Showing papers in "Statistical Methodology in 2008"

Abstract: The statistical analysis of the soon to come Planck satellite CMB data will help set tighter bounds on major cosmological parameters. On the way, a number of practical difficulties need to be tackled, notably that several other astrophysical sources emit radiation in the frequency range of CMB observations. Some level of residual contributions, most significantly in the galactic region and at the locations of strong radio point sources will unavoidably contaminate the estimated spherical CMB map. Masking out these regions is common practice but the gaps in the data need proper handling. In order to restore the stationarity of a partly incomplete CMB map and thus lower the impact of the gaps on non-local statistical tests, we developed an inpainting algorithm on the sphere based on a sparse representation of the data, to fill in and interpolate across the masked regions. c 2007 Elsevier B.V. All rights reserved.

Abstract: In the last decade, the study of cosmic microwave background (CMB) data has become one of the most powerful tools for studying and understanding the Universe. More precisely, measuring the CMB power spectrum leads to the estimation of most cosmological parameters. Nevertheless, accessing such precious physical information requires extracting several different astrophysical components from the data. Recovering those astrophysical sources (CMB, Sunyaev–Zel’dovich clusters, galactic dust) thus amounts to a component separation problem which has already led to an intensive activity in the field of CMB studies. In this paper, we introduce a new sparsity-based component separation method coined Generalized Morphological Component Analysis (GMCA). The GMCA approach is formulated in a Bayesian maximum a posteriori (MAP) framework. Numerical results show that this new source recovery technique performs well compared to state-of-the-art component separation methods already applied to CMB data.

Abstract: Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.

Abstract: In this paper, step-stress partially accelerated life tests are considered when the lifetime of a product follows a Burr type XII distribution. Based on type I censoring, the maximum likelihood estimates (MLEs) are obtained for the distribution parameters and acceleration factor. In addition, asymptotic variance and covariance matrix of the estimators are given. An iterative procedure is used to obtain the estimators numerically using Mathcad (2001). Furthermore, confidence intervals of the estimators are presented. Simulation results are carried out to study the precision of the MLEs for the parameters involved.

Abstract: This note considers the variance estimation for population size estimators based on capture–recapture experiments Whereas a diversity of estimators of the population size has been suggested, the question of estimating the associated variances is less frequently addressed This note points out that the technique of conditioning can be applied here successfully which also allows us to identify sources of variation: the variance due to estimation of the model parameters and the binomial variance due to sampling nn units from a population of size NN It is applied to estimators typically used in capture–recapture experiments in continuous time including the estimators of Zelterman and Chao and improves upon previously used variance estimators In addition, knowledge of the variances associated with the estimators by Zelterman and Chao allows the suggestion of a new estimator as the weighted sum of the two The decomposition of the variance into the two sources allows also a new understanding of how resampling techniques like the Bootstrap could be used appropriately Finally, the sample size question for capture–recapture experiments is addressed Since the variance of population size estimators increases with the sample size, it is suggested to use relative measures such as the observed-to-hidden ratio or the completeness of identification proportion for approaching the question of sample size choice

Abstract: Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. Facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong “staircase” artifacts. In this paper, we propose to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar. The Bi-Haar filter bank is normalized such that the p -values of Bi-Haar coefficients ( p B H ) provide good approximation to those of Haar ( p H ) for high-intensity settings or large scales; for low-intensity settings and small scales, we show that p B H are essentially upper-bounded by p H . Thus, we may apply the Haar-based HTs to Bi-Haar coefficients to control a prefixed false positive rate. By doing so, we benefit from the regular Bi-Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity. A Fisher-approximation-based threshold implementing the HTs is also established. The efficiency of this method is illustrated on an example of hyperspectral-source-flux estimation.

Abstract: Given observations originating from a mixture distribution f [ x ; Q ( λ ) ] where the kernel f is known and the mixing distribution Q is unknown, we consider estimating a functional θ ( Q ) of Q . A natural estimator of such a functional can be obtained by substituting Q with its nonparametric maximum likelihood estimator (NPMLE), denoted here as Q ˆ . We demonstrate however, that the plug-in estimator θ ( Q ˆ ) can be unstable or substantially biased due to large variability of Q ˆ or structural properties of the parameter space of λ . In this paper we propose using a partial prior for Q to improve the estimation in motivating examples. In particular we propose an empirical Bayes estimation method based on an exponential prior, and show its effectiveness in improving estimation in motivating examples of binomial mixture.

Abstract: The sampling distributions are generally unavailable in exact form and are approximated either in terms of the asymptotic distributions, or their correction using expansions such as Edgeworth, Laguerre or Cornish–Fisher; or by using transformations analogous to that of Wilson and Hilferty. However, when theoretical routes are intractable, in this electronic age, the sampling distributions can be reasonably approximated using empirical methods. The point is illustrated using the null distribution of Hoeffding’s test of bivariate independence which is important because of its consistency against all dependence alternatives. For constructing the approximations we employ two Weibull extensions, the generalized Weibull and the exponentiated Weibull families, which contain a rich variety of density shapes and tail lengths, and have their distribution functions and quantile functions available in closed form, making them convenient for obtaining the necessary percentiles and p -values. Both approximations are seen to be excellent in terms of accuracy, but that based on the generalized Weibull is more portable.

Abstract: The forecasting stage in the analysis of a univariate threshold-autoregressive model, with exogenous threshold variable, has been developed in this paper via the computation of the so-called predictive distributions. The procedure permits one to forecast simultaneously the response and exogenous variables. An important issue in this work is the treatment of eventual missing observations present in the two time series before obtaining forecasts.

Abstract: Wald’s [A. Wald, Sequential Analysis, Wiley, New York, 1947] sequential probability ratio test (SPRT) and group sequential probability ratio test (GSPRT) remain relevant in addressing a wide range of practical problems. The area of clinical trials owes a great debt to the theory and methodologies of SPRT and GSPRT. In recent years, there has been a surge of practical applications of these methodologies in many areas including low frequency sonar detection, tracking of signals, early detection of abrupt changes in signals, computer simulations, agricultural sciences, pest management, educational testing, economics, and finance. But, obviously, there are circumstances where sampling one observation at a time may not be practical. In contexts of monitoring “inventory” or “queues”, observations may appear sequentially but in groups where the group sizes may be ideally treated as random variables themselves. For example, one may sequentially record the number of stopped cars ( M i ) and the number of cars ( ∑ j = 1 M i X i j ) without “working brake lights” when a traffic signal changes from green to red, i = 1 , 2 , … . This can be easily accomplished since every “working brake light” must glow bright red when a driver applies brakes. In this example, one notes that (i) it may be reasonable to model M i ’s as random, but (ii) it would appear impossible to record data sequentially on the status of brake lights (that is, X i j = 0 or 1) individually for car #1, and then for car #2, and so on when a group of cars rush to stop at an intersection! In order to address these kinds of situations, we start with an analog of the concept of a best “fixed-sample-size” test based on data { M i , X i 1 , … , X i M i , i = 1 , … , k } . Then, a random sequential probability ratio test (RSPRT) is developed for deciding between a simple null hypothesis and a simple alternative hypothesis with preassigned Type I and II errors α , β . The RSPRT and the best “fixed-sample-size” test with k ≡ k min associated with the same errors α , β are compared. Illustrations of RSPRT include a one-parameter exponential family followed by substantive computer simulations that lead to a broad set of guidelines. Two real applications and data analysis are highlighted.

Abstract: While at least some standard graphical tools do exist for cardinal time series analysis, little research effort has been given directed towards the visualization of categorical time series. The repertoire of such visual methods is nearly exclusively restricted to few isolated proposals from computer science and biology. This article aims at presenting a toolbox of known and newly developed approaches for analysing given categorical time series data visually. Among these tools, especially the rate evolution graph, the circle transformation, pattern histograms and control charts are promising.

Abstract: Different priors have been suggested to reflect spatial dependence in area health outcomes or in spatial regression residuals. However, to assume that residuals demonstrate spatial clustering only is a strong prior belief and alternatives have been suggested. A scheme suggested by Leroux et al. [B. Leroux, X. Lei, N. Breslow, Estimation of disease rates in small areas: A new mixed model for spatial dependence, in: M. Halloran, D. Berry (Eds.), Statistical Models in Epidemiology, the Environment and Clinical Trials, Springer-Verlag, New York, 1999, pp. 135–178] involves a single set of random effects and a spatial correlation parameter with extreme values corresponding to pure spatial and pure unstructured residual variation. This paper considers a spatially adaptive extension of that prior to reflect the fact that the appropriate mix between local and global smoothing may not be constant across the region being studied. Local smoothing will not be indicated when an area is disparate from its neighbours (e.g. in terms of social or environmental risk factors for the health outcome being considered). The prior for varying spatial correlation parameters may be based on a regression structure which includes possible observed sources of disparity between neighbours. A case study considers probabilities of long term illness in 133 small areas in NE London, with disparities based on a measure of socio-economic deprivation.

Abstract: We present a method based on a local regularity analysis for detecting and removing artefact signatures in noisy interferometric signals. Using Holder and wavelet transform modulus maxima lines analysis (WTMML) [S. Mallat, W. Hwang, Singularities detection and processing with wavelets, IEEE Transaction on Information Theory 38 (1992) 617–643] in suitably selected regions of the time-scale half-plane, we can estimate the regularity degree of the signal. Glitches that are considered as a discontinuity on the signal show Holder component lower than a fixed threshold defined for a continuous signal. After detection and signature removal, the signal is then locally reconstructed using Mallat reconstruction formulae. The method has been tested with Herschel SPIRE FTS proto-flight model calibration observations and shows remarkable results. Optimization of the detection parameters has been performed on the correlation coefficient, the scale domain for Holder exponent estimation and reconstruction for SPIRE FTS signals.

Abstract: We study the use of a Scheffe-style simultaneous confidence band as applied to low-dose risk estimation with quantal response data. We consider two formulations for the dose-response risk function, an Abbott-adjusted Weibull model and an Abbott-adjusted log-logistic model. Using the simultaneous construction, we derive methods for estimating upper confidence limits on predicted extra risk and, by inverting the upper bands on risk, lower bounds on the benchmark dose, or BMD, at a specific level of ‘benchmark risk’. Monte Carlo evaluations explore the operating characteristics of the simultaneous limits.

Abstract: In the present paper we introduce a partially sequential sampling procedure to develop a nonparametric method for simultaneous testing. Our work, as in [U. Bandyopadhyay, A. Mukherjee, B. Purkait, Nonparametric partial sequential tests for patterned alternatives in multi-sample problems, Sequential Analysis 26 (4) (2007) 443–466], is motivated by an interesting investigation related to arsenic contamination in ground water. Here we incorporate the idea of multiple hypotheses testing as in [Y. Benjamini, T. Hochberg, Controlling the false discovery rate: A practical and powerful approach to multiple testing, Journal of Royal Statistical Society B 85 (1995) 289–300] in a typical way. We present some Monte Carlo studies related to the proposed procedure. We observe that the proposed sampling design minimizes the expected sample sizes in different situations. The procedure as a whole effectively describes the testing under dual pattern alternatives. We indicate in brief some large sample situations. We also present detailed analysis of a geological field survey data.

Abstract: An important goal of research involving gene expression data for outcome prediction is to establish the ability of genomic data to define clinically relevant risk factors. Recent studies have demonstrated that microarray data can successfully cluster patients into low- and high-risk categories. However, the need exists for models which examine how genomic predictors interact with existing clinical factors and provide personalized outcome predictions. We have developed clinico-genomic tree models for survival outcomes which use recursive partitioning to subdivide the current data set into homogeneous subgroups of patients, each with a specific Weibull survival distribution. These trees can provide personalized predictive distributions of the probability of survival for individuals of interest. Our strategy is to fit multiple models; within each model we adopt a prior on the Weibull scale parameter and update this prior via Empirical Bayes whenever the sample is split at a given node. The decision to split is based on a Bayes factor criterion. The resulting trees are weighted according to their relative likelihood values and predictions are made by averaging over models. In a pilot study of survival in advanced stage ovarian cancer we demonstrate that clinical and genomic data are complementary sources of information relevant to survival, and we use the exploratory nature of the trees to identify potential genomic biomarkers worthy of further study.

Abstract: In the present paper, minimum Hellinger distance estimates for parameters of a bilinear time series model are presented. The probabilistic properties such as stationarity, existence of moments of the stationary distribution and strong mixing property of the model are well known (see for instance [J. Liu, A note on causality and invertibility of a general bilinear time series model, Adv. Appl. Probab. 22 (1990) 247–250; J. Liu, P.J. Brockwell, On the general bilinear time series model, J. Appl. Probab. 25 (1988) 553–564; D.T. Pham, The mixing property of bilinear and generalised random coefficients autoregressive models, Stoch. Process Appl. 23 (1986) 291–300]). We establish, under some mild conditions, the consistency and the asymptotic normality of the minimum Hellinger distance estimates of the parameters of the model.

Abstract: The Escoufier operators are used to obtain a geometrical representation for the studies in a series. A model is developed assuming the normality of the vec of those operators. It is shown how to validate such a model. An application to the results of local elections in mainland Portugal is used for assessing the proposed model.

Abstract: This paper deals with the queue size distribution of an M x / G / 1 queue with a random set-up time and with a Bernoulli vacation schedule under a restricted admissibility policy. This generalizes the model studied by Madan and Choudhury [Sankhya 66 (2004) 175–193].

Abstract: The kernel smoothed Nelson–Aalen estimator has been well investigated, but is unsuitable when some of the censoring indicators are missing. A representation introduced by Dikta, however, facilitates hazard estimation when there are missing censoring indicators. In this article, we investigate (i) a kernel smoothed semiparametric hazard estimator and (ii) a kernel smoothed “pre-smoothed” Nelson–Aalen estimator. We derive the asymptotic normality of the proposed estimators and compare their asymptotic variances.

Abstract: It is not uncommon to encounter a randomized clinical trial (RCT) in which each patient is treated with several courses of therapies and his/her response is taken after treatment with each course because of the nature of a treatment design for a disease. On the basis of a simple multiplicative risk model proposed elsewhere for repeated binary measurements, we derive the maximum likelihood estimator (MLE) for the proportion ratio ( PR ) of responses between two treatments in closed form without the need of modeling the complicated relationship between patient’s compliance and patient’s response. We further derive the asymptotic variance of the MLE and propose an asymptotic interval estimator for the PR using the logarithmic transformation. We also consider two other asymptotic interval estimators. One is derived from the principle of Fieller’s Theorem and the other is derived by using the randomization-based approach suggested elsewhere. To evaluate and compare the finite-sample performance of these interval estimators, we apply the Monte Carlo simulation. We find that the interval estimator using the logarithmic transformation of the MLE consistently outperforms the other two estimators with respect to efficiency. This gain in efficiency can be substantial especially when there are patients not complying with their assigned treatments. Finally, we employ the data regarding the trial of using macrophage colony stimulating factor (M-CSF) over three courses of intensive chemotherapies to reduce febrile neutropenia incidence for acute myeloid leukemia patients to illustrate the use of these estimators.

Abstract: In astronomy multiple images are frequently obtained at the same position of the sky for follow-up coaddition as it helps one go deeper and look for fainter objects. With large scale panchromatic synoptic surveys becoming more common, image co-addition has become even more necessary as new observations start to get compared with coadded fiducial sky in real time. The standard coaddition techniques have included straight averages, variance weighted averages, medians etc. A more sophisticated nonlinear response chi-square method is also used when it is known that the data are background noise limited and the point spread function is homogenized in all channels. A more robust object detection technique capable of detecting faint sources, even those not seen at all epochs which will normally be smoothed out in traditional methods, is described. The analysis at each pixel level is based on a formula similar to Mahalanobis distance. c 2008 Elsevier B.V. All rights reserved.

Abstract: A novel framework for the estimation of multiple frequencies from irregularly sampled time series is proposed. Spectral analysis is addressed as an under-determined linear inverse problem, where the spectrum is discretised on an arbitrarily thin frequency grid. Regularisation is obtained by taking into account the prior sparseness of the solution, as we focus on line spectra estimation. Such a global approach - all parameters are estimated jointly - is an efficientalternative to usual sequential prewhitening methods, especially in case of strong sampling aliases perturbating the Fourier spectrum. Two regularisation methods are proposed within the Bayesian framework. A first method considers regularisation through the minimisation of a penalised least-squares criterion. In particular, penalisation by the l 1 -norm of the complex spectral amplitudes is studied, that shows a satisfactory behavior for sparseness modeling. An efficient optimisation algorithm is proposed, that allows a very high spectral resolution at low computational cost. A second method addresses probabilistic regularisation by modeling the spectral amplitudes as the realisation of a Bernoulli-Gaussian process. Bayesian estimation is then addressed using MCMC methods, which enable a fully unsupervised procedure. The probabilistic interpretation of the estimator, combined with variance information for each estimated parameter, then provides confidence levels on the detection of every spectral component, as well as uncertainties on the corresponding frequencies, amplitudes and phases. Both methods should be intensively tested on real data sets by astronomers. An implementation of the first one is already available online, and so will be the second one soon.

Abstract: Let X ( 1 , n , m 1 , k ) , X ( 2 , n , m 2 , k ) , … , X ( n , n , m , k ) be n generalized order statistics from a continuous distribution F which is strictly increasing over ( a , b ) , − ∞ ≤ a b ≤ ∞ , the support of F . Let g be an absolutely continuous and monotonically increasing function in ( a , b ) with finite g ( a + ) , g ( b − ) and E ( g ( X ) ) . Then for some positive integer s , 1 s ≤ n , we give characterization of distributions by means of E [ 1 s − 1 ∑ r = 1 s − 1 g ( X ( r , n , m , k ) ) | X ( s , n , m s , k ) = x ] = g ( x ) + g ( a + ) 2 , ∀ x ∈ ( a , b ) .

Abstract: The polar plumes are very fine structures of the solar K-corona lying around the poles and visible during the period of minimum of activity. These poorly known structures are linked with the solar magnetic field and with numerous coronal phenomena such as the fast solar wind and the coronal holes. The SOHO space mission has provided some continuous observations to high cadence (each 10 min). From these observations the images of the K-corona have been derived and preprocessed with an adapted anisotropic filtering. Then, a peculiar type of sinogram called Time Intensity Diagram (TID) has been built. It is adapted to the evolution of polar plumes with the time. A multiresolution wavelet analysis of the TID has then revealed that the spatial distribution of the polar plumes as well as their temporal evolution were fractal. The present study consists in simulating polar plumes by forward modeling techniques in order to validate several assumptions concerning their nature and their temporal evolution. Our work involves two main steps. The first one concerns the simulation of polar plumes and the computation of their corresponding TID. The second one concerns the estimation of analysis criteria in order to compare the original TID and the simulated ones. Static and dynamic models were both used in order to confirm the fractal behavior of the temporal evolution of plumes. The most recent and promising model is based on a Hidden Markov Tree. It allows us to control the fractal parameters of the TID.

Abstract: Consider a parallel system with n independent components. Assume that the lifetime of the j th component follows an exponential distribution with a constant but unknown parameter λ j , 1 ≤ j ≤ n . We test r j components of type- j for failure and compute the total time T j of r j failures for the j th component. Based on T = ( T 1 , T 2 , … , T n ) and r = ( r 1 , r 2 , … , r n ) , we derive optimal reliability test plans which ensure the usual probability requirements on system reliability. Further, we solve the associated nonlinear integer programming problem by a simple enumeration of integers over the feasible range. An algorithm is developed to obtain integer solutions with minimum cost. Finally, some examples have been discussed for various levels of producer’s and consumer’s risk to illustrate the approach. Our optimal plans lead to considerable savings in costs over the available plans in the literature.

Abstract: A distribution on the unit sphere is generated by conditioning a normal mixture distribution with an inverse gamma distributed weighting function. It can be regarded as the generalized symmetric Laplace distribution on the unit sphere. The density involves a modified Bessel function of the third kind which can be approximated by other simpler functions in certain limiting cases. As a consequence, the von Mises–Fisher, cardioid and Jones–Pewsey distributions are limiting cases of the new distribution. No closed form expressions exist for the roots of the likelihood equations. However, given the normal mixture structure of the distribution, we propose an E–M-algorithm-based approach for finding the maximum-likelihood estimates of the parameters which assumes the weights in the mixture to be missing data. The modeling capabilities of the proposed distribution are illustrated by fitting it and some of its competitors to two circular data sets.

Abstract: We consider nonparametric estimation of cure-rate based on mixture model under Case-1 interval censoring. We show that the nonparametric maximum-likelihood estimator (NPMLE) of cure-rate is non-unique as well as inconsistent, and propose two estimators based on the NPMLE of the distribution function under this censoring model. We present a cross-validation method for choosing a ‘cut-off’ point needed for the estimators. The limiting distributions of the latter are obtained using extreme-value theory. Graphical illustration of the procedures based on simulated data is provided.

Abstract: We study the suitability of different modelling methods for joint prediction of mean and variance based on large data sets. We review the approaches to the modelling of conditional variance function that are capable of handling a problem where conditional variance depends on about 10 explanatory variables and training dataset consists of 100 000 observations. We present a promising approach for neural network modelling of mean and dispersion. We compare different approaches in predicting the mechanical properties of steel in two case data sets collected from the production line of a steel plate mill. As a conclusion we give some recommendations concerning the modelling of conditional variance in large datasets.

Abstract: We propose a new test for the two-sample bivariate location problem. The proposed test statistic has a U -statistic representation with a degenerate kernel. The limiting distribution is found for the proposed test statistic. The power of the test is compared using Monte Carlo simulation to the tests of Blumen [I. Blumen, A new bivariate sign-test for location, Journal of the American Statistical Association 53 (1958) 448–456], Mardia [K.V. Mardia, A non-parametric test for the bivariate two-sample location problem, Journal of the Royal Statistical Society, Series B 29 (1967) 320–342], Peters and Randles [D. Peters, R.H. Randles, A bivariate signed-rank test for the two-sample location problem, Journal of the Royal Statistical Society, Series B 53 (1991) 493–504], LaRocque, Tardif and van Eeden [D. LaRocque, S. Tardif, C. van Eeden, An affine-invariant generalization of the Wilcoxon signed-rank test for the bivariate location problem, Australian and New Zealand Journal of Statistics 45 (2003) 153–165], and Baringhaus and Franz [L. Baringhaus, C. Franz, On a new multivariate two-sample test, Journal of Multivariate Analysis 88 (2004) 190–206]. Under the bivariate normal and bivariate t distributions the proposed test was more powerful than the competitors for almost every change in location. Under the other distributions the proposed test reached the desired power of one at a faster rate than the other tests in the simulation study. Application of the test is presented using bivariate data from a synthetic and a real-life data set.