Multivariate Behrens-Fisher problem with missing data

Inference about the difference between two normal mean vectors when the covariance matrices are unknown and arbitrary is considered. Assuming that the incomplete data are of monotone pattern, a pivotal quantity, similar to the Hotelling T² statistic, is proposed. A satisfactory moment approximation to the distribution of the pivotal quantity is derived. Hypothesis testing and confidence estimation based on the approximate distribution are outlined. The accuracy of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.     

Improving estimation in speckled imagery

We propose an analytical bias correction for the maximum likelihood estimators of theG ₁ ⁰ distribution. This distribution is a very powerful tool for speckled imagery analysis, since it is capable of describing a wide range of target roughness. We compare the performance of the corrected estimators with the corresponding original version using Monte Carlo simulation. This second-order bias correction leads to estimators which are better from both the bias and mean square error criteria.     

Improved techniques for parametric and nonparametric evaluations of the first‐passage time for degradation processes

For degradation data in reliability analysis, estimation of the first‐passage time (FPT) distribution to a threshold provides valuable information on reliability characteristics. Recently, Balakrishnan and Qin (2019; Applied Stochastic Models in Business and Industry, 35:571–590) studied a nonparametric method to approximate the FPT distribution of such degradation processes if the underlying process type is unknown. In this article, we propose some improved techniques based on saddlepoint approximation, which enhance those existing methods. Numerical examples and Monte Carlo simulation studies are used to illustrate the advantages of the proposed techniques. Limitations of the improved techniques are discussed and some possible solutions to such are proposed. Some concluding remarks and practical recommendations are provided based on the results.     

New fast divide‐and‐conquer algorithms for the symmetric tridiagonal eigenvalue problem

In this paper, two accelerated divide‐and‐conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N²r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy‐like matrices and are off‐diagonally low‐rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low‐rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off‐diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. Copyright © 2016 John Wiley & Sons, Ltd.     

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??Kolmogorov-Smirnov (KS), Cramer-von Mises (CM) and Anderson-Darling (AD) test, which are based on empirical distribution function (EDF), are well-known statistics in testing univariate normality. In this paper, we focus on the high dimensional case and propose a family of generalized EDF based statistics to test the high-dimensional normal distribution by reducing the dimension of the variable. Not only can we approximate the corresponding critical values of three statistics by Monte Carlo method, we also can investigate the approximate distributions of proposed statistics based on approximate formulas in univariate case under null hypothesis. The Monte Carlo simulation is carried out to demonstrate that the performance of proposed statistics is more competitive than existing methods under some alternative hypotheses. Finally, the proposed tests are applied to real data to illustrate their utility.     

Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis–Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the $n$th step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

Testing Epidemic Changes of Infinite Dimensional Parameters

To detect epidemic change in the mean of a sample of size n of random elements in a Banach space, we introduce new test statistics DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length log^γ n, γ > 1. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrices of random vectors. Final version 27 October 2004     

Multilevel Monte Carlo for stochastic differential equations with additive fractional noise

We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order ε with a computational effort of order ε ⁻².     

An improved Monte Carlo factorization algorithm

Pollard's Monte Carlo factorization algorithm usually finds a factor of a composite integerN inO(N ^1/4) arithmetic operations. The algorithm is based on a cycle-finding algorithm of Floyd. We describe a cycle-finding algorithm which is about 36 percent faster than Floyd's (on the average), and apply it to give a Monte Carlo factorization algorithm which is similar to Pollard's but about 24 percent faster.     

Testing significance of a mean vector—A possible alternative to Hotelling'sT 2

In problems involving multivariate measurements experimental considerations often indicate grouping of variables into subsets ordered according to their importance. In such situations, the problems such as comparison of two mean vectors and profile analysis may be treated by Hotelling'sT ²-test adapted along the lines of the step-wise procedure of J. Roy [10], or the well known test for additional information due to Rao [9]. In this paper we study a modification of the step-wise procedure obtained by combining the component tests. The exact Bahadur slopes of resulting procedures are computed and it is shown that the procedure based upon Fisher's combination method is asymptotically equivalent to Hotelling'sT ². A Monte Carlo study suggests that even in small samples the power functions of the new method and Hotelling'sT ²-test are practically equivalent. Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant No. AFOSR-77-3360. The United States Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation hereon.     

Generation of the Exact Distribution and Simulation of Matched Nucleotide Sequences on a Phylogenetic Tree

Nucleotide sequences are often generated by Monte Carlo simulations to address complex evolutionary or analytic questions but the simulations are rarely described in sufficient detail to allow the research to be replicated. Here we briefly review the Markov processes of substitution in a pair of matching (homologous) nucleotide sequences and then extend it to k matching nucleotide sequences. We describe calculation of the joint distribution of nucleotides of two matching sequences. Based on this distribution, we give a method for simulation of the divergence matrix for n sites using the multinomial distribution. This is then extended to the joint distribution for k nucleotide sequences and the corresponding 4 ^k divergence array, generalizing Felsenstein (Journal of Molecular Evolution, 17, 368–376, 1981), who considered stationary, homogeneous and reversible processes on trees. We give a second method to generate matched sequences that begins with a random ancestral sequence and applies a continuous Markov process to each nucleotide site as in Rambaut and Grassly (Computer Applications in the Biosciences, 13, 235–238, 1997); further, we relate this to an equivalent approach based on an embedded Markov chain. Finally, we describe an approximate method that was recently implemented in a program developed by Jermiin et al. (Applied Bioinformatics, 2, 159–163, 2003). The three methods presented here cater for different computational and mathematical limitations and are shown in an example to produce results close to those expected on theoretical grounds. All methods are implemented using functions in the S-plus or R languages.     

Dense Fast Random Projections and Lean Walsh Transforms

Random projection methods give distributions over k×d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors x _i ∈ℝ ^d the resulting vectors Ψx _i ∈ℝ ^k approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random ±1 diagonal matrix) is a good random projector for a subset of vectors in ℝ ^d . Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms of running time, the best known current result (due to Matousek) under comparable assumptions.     

Low-rank Orthogonal Decompositions for Information Retrieval Applications

Current methods to index and retrieve documents from databases usually depend on a lexical match between query terms and keywords extracted from documents in a database. These methods can produce incomplete or irrelevant results due to the use of synonyms and polysemus words. The association of terms with documents (or implicit semantic structure) can be derived using large sparse {\it term-by-document} matrices. In fact, both terms and documents can be matched with user queries using representations in k-space (where 100 ≤ k ≤ 200) derived from k of the largest approximate singular vectors of these term-by-document matrices. This completely automated approach called latent semantic indexing or LSI, uses subspaces spanned by the approximate singular vectors to encode important associative relationships between terms and documents in k-space. Using LSI, two or more documents may be closeto each other in k-space (and hence meaning) yet share no common terms. The focus of this work is to demonstrate the computational advantages of exploiting low-rank orthogonal decompositions such as the ULV (or URV) as opposed to the truncated singular value decomposition (SVD) for the construction of initial and updated rank-k subspaces arising from LSI applications.     

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.     

Randomized approximation scheme and perfect sampler for closed Jackson networks with multiple servers

In this paper, we propose a fully polynomial-time randomized approximation scheme (FPRAS) for a closed Jackson network. Our algorithm is based on the Markov chain Monte Carlo (MCMC) method. Thus our scheme returns an approximate solution, for which the size of the error satisfies a given bound. To our knowledge, this algorithm is the first polynomial time MCMC algorithm for closed Jackson networks with multiple servers. We propose two new ergodic Markov chains, both of which have a unique stationary distribution that is the product form solution for closed Jackson networks. One of them is for an approximate sampler, and we show that it mixes rapidly. The other is for a perfect sampler based on the monotone coupling from the past (CFTP) algorithm proposed by Propp and Wilson, and we show that it has a monotone update function.     

A Family of Estimators of Finite-Population Distribution Function Using Auxiliary Information

This paper considers the problem of estimating the finite-population distribution function and quantiles with the use of auxiliary information at the estimation stage of a survey. We propose the families of estimators of the distribution function of the study variate y using the knowledge of the distribution function of the auxiliary variate x. In addition to ratio, product and difference type estimators, many other estimators are identified as members of the proposed families. For these families the approximate variances are derived, and in addition, the optimum estimator is identified along with its approximate variance. Estimators based on the estimated optimum values of the unknown parameters used to minimize the variance are also given with their properties. Further, the family of estimators of a finite-population distribution function using two-phase sampling is given, and its properties are investigated.      

A non-negative matrix factorization model based on the zero-inflated Tweedie distribution

Non-negative matrix factorization (NMF) is a technique of multivariate analysis used to approximate a given matrix containing non-negative data using two non-negative factor matrices that has been applied to a number of fields. However, when a matrix containing non-negative data has many zeroes, NMF encounters an approximation difficulty. This zero-inflated situation occurs often when a data matrix is given as count data, and becomes more challenging with matrices of increasing size. To solve this problem, we propose a new NMF model for zero-inflated non-negative matrices. Our model is based on the zero-inflated Tweedie distribution. The Tweedie distribution is a generalization of the normal, the Poisson, and the gamma distributions, and differs from each of the other distributions in the degree of robustness of its estimated parameters. In this paper, we show through numerical examples that the proposed model is superior to the basic NMF model in terms of approximation of zero-inflated data. Furthermore, we show the differences between the estimated basis vectors found using the basic and the proposed NMF models for \(\beta \) divergence by applying it to real purchasing data.     

Semiparametric quantile regression with random censoring

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.