On the accuracy and cost of numerical integration in several variables ∗

Identical numerical integration experiments are performed on a CYBER 205 and an IBM 3081 in order to gauge the relative performance of several methods of integration. The methods employed are the general methods of Gauss-Legendre, iterated Gauss-Legendre, Newton-Cotes, Romberg and Monte Carlo as well as three methods, due to Owen, Dutt, and Clark respectively, for integrating the normal density. The bi- and trivariate normal densities and four other functions are integrated; the latter four have integrals expressible in closed form and some of them can be parameterized to exhibit singularities or highly periodic behavior. The various Gauss-Legendre methods tend to be most accurate (when applied to the normal density they are even more accurate than the special purpose methods designed for the normal) and while they are not the fastest, they are at least competitive. In scalar mode the CYBER is about 2-6 times faster than the IBM 3081 and the speed advantage of vectorised to scalar mode ranges from 6 to 15. Large scale econometric problems of the probit type should now be routinely soluble.     

On tests of symmetry,marginal homogeneity and quasi-symmetry in two-way contingency tables based on minimum φ-divergence estimator with constraints

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ₂ for estimation and the best function φ₁ for testing.     

Optimal determination of the parameters of some biased estimators using genetic algorithm

In this paper, some new algorithms for estimating the biasing parameters of the ridge, Liu and two-parameter estimators are introduced with the help of genetic algorithm (GA). The proposed algorithms are based on minimizing some statistical measures such as mean square error (MSE), mean absolute error (MAE) and mean absolute prediction error (MAPE). At the same time, the new algorithms allow one to keep the condition number and variance inflation factors to be less than or equal to ten by means of the GA. A numerical example is presented to show the utility of the new algorithms. In addition, an extensive Monte Carlo experiment is conducted. The numerical findings prove that the proposed algorithms enable to eliminate the problem of multicollinearity and minimize the MSE, MAE and MAPE.     

Uniform Test for Predictive Regression With AR Errors

Testing predictability is of importance in economics and finance. Based on a predictive regression model with independent and identically distributed errors, some uniform tests have been proposed in the literature without distinguishing whether the predicting variable is stationary or nearly integrated. In this article, we extend the empirical likelihood methods of Zhu, Cai, and Peng with independent errors to the case of an AR error process. Again, the proposed new tests do not need to know whether the predicting variable is stationary or nearly integrated, and whether it has a finite variance or an infinite variance. A simulation study shows the new methodologies perform well in finite sample.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

Robust bootstrap estimates in heteroscedastic semi-varying coefficient models and applications in analyzing Australia CPI data

This article deals with the estimation of the parametric component, which is of primary interest, in the heteroscedastic semi-varying coefficient models. Based on the bootstrap technique, we present a procedure for estimating the parameters, which can provide a reliable approximation to the asymptotic distribution of the profile least-square (PLS) estimator. Furthermore, a bootstrap-type estimator of covariance matrix is developed, which is proved to be a consistent estimator of the covariance matrix. Moreover, some simulation experiments are conducted to evaluate the finite sample performance for the proposed methodology. Finally, the Australia CPI dataset is analyzed to demonstrate the application of the methods.     

Empirical Likelihood Inference for the Rao‐Hartley‐Cochran Sampling Design

The Hartley‐Rao‐Cochran sampling design is an unequal probability sampling design which can be used to select samples from finite populations. We propose to adjust the empirical likelihood approach for the Hartley‐Rao‐Cochran sampling design. The approach proposed intrinsically incorporates sampling weights, auxiliary information and allows for large sampling fractions. It can be used to construct confidence intervals. In a simulation study, we show that the coverage may be better for the empirical likelihood confidence interval than for standard confidence intervals based on variance estimates. The approach proposed is simple to implement and less computer intensive than bootstrap. The confidence interval proposed does not rely on re‐sampling, linearization, variance estimation, design‐effects or joint inclusion probabilities.     

Empirical Bayes estimation in continuous one-parameter exponential families under associated samples

In this paper, we study the empirical Bayes (EB) estimation in continuous one-parameter exponential families under negatively associated (NA) samples and positively associated (PA) samples. Under certain regularity conditions, it is shown that the convergence rates of proposed EB estimators under NA or PA samples are the same as those of EB estimators under independent observations, which significantly improve the existing results in EB estimation under associated samples.     

Revisiting retesting in the estimation of proportions by group testing

In the estimation of a proportion p by group testing (pooled testing), retesting of units within positive groups has received little attention due to the minimal gain in precision compared to testing additional units. If acquisition of additional units is impractical or too expensive, and testing is not destructive, we show that retesting can be a useful option. We propose the retesting of a random grouping of units from positive groups, and compare it with nested halving procedures suggested by others. We develop an estimator of p for our proposed method, and examine its variance properties. Using simulation we compare retesting methods across a range of group testing situations, and show that for most realistic scenarios, our method is more efficient.     

Wavelet Estimator in Nonparametric Regression Model with Dependent Error’s Structure

In this article, we consider a nonparametric regression model with replicated observations based on the dependent error’s structure, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. The moment consistency, the strong consistency, strong convergence rate and asymptotic normality of wavelet estimator are established under suitable conditions. A simulation study is undertaken to assess the finite sample performance of the proposed method.