined for the vector parameter in a partial linear model when the errors are dependent. It is suspected that some additional linear constraints may hold on to the whole parameter space. The estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. The risk functions of the proposed estimators are derived under balanced loss function. Finally, the performance of the new estimators is evaluated by a simulated data set.

Semiparametric Ridge Regression Approach in Partially Linear Models

In this article, we introduce a semiparametric ridge regression estimator for the vector-parameter in a partial linear model. It is also assumed that some additional artificial linear restrictions are imposed to the whole parameter space and the errors are dependent. This estimator is a generalization of the well-known restricted least-squares estimator and is confined to the (affine) subspace which is generated by the restrictions. Asymptotic distributional bias and risk are also derived and the comparison result is then given.