Nonparametric rank regression for analyzing water quality concentration data with multiple detection limits

Environmental data usually include measurements, such as water quality data, which fall below detection limits, because of limitations of the instruments or of certain analytical methods used. The fact that some responses are not detected needs to be properly taken into account in statistical analysis of such data. However, it is well-known that it is challenging to analyze a data set with detection limits, and we often have to rely on the traditional parametric methods or simple imputation methods. Distributional assumptions can lead to biased inference and justification of distributions is often not possible when the data are correlated and there is a large proportion of data below detection limits. The extent of bias is usually unknown. To draw valid conclusions and hence provide useful advice for environmental management authorities, it is essential to develop and apply an appropriate statistical methodology. This paper proposes rank-based procedures for analyzing non-normally distributed data collected at different sites over a period of time in the presence of multiple detection limits. To take account of temporal correlations within each site, we propose an optimal linear combination of estimating functions and apply the induced smoothing method to reduce the computational burden. Finally, we apply the proposed method to the water quality data collected at Susquehanna River Basin in United States of America, which dearly demonstrates the advantages of the rank regression models.

Rank regression for analysis of clustered data: A natural induced smoothing approach

We consider rank regression for clustered data analysis and investigate the induced smoothing method for obtaining the asymptotic covariance matrices of the parameter estimators. We prove that the induced estimating functions are asymptotically unbiased and the resulting estimators are strongly consistent and asymptotically normal. The induced smoothing approach provides an effective way for obtaining asymptotic covariance matrices for between- and within-cluster estimators and for a combined estimator to take account of within-cluster correlations. We also carry out extensive simulation studies to assess the performance of different estimators. The proposed methodology is substantially Much faster in computation and more stable in numerical results than the existing methods. We apply the proposed methodology to a dataset from a randomized clinical trial.

Weighted rank regression for clustered data analysis

We consider ranked-based regression models for clustered data analysis. A weighted Wilcoxon rank method is proposed to take account of within-cluster correlations and varying cluster sizes. The asymptotic normality of the resulting estimators is established. A method to estimate covariance of the estimators is also given, which can bypass estimation of the density function. Simulation studies are carried out to compare different estimators for a number of scenarios on the correlation structure, presence/absence of outliers and different correlation values. The proposed methods appear to perform well, in particular, the one incorporating the correlation in the weighting achieves the highest efficiency and robustness against misspecification of correlation structure and outliers. A real example is provided for illustration.

Rank-based regression for analysis of repeated measures

We consider rank-based regression models for repeated measures. To account for possible withinsubject correlations, we decompose the total ranks into between- and within-subject ranks and obtain two different estimators based on between- and within-subject ranks. A simple perturbation method is then introduced to generate bootstrap replicates of the estimating functions and the parameter estimates. This provides a convenient way for combining the corresponding two types of estimating function for more efficient estimation.

Induced smoothing for rank regression with censored survival times

Adaptions of weighted rank regression to the accelerated failure time model for censored survival data have been successful in yielding asymptotically normal estimates and flexible weighting schemes to increase statistical efficiencies. However, for only one simple weighting scheme, Gehan or Wilcoxon weights, are estimating equations guaranteed to be monotone in parameter components, and even in this case are step functions, requiring the equivalent of linear programming for computation. The lack of smoothness makes standard error or covariance matrix estimation even more difficult. An induced smoothing technique overcame these difficulties in various problems involving monotone but pure jump estimating equations, including conventional rank regression. The present paper applies induced smoothing to the Gehan-Wilcoxon weighted rank regression for the accelerated failure time model, for the more difficult case of survival time data subject to censoring, where the inapplicability of permutation arguments necessitates a new method of estimating null variance of estimating functions. Smooth monotone parameter estimation and rapid, reliable standard error or covariance matrix estimation is obtained.

Standard errors and covariance matrices for smoothed rank estimators

A 'pseudo-Bayesian' interpretation of standard errors yields a natural induced smoothing of statistical estimating functions. When applied to rank estimation, the lack of smoothness which prevents standard error estimation is remedied. Efficiency and robustness are preserved, while the smoothed estimation has excellent computational properties. In particular, convergence of the iterative equation for standard error is fast, and standard error calculation becomes asymptotically a one-step procedure. This property also extends to covariance matrix calculation for rank estimates in multi-parameter problems. Examples, and some simple explanations, are given.

Closed-form solutions for low-rank non-rigid reconstruction

Recovering the motion of a non-rigid body from a set of monocular images permits the analysis of dynamic scenes in uncontrolled environments. However, the extension of factorisation algorithms for rigid structure from motion to the low-rank non-rigid case has proved challenging. This stems from the comparatively hard problem of finding a linear “corrective transform” which recovers the projection and structure matrices from an ambiguous factorisation. We elucidate that this greater difficulty is due to the need to find multiple solutions to a non-trivial problem, casting a number of previous approaches as alleviating this issue by either a) introducing constraints on the basis, making the problems nonidentical, or b) incorporating heuristics to encourage a diverse set of solutions, making the problems inter-dependent. While it has previously been recognised that finding a single solution to this problem is sufficient to estimate cameras, we show that it is possible to bootstrap this partial solution to find the complete transform in closed-form. However, we acknowledge that our method minimises an algebraic error and is thus inherently sensitive to deviation from the low-rank model. We compare our closed-form solution for non-rigid structure with known cameras to the closed-form solution of Dai et al. [1], which we find to produce only coplanar reconstructions. We therefore make the recommendation that 3D reconstruction error always be measured relative to a trivial reconstruction such as a planar one.

Temporal genetic structure patterns in tropical maize populations under reciprocal recurrent selection.

In order to investigate the effect of long term recurrent selection on the pattern of gene diversity, thirty randomly-selected individuals from the progenitors (p) and four selection cycles (C0, C3, C6 and C11) were sampled for DNA analysis from the tropical maize (Zea mays L.) breeding populations, Atherton 1 (AT1) and Atherton 2 (AT2). Fifteen polymorphic Simple Sequence Repeat markers amplified a total of 284 and 257 alleles in AT1 and AT2 populations, respectively. Reductions in the number of alleles were observed at advanced selection cycles. About 11 and 12% of the alleles in AT1 and AT2 populations respectively, were near to fixation. However, a higher number of alleles (37% in AT1 and 33% in AT2) were close to extinction. Fisher's exact test and analysis of molecular variance (AMOVA) showed significant population differentiations. Gene diversity estimates and AMOVA revealed increased genetic differentiations at the expense of loss of heterozygosity. Population differentiations were mainly due to fixation of complementary alleles at a locus in the two breeding populations. The estimates of effective population at an advanced selection cycle were close to the population size predicted by the breeding method.

Reciprocal and advanced generation hybrids between Corymbia citriodora and C-torelliana: forestry breeding and the risk of gene flow

Corymbia F1 hybrids have high potential for plantation forestry; however, little is known of their reproductive biology and potential for genetic pollution of native Corymbia populations. This study aims to quantify the influence of reproductive isolating barriers on the success of novel reciprocal and advanced generation Corymbia hybrids. Two maternal taxa, Corymbia citriodora subsp. citriodora and Corymbia torelliana, were pollinated using five paternal taxa, C. citriodora subsp. citriodora, C. torelliana, one C. torelliana x C. citriodora subsp. citriodora hybrid and two C. torelliana x C. citriodora subsp. variegata hybrids. Pollen tube, embryo and seed development were assessed. Reciprocal hybridisation between C. citriodora subsp. citriodora and C. torelliana was successful. Advanced generation hybrids were also created when C. citriodora subsp. citriodora or C. torelliana females were backcrossed with F1 hybrid taxa. Prezygotic reproductive isolation was identified via reduced pollen tube numbers in the style and reduced numbers of ovules penetrated by pollen tubes. Reproductive isolation was weakest within the C. citriodora subsp. citriodora maternal taxon, with two hybrid backcrosses producing equivalent capsule and seed yields to the intraspecific cross. High hybridising potential was identified between all Corymbia species and F1 taxa studied. This provides opportunities for advanced generation hybrid breeding, allowing desirable traits to be amplified. It also indicates risks of gene flow between plantation and native Corymbia populations.

Reverse Channel Training For Reciprocal Mimo Systems With Spatial Multiplexing

This paper investigates the problem of designing reverse channel training sequences for a TDD-MIMO spatial-multiplexing system. Assuming perfect channel state information at the receiver and spatial multiplexing at the transmitter with equal power allocation to them dominant modes of the estimated channel, the pilot is designed to ensure an stimate of the channel which improves the forward link capacity. Using perturbation techniques, a lower bound on the forward link capacity is derived with respect to which the training sequence is optimized. Thus, the reverse channel training sequence makes use of the channel knowledge at the receiver. The performance of orthogonal training sequence with MMSE estimation at the transmitter and the proposed training sequence are compared. Simulation results show a significant improvement in performance.

Rank-Augmented LU-Algorithm for Computing Generalized Matrix Inverses

A rank-augmnented LU-algorithm is suggested for computing a generalized inverse of a matrix. Initially suitable diagonal corrections are introduced in (the symmetrized form of) the given matrix to facilitate decomposition; a backward-correction scheme then yields a desired generalized inverse.

Bootstrap and Fast Double Bootstrap Tests of Cointegration Rank with Financial Time Series

The likelihood ratio test of cointegration rank is the most widely used test for cointegration. Many studies have shown that its finite sample distribution is not well approximated by the limiting distribution. The article introduces and evaluates by Monte Carlo simulation experiments bootstrap and fast double bootstrap (FDB) algorithms for the likelihood ratio test. It finds that the performance of the bootstrap test is very good. The more sophisticated FDB produces a further improvement in cases where the performance of the asymptotic test is very unsatisfactory and the ordinary bootstrap does not work as well as it might. Furthermore, the Monte Carlo simulations provide a number of guidelines on when the bootstrap and FDB tests can be expected to work well. Finally, the tests are applied to US interest rates and international stock prices series. It is found that the asymptotic test tends to overestimate the cointegration rank, while the bootstrap and FDB tests choose the correct cointegration rank.

The Power of Bootstrap Tests of Cointegration Rank with Financial Time Series

Bootstrap likelihood ratio tests of cointegration rank are commonly used because they tend to have rejection probabilities that are closer to the nominal level than the rejection probabilities of the correspond- ing asymptotic tests. The e¤ect of bootstrapping the test on its power is largely unknown. We show that a new computationally inexpensive procedure can be applied to the estimation of the power function of the bootstrap test of cointegration rank. The bootstrap test is found to have a power function close to that of the level-adjusted asymp- totic test. The bootstrap test estimates the level-adjusted power of the asymptotic test highly accurately. The bootstrap test may have low power to reject the null hypothesis of cointegration rank zero, or underestimate the cointegration rank. An empirical application to Euribor interest rates is provided as an illustration of the findings.