Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation

Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Second, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi-Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.     

Maximizing Complex Likelihoods via Directed Stochastic Searching Algorithm

In this article, a directed stochastic searching algorithm is defined. It is a root or optimal parameter searching algorithm with stochastic searching directions. This algorithm is especially relevant when the objective function is complex or is observed with errors. We prove that the resulting roots or estimators have well-controlled biases under certain conditions. We examine the proposed method by finding the maximum likelihood estimates for which the corresponding likelihood function has or does not have a closed-form representation in both the simulations and the real cases. Finally, the limitations and the consequences when multiple solutions exist are addressed.     

Estimating monotone convex functions via sequential shape modification

We propose a sequential method to estimate monotone convex functions that consists of: (i) monotone regression via solving a constrained least square (LS) problem and (ii) convexification of the monotone regression estimate via solving a uniform approximation problem with associated constraints. We show that this method is faster than the constrained LS method. The ratio of computation time increases as data size increases. Moreover, we show that, under an appropriate smoothness condition, the uniform convergence rate achieved by the proposed method is nearly comparable to the best achievable rate for a non-parametric estimate which ignores the shape constraint. Simulation studies show that our method is comparable to the constrained LS method in estimation error. We illustrate our method by analysing ground water level data of wells in Korea.     

Quantile regression via iterative least squares computations

We present an estimating framework for quantile regression where the usual L ₁-norm objective function is replaced by its smooth parametric approximation. An exact path-following algorithm is derived, leading to the well-known ‘basic’ solutions interpolating exactly a number of observations equal to the number of parameters being estimated. We discuss briefly possible practical implications of the proposed approach, such as early stopping for large data sets, confidence intervals, and additional topics for future research.     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.     

Polynomial Spline Estimation for a Generalized Additive Coefficient Model

Abstract.  We study a semiparametric generalized additive coefficient model (GACM), in which linear predictors in the conventional generalized linear models are generalized to unknown functions depending on certain covariates, and approximate the non-parametric functions by using polynomial spline. The asymptotic expansion with optimal rates of convergence for the estimators of the non-parametric part is established. Semiparametric generalized likelihood ratio test is also proposed to check if a non-parametric coefficient can be simplified as a parametric one. A conditional bootstrap version is suggested to approximate the distribution of the test under the null hypothesis. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed methods. We further apply the proposed model and methods to a data set from a human visceral Leishmaniasis study conducted in Brazil from 1994 to 1997. Numerical results outperform the traditional generalized linear model and the proposed GACM is preferable.     

On the inapproximability of the exemplar conserved interval distance problem of genomes

In this paper we present two main results about the inapproximability of the exemplar conserved interval distance problem of genomes. First, we prove that it is NP-complete to decide whether the exemplar conserved interval distance between any two genomes is zero or not. This result implies that the exemplar conserved interval distance problem does not admit any approximation in polynomial time, unless P=NP. In fact, this result holds, even when every gene appears in each of the given genomes at most three times. Second, we strengthen the first result under a weaker definition of approximation, called weak approximation. We show that the exemplar conserved interval distance problem does not admit any weak approximation within a super-linear factor of , where m is the maximal length of the given genomes. We also investigate polynomial time algorithms for solving the exemplar conserved interval distance problem when certain constrains are given. We prove that the zero exemplar conserved interval distance problem of two genomes is decidable in polynomial time when one genome is O(log n)-spanned. We also prove that one can solve the constant-sized exemplar conserved interval distance problem in polynomial time, provided that one genome is trivial.     

Two-step variable selection in partially linear additive models with time series data

Lots of semi-parametric and nonparametric models are used to fit nonlinear time series data. They include partially linear time series models, nonparametric additive models, and semi-parametric single index models. In this article, we focus on fitting time series data by partially linear additive model. Combining the orthogonal series approximation and the adaptive sparse group LASSO regularization, we select the important variables between and within the groups simultaneously. Specially, we propose a two-step algorithm to obtain the grouped sparse estimators. Numerical studies show that the proposed method outperforms LASSO method in both fitting and forecasting. An empirical analysis is used to illustrate the methodology.     

Confidence intervals for a two-parameter exponential distribution: One- and two-sample problems

The problems of interval estimating the mean, quantiles, and survival probability in a two-parameter exponential distribution are addressed. Distribution function of a pivotal quantity whose percentiles can be used to construct confidence limits for the mean and quantiles is derived. A simple approximate method of finding confidence intervals for the difference between two means and for the difference between two location parameters is also proposed. Monte Carlo evaluation studies indicate that the approximate confidence intervals are accurate even for small samples. The methods are illustrated using two examples.