On empirical likelihood for linear models with missing responses

Suppose that we have a linear regression model Y=X^′β+_ν0(X)ε$Y=X^{\prime }\beta +{\nu }_0(X)\epsilon$ with random error ε$\epsilon$, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random (MAR). In this paper, based on the ‘complete’ data set for Y after inverse probability weighted imputation, we construct empirical likelihood statistics on EY   and β$\beta$ which have the χ²${\chi }^2$-type limiting distributions under some new conditions compared with Xue (2009). Our results broaden the applicable scope of the approach combined with Xue (2009).     

Random Weighting Estimation of Confidence Intervals for Quantiles

This paper presents a new random weighting method for confidence interval estimation for the sample ‐quantile. A theory is established to extend ordinary random weighting estimation from a non‐smoothed function to a smoothed function, such as a kernel function. Based on this theory, a confidence interval is derived using the concept of backward critical points. The resultant confidence interval has the same length as that derived by ordinary random weighting estimation, but is distribution‐free, and thus it is much more suitable for practical applications. Simulation results demonstrate that the proposed random weighting method has higher accuracy than the Bootstrap method for confidence interval estimation.     

Random vs. systematic non-response in group-level research

This study investigates whether, in the context of group-level research, the effect of random non-response pattern, in biasing group-level scores, would be more salient than the effect of systematic non-response pattern with all other factors (i.e. group size, within-group variability, between-group variability and response rate) held constant. The findings of the Monte Carlo simulation study are counter-intuitive and reveal that the amount of error in the group-level variable is indeed the largest under the conditions of random non-response pattern.     

Knot selection for least-squares and penalized splines

Two new stochastic search methods are proposed for optimizing the knot locations and/or smoothing parameters for least-squares or penalized splines. One of the methods is a golden-section-augmented blind search, while the other is a continuous genetic algorithm. Monte Carlo experiments indicate that the algorithms are very successful at producing knot locations and/or smoothing parameters that are near optimal in a squared error sense. Both algorithms are amenable to parallelization and have been implemented in OpenMP and MPI. An adjusted GCV criterion is also considered for selecting both the number and location of knots. The method performed well relative to MARS in a small empirical comparison.     

Data-driven sensitivity analysis to detect missing data mechanism with applications to structural equation modelling

Missing data are a common problem in almost all areas of empirical research. Ignoring the missing data mechanism, especially when data are missing not at random (MNAR), can result in biased and/or inefficient inference. Because MNAR mechanism is not verifiable based on the observed data, sensitivity analysis is often used to assess it. Current sensitivity analysis methods primarily assume a model for the response mechanism in conjunction with a measurement model and examine sensitivity to missing data mechanism via the parameters of the response model. Recently, Jamshidian and Mata (Post-modelling sensitivity analysis to detect the effect of missing data mechanism, Multivariate Behav. Res. 43 (2008), pp. 432–452) introduced a new method of sensitivity analysis that does not require the difficult task of modelling the missing data mechanism. In this method, a single measurement model is fitted to all of the data and to a sub-sample of the data. Discrepancy in the parameter estimates obtained from the the two data sets is used as a measure of sensitivity to missing data mechanism. Jamshidian and Mata describe their method mainly in the context of detecting data that are missing completely at random (MCAR). They used a bootstrap type method, that relies on heuristic input from the researcher, to test for the discrepancy of the parameter estimates. Instead of using bootstrap, the current article obtains confidence interval for parameter differences on two samples based on an asymptotic approximation. Because it does not use bootstrap, the developed procedure avoids likely convergence problems with the bootstrap methods. It does not require heuristic input from the researcher and can be readily implemented in statistical software. The article also discusses methods of obtaining sub-samples that may be used to test missing at random in addition to MCAR. An application of the developed procedure to a real data set, from the first wave of an ongoing longitudinal study on aging, is presented. Simulation studies are performed as well, using two methods of missing data generation, which show promise for the proposed sensitivity method. One method of missing data generation is also new and interesting in its own right.     

Small area estimation under random regression coefficient models

Statistical agencies are interested to report precise estimates of linear parameters from small areas. This goal can be achieved by using model-based inference. In this sense, random regression coefficient models provide a flexible way of modelling the relationship between the target and the auxiliary variables. Because of this, empirical best linear unbiased predictor (EBLUP) estimates based on these models are introduced. A closed-formula procedure to estimate the mean-squared error of the EBLUP estimators is also given and empirically studied. Results of several simulation studies are reported as well as an application to the estimation of household normalized net annual incomes in the Spanish Living Conditions Survey.     

A New Test of Randomness for Lehmer Generators Based on the Manhattan Distance Between Pairs of Consecutive Random Numbers

This article considers the Marsaglia effect by proposing a new test of randomness for Lehmer random number generators. Our test is based on the Manhattan distance criterion between consecutive pairs of random numbers rather than the usually adopted Euclidian distance. We derive the theoretical distribution functions for the Manhattan distance for both overlapping (two dimensional) as well as non-overlapping cases. Extensive goodness-of-fit testing as well as empirical experimentation provides ample proof of the merits of the proposed criterion.     

Stability of the roots of random algebraic polynomials

In this paper we consider the behavior of the roots of random algebraic polynomials. A code was developed which generates a sample of random algebraic polynomials, calculates the roots of each sample polynomial, and then calculates the averages of the roots. Finally, the roots of the deterministic algebraic polynomial whose coefficients are the averages of the sample coefficients are calculated. These data are then tabulated and graphically displayed. The relationship between the averages of the roots of the sample polynomials and the roots of the average polynomial is discussed.     

Randomized heuristic algorithms for orthogonal projection of a point onto a set

Abstract

The problem of orthogonal projection of a point onto a set is an essential problem of computational geometry. This problem has many practical applications in different areas such as robotics, computer graphics and so on. In the present paper three algorithms for solving this problem are proposed. This algorithms are based on the idea of heuristic random search. Numerical experiments illustrating the work of the proposed methods are presented.