Parameter estimation in semi-linear models using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood and maximum profile-marginal likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.     

Applications of asymptotic confidence intervals with continuity corrections for asymmetric comparisons in noninferiority trials

A large‐sample problem of illustrating noninferiority of an experimental treatment over a referent treatment for binary outcomes is considered. The methods of illustrating noninferiority involve constructing the lower two‐sided confidence bound for the difference between binomial proportions corresponding to the experimental and referent treatments and comparing it with the negative value of the noninferiority margin. The three considered methods, Anbar, Falk–Koch, and Reduced Falk–Koch, handle the comparison in an asymmetric way, that is, only the referent proportion out of the two, experimental and referent, is directly involved in the expression for the variance of the difference between two sample proportions. Five continuity corrections (including zero) are considered with respect to each approach. The key properties of the corresponding methods are evaluated via simulations. First, the uncorrected two‐sided confidence intervals can, potentially, have smaller coverage probability than the nominal level even for moderately large sample sizes, for example, 150 per group. Next, the 15 testing methods are discussed in terms of their Type I error rate and power. In the settings with a relatively small referent proportion (about 0.4 or smaller), the Anbar approach with Yates’ continuity correction is recommended for balanced designs and the Falk–Koch method with Yates’ correction is recommended for unbalanced designs. For relatively moderate (about 0.6) and large (about 0.8 or greater) referent proportion, the uncorrected Reduced Falk–Koch method is recommended, although in this case, all methods tend to be over‐conservative. These results are expected to be used in the design stage of a noninferiority study when asymmetric comparisons are envisioned. Copyright © 2013 John Wiley & Sons, Ltd.     

An online estimation scheme for a Hull–White model with HMM-driven parameters

This paper considers the implementation of a mean-reverting interest rate model with Markov-modulated parameters. Hidden Markov model filtering techniques in Elliott (1994, Automatica, 30:1399–1408) and Elliott et al. (1995, Hidden Markov Models: Estimation and Control. Springer, New York) are employed to obtain optimal estimates of the model parameters via recursive filters of auxiliary quantities of the observation process. Algorithms are developed and implemented on a financial dataset of 30-day Canadian Treasury bill yields. We also provide standard errors for the model parameter estimates. Our analysis shows that within the dataset and period studied, a model with two regimes is sufficient to describe the interest rate dynamics on the basis of very small prediction errors and the Akaike information criterion.     

Estimation in singular partitioned,reduced or transformed linear models

A singular partitioned linear model, i.e. the singular model comprising the main parameters and the nuisance parameters, can be reduced, or transformed to the form in which only linear functions concerning main parameters are involved. In the paper some properties of the best linear unbiased estimators of these functions following from these models are considered.     

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n^?1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.     

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.     

Local linear estimation for covariate-adjusted varying-coefficient models

We consider local linear estimation of varying-coefficient models in which the data are observed with multiplicative distortion which depends on an observed confounding variable. At first, each distortion function is estimated by non parametrically regressing the absolute value of contaminated variable on the confounder. Secondly, the coefficient functions are estimated by the local least square method on the basis of the predictors of latent variables, which are obtained in terms of the estimated distorting functions. We also establish the asymptotic normality of our proposed estimators and discuss the inference about the distortion function. Simulation studies are carried out to assess the finite sample performance of the proposed estimators and a real dataset of Pima Indians diabetes is analyzed for illustration.     

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.     

Weighted discrepancies and maximum likelihood estimation for discrete distributions

The paper shows that many estimation methods, including ML, moments, even-points, empirical c.f. and minimum chi-square, can be regarded as scoring procedures using weighted sums of the discrepancies between observed and expected frequencies The nature of the weights is investigated for many classes of distributions; the study of approximations to the weights clarifies the relationships between estimation methods, and also leads to useful formulae for initial values for ML iteration.     

Numerical treatment of restricted gauss-markov model 1

The singular value decomposition (SVD) has been widely used in the ordinary linear model and other statistical problems. In this paper, we shall introduce the generalized singular value decomposition (GSVD) of any two matrices X and H having the same number of columns to moti-vate the numerical treatment of large scale restricted Gauss-Markov model (y,Xβ\Hβ = r,σ²1), a situation to reveal the relationship (or restriction) existing among the parameters of the model. Many approaches to restricted linear model are already available. Those approaches apply the generalized inverse of matrices and emphasize the the-oretical solution of the problem rather than the development of efficient and numerical stable algorithm for the computation of estimators. The possible merit of the method present here might lie in the facts that they directly lead to an efficient, numerically stable and easily programmed algorithm for