Unrestricted Estimator Research Articles

Preliminary Test Estimation for Parallel 2-Sampling in Autoregressive Model

The purpose of this paper is to discuss the problem of estimation and testing the equality of two autoregressive parameters of two first-order autoregressive processes AR(1), where for each process, the observations are made at different time points. The primary interest is to propose the testing procedures for the homogeneity of autocorrelation parameters ρ1 and ρ2. Furthermore, we are interested in estimating ρ1 under uncertain and weak prior information about the possible equality of ρ1 and ρ2, though we may not have full confidence in the tenacity of this information. A large sample test for the homogeneity of the parameters is developed. Pooled “P” (or restricted estimator) and preliminary test “PT” estimators are proposed, and their properties are investigated and compared with the unrestricted estimator “UE” of ρ1.

Shrinkage Estimation and Forecasting in Dynamic Regression Models Under Structural Instability

Abstract This paper introduces a Stein-like shrinkage method for estimating slope coefficients and forecasting in first order dynamic regression models under structural breaks. The model allows for unit root and non-stationary regressors. The proposed shrinkage estimator is a weighted average of a restricted estimator that ignores the break in the slope coefficients, and an unrestricted estimator that uses the observations within each regime. The restricted estimator is the most efficient estimator but inconsistent when there is a break. However, the unrestricted estimator is consistent but not efficient. Therefore, the proposed shrinkage estimator balances the trade-off between the bias and variance efficiency of the restricted estimator. The averaging weight is proportional to the weighted distance of the restricted estimator, and the unrestricted estimator. We derive the analytical large-sample approximation of the bias, mean squared error, and risk for the shrinkage estimator, the unrestricted estimator, and the restricted estimator. We show that the risk of the shrinkage estimator is lower than the risk of the unrestricted estimator under any break size and break points. Moreover, we extend the results for the model with a unit root and non-stationary regressors. We evaluate the finite sample performance of our proposed method via extensive simulation study, and empirically in forecasting output growth.

Improved estimators in bell regression model with application

In this paper, we propose the application of shrinkage strategies to estimate coefficients in the Bell regression models when prior information about the coefficients is available. The Bell regression models are well-suited for modelling count data with multiple covariates. Furthermore, we provide a detailed explanation of the asymptotic properties of the proposed estimators, including asymptotic biases and mean squared errors. To assess the performance of the estimators, we conduct numerical studies using Monte Carlo simulations and evaluate their simulated relative efficiency. The results demonstrate that the suggested estimators outperform the unrestricted estimator when prior information is taken into account. Additionally, we present an empirical application to demonstrate the practical utility of the suggested estimators.

Improved estimation in a multivariate regression with measurement error

In this paper, we study the estimation problem about the regression coefficients of a multivariate regression model with measurement errors under some uncertain restrictions. Specifically, we propose the unrestricted estimator (UE) and three restricted estimators (REs), and prove that they are all consistent for the true coefficients. We derive the asymptotic distributions of the proposed estimators under the sequence of local alternative restrictions. We also propose shrinkage estimators (SEs) to address the problem of the uncertainty of the restrictions. In addition, we establish the asymptotic distributional risk (ADR) of the proposed estimators and compare the risk performance of these estimators. It is established that the REs perform better than the UE only near the restriction, while they perform poorly as one moves farther away from the restriction. We also prove that SEs dominate the UE. These theoretical results are confirmed by simulations.

Structural breaks in seemingly unrelated regression models

AbstractThis paper develops an efficient Stein-like shrinkage estimator for estimating slope parameters under structural breaks in seemingly unrelated regression models, which is then used for forecasting. The proposed method is a weighted average of two estimators: a restricted estimator that estimates the parameters under the restriction of no break in the coefficients, and an unrestricted estimator that considers break points and estimates the parameters using the observations within each regime. It is established that the asymptotic risk of the Stein-like shrinkage estimator is smaller than that of the unrestricted estimator, which is the method typically used to estimate the slope coefficients under structural breaks. Furthermore, this paper proposes an averaging minimal mean squared error estimator in which the averaging weight is derived by minimizing its asymptotic risk. Insights from the theoretical analysis are demonstrated in Monte Carlo simulations and through an empirical example of forecasting output growth of G7 countries.

James-Stein type estimators in beta regression model: simulation and application

Recently, the beta regression model has been used in several fields of science to model data in the form of rate or proportion. In this paper, some novel and improved methods to estimate parameters in the beta regression model are proposed. We consider a sub-space on the regression coefficients of the beta regression model and combine the unrestricted and restricted estimators then we present Stein-type and preliminary estimators. We develop the expressions for the proposed estimators' asymptotic biases and their quadratic risks. Numerical studies through Monte Carlo simulations are used to evaluate the performance of the proposed estimators in terms of their simulated relative efficiency. The results show that the proposed estimators outperform the unrestricted estimator when the restrictions hold. Finally, an empirical application is given to show how useful the proposed estimators are in the practical area.

Linear Shrinkage and Shrinkage Pretest Strategies in Partially Linear Models

In this paper, we improved the efficiency of parameter estimation in partially linear models, where subspace information is available. We proposed linear shrinkage and shrinkage pretest estimation strategies. The asymptotic distributional risk of the proposed estimators was examined. We also conducted a Monte Carlo simulation to evaluate the risk performance of the estimators. The proposed estimators performed better than the unrestricted estimator. A real data example was used to illustrated the application of the proposed estimators.

Liu-type shrinkage strategies in zero-inflated negative binomial models with application to Expenditure and Default Data

In modeling count data with overdispersion and extra zeros, zero-inflated negative binomial (ZINB) regression model is useful. In a regression model, the multicollinearity problem arises when there are some high correlations between predictor variables. This problem leads to the maximum likelihood method will not be an efficient estimator. The ridge and Liu-type estimators have been proposed to combat the multicollinearity problem so that the Liu-type estimator is better. In this paper, we proposed the Liu-type shrinkage estimators, namely linear shrinkage, preliminary test, shrinkage preliminary test, Stein-type, and positive Stein-type Liu estimators to estimate the count parameters in the ZINB model, when some of the predictor variables have not a significant effect to predict the response variable so that a sub-model may be sufficient. The asymptotic distributional biases and variances of the proposed estimators are nicely demonstrated. We also compared the performance of the Liu-type shrinkage estimators along with the Liu-type unrestricted estimator by using an extensive Monte Carlo simulation study. The results show that the performances of the proposed estimators are superior to those based on Liu-type unrestricted estimators. We also applied the proposed estimation methods to Expenditure and Default Data.

Efficient estimation method for generalized ARFIMA models

This paper focuses on pretest and shrinkage estimation strategies for generalized autoregressive fractionally integrated moving average (GARFIMA) models when some of the regression parameters are possible to restrict to a subspace. These estimation strategies are constructed on the assumption that some covariates are not statistically significant for the response. To define the pretest and shrinkage estimators, we fit two models: one includes all the covariates and the others are subject to linear constraint based on the auxiliary information of the insignificant covariates. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We enlighten the statistical properties of the shrinkage and pretest estimators in terms of asymptotic bias and risk. We examine the comparative performance of pretest and shrinkage estimators with respect to unrestricted maximum partial likelihood estimator (UMPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the UMPLE when the number of significant covariates exceeds two. Monte Carlo simulations are conducted to examine the relative performance of the proposed estimators to the UMPLE. An empirical application is used for the usefulness of our proposed estimation strategies.

The risk of tensor Stein-rules in elliptically contoured distributions

In this paper, we consider an estimation problem of a tensor parameter of a random tensor which follows elliptically contoured distribution. We establish the risk functions of a class of tensor shrinkage estimators of the mean parameter tensor of a random tensor which follows an elliptically contoured distribution. In particular, we generalize some recent findings in three ways. First, the problem considered extends the one about a matrix estimation. Second, we generalize some recent identities which are useful in establishing the risk of tensor shrinkage estimators. Third, we derive a more general sufficient condition for the tensor shrinkage estimators to dominate the unrestricted estimator. The additional novelty of the derived results consists in the fact that, on the top of the complexity related to tensor random fact, the case considered here takes into account a possible correlation between the shrinking factor and the restricted estimator. Finally, in order to illustrate the applications of the proposed method, we present some simulation results and a real dataset analysis.

Improved estimation in tensor regression with multiple change-points

In this paper, we consider an estimation problem about the tensor coefficient in a tensor regression model with multiple and unknown change-points. We generalize some recent findings in five ways. First, the problem studied is more general than the one in context of a matrix parameter with multiple change-points. Second, we develop asymptotic results of the tensor estimators in the context of a tensor regression with unknown change-points. Third, we construct a class of shrinkage tensor estimators that encompasses the unrestricted estimator (UE) and the restricted estimator (RE). Fourth, we generalize some identities which are crucial in deriving the asymptotic distributional risk (ADR) of the tensor estimators. Fifth, we show that the proposed shrinkage estimators perform better than the UE. The additional novelty of the established results consists in the fact that the dependence structure of the errors is as weak as that of an L2-mixingale. Finally, the theoretical results are corroborated by the simulation findings and our methods are applied to analyse MRI and fMRI datasets.

Improved shrinkage estimators in zero-inflated negative binomial regression model

‎Zero-inflated negative binomial model is an appropriate choice to model count response variables with excessive zeros and over-dispersion simultaneously. ‎This paper addressed parameter estimation in the zero-inflated negative binomial model when there are many parameters, ‎so that some of them have not influence on the response variable. ‎We proposed parameter estimation based on the linear shrinkage, ‎pretest, ‎shrinkage pretest, ‎Stein-type, ‎and positive Stein-type estimators. ‎We obtained the asymptotic distributional biases and risks of the suggested estimators theoretically. ‎‎We also conducted a Monte Carlo simulation study ‎to compare the performance of each estimator with the unrestricted estimator using simulated relative efficiency (SRE) criterion.‎‎ ‎The results reveal that the SREs of proposed estimators are higher than the unrestricted estimator. ‎The suggested estimators were applied to the wildlife fish data to appraise their performance.

Estimation of two ordered quantile residual life functions based on mixtures

This paper investigates some properties of the mixture models from the quantile residual life perspective. It is shown that a mixture model is bounded by its components via the quantile residual life. We investigate how mixture models are ordered in terms of the quantile residual life function when their components are ordered. Besides, we prove that the limiting quantile residual life of a mixture is similar to that of the greatest component at infinity. Based on these results, it is possible to construct estimators of two quantile residual life functions subject to an order restriction. Such estimators are shown to be strongly uniformly consistent and asymptotically unbiased. We develop the weak convergence theory for these estimators. Simulations seem to indicate that both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of the mean squared error, uniformly at all quantiles, and for a variety of distributions.

Isotonic regression discontinuity designs

This paper studies the estimation and inference for the isotonic regression at the boundary point, an object that is particularly interesting and required in the analysis of monotone regression discontinuity designs. We show that the isotonic regression is inconsistent in this setting and derive the asymptotic distributions of boundary corrected estimators. Interestingly, the boundary corrected estimators can be bootstrapped without subsampling or additional nonparametric smoothing which is not the case for the interior point. The Monte Carlo experiments indicate that shape restrictions can improve dramatically the finite-sample performance of unrestricted estimators. Lastly, we estimate the causal effect of incumbency in U.S. House elections via the isotonic regression discontinuity design.

Optimal shrinkage estimations in partially linear single-index models for binary longitudinal data

This paper focuses on the optimal estimation strategies of partially linear single-index models (PLSIM) for binary longitudinal data. Fitting model between the response and covariates may cause complexity and the linear terms may not be adequate to represent the relationship. In this situation, the PLSIM containing both linear and nonlinear terms is preferable. The objective of this paper is to develop optimal estimation strategies such as, pretest and shrinkage methods, for the analysis of binary longitudinal data under the PLSIM where some regression parameters are subject to restrictions. We estimate the nonparametric component using kernel estimating equations, and then use profile estimating equations to estimate the unrestricted and restricted estimators. To apply the pretest and shrinkage methods, we fit two models: one includes all covariates and the other restricts the regression parameters based on the auxiliary information. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We also derive the asymptotic properties of the estimators in terms of biases and risks. Monte Carlo simulations are also conducted to examine the relative performance of the proposed estimators to the unrestricted estimator. An empirical application is also be used to illustrate the usefulness of our methodology.

Inference problem in generalized fractional Ornstein–Uhlenbeck processes with change-point

In this paper, we study an inference problem in generalized fractional Ornstein–Uhlenbeck (O–U) processes with an unknown change-point when the drift parameter is suspected to satisfy some constraints. The constraint considered is very general and, the testing problem studied generalizes a very recent inference problem in generalized O–U processes. We derive the unrestricted estimator (UE) and the restricted estimator (RE) and we establish the asymptotic properties of the UE and RE. We also propose some shrinkage-type estimators (SEs) as well as a test for testing the constraint. Finally, we derive the asymptotic power of the proposed test and we study the relative risk dominance of the proposed estimators.

Modeling I(2) Processes Using Vector Autoregressions Where the Lag Length Increases with the Sample Size

In this paper the theory on the estimation of vector autoregressive (VAR) models for I(2) processes is extended to the case of long VAR approximation of more general processes. Hereby the order of the autoregression is allowed to tend to infinity at a certain rate depending on the sample size. We deal with unrestricted OLS estimators (in the model formulated in levels as well as in vector error correction form) as well as with two stage estimation (2SI2) in the vector error correction model (VECM) formulation. Our main results are analogous to the I(1) case: We show that the long VAR approximation leads to consistent estimates of the long and short run dynamics. Furthermore, tests on the autoregressive coefficients follow standard asymptotics. The pseudo likelihood ratio tests on the cointegrating ranks (using the Gaussian likelihood) used in the 2SI2 algorithm show under the null hypothesis the same distributions as in the case of data generating processes following finite order VARs. The same holds true for the asymptotic distribution of the long run dynamics both in the unrestricted VECM estimation and the reduced rank regression in the 2SI2 algorithm. Building on these results we show that if the data is generated by an invertible VARMA process, the VAR approximation can be used in order to derive a consistent initial estimator for subsequent pseudo likelihood optimization in the VARMA model.

A ridge to homogeneity for linear models

In some heavily parameterized models, one may benefit from shifting some of parameters towards a common target. We consider shrinkage towards an equal parameter value that balances between unrestricted estimation (i.e. allowing full heterogeneity) and estimation under equality restriction (i.e. imposing full homogeneity). The penalty parameter of such ridge regression estimator is tuned using leave-one-out cross-validation. The reduction in predictive mean squared error tends to increase with the dimensionality of the parameter set. We illustrate the benefit of such shrinkage with a few stylized examples. We also work out an example of a heterogeneous panel model, including estimation on real data.

Exact likelihood-ratio tests for joint type-II censored exponential data

ABSTRACTSeveral results based on the likelihood-ratio test have been discussed in the literature for testing the scale parameters of two exponential distributions under complete and different censored forms of data, many of which are asymptotic in nature. However, exact likelihood-ratio tests have been derived only for the case of complete and independent Type-II censored samples. In this paper, we first develop exact inference using the likelihood-ratio test based on two exponential samples under joint Type-II censoring. We then obtain the exact critical value as well as the power function by using both unrestricted and restricted maximum likelihood estimators (MLEs) in the likelihood-ratio test statistics. Based on these results, we discuss the procedure for designing an optimal life-testing experiment in this set-up. Finally, some numerical examples are presented for illustrating all the results developed here.

Measuring Uncertainty Under Prior Information

There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: $(i)$ i.i.d. sampling, $(ii)$ record values, $(iii)$ Type-II censoring and $(iv)$ progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.