Sensitivity analysis approaches to high-dimensional screening problems at low sample size

Sensitivity analysis is an essential tool in the development of robust models for engineering, physical sciences, economics and policy-making, but typically requires running the model a large number of times in order to estimate sensitivity measures. While statistical emulators allow sensitivity analysis even on complex models, they only perform well with a moderately low number of model inputs: in higher dimensional problems they tend to require a restrictively high number of model runs unless the model is relatively linear. Therefore, an open question is how to tackle sensitivity problems in higher dimensionalities, at very low sample sizes. This article examines the relative performance of four sampling-based measures which can be used in such high-dimensional nonlinear problems. The measures tested are the Sobol' total sensitivity indices, the absolute mean of elementary effects, a derivative-based global sensitivity measure, and a modified derivative-based measure. Performance is assessed in a ‘screening’ context, by assessing the ability of each measure to identify influential and non-influential inputs on a wide variety of test functions at different dimensionalities. The results show that the best-performing measure in the screening context is dependent on the model or function, but derivative-based measures have a significant potential at low sample sizes that is currently not widely recognised.     

A Stochastic Simulation Approach to Model Selection for Stochastic Volatility Models

Stochastic volatility models have been widely appreciated in empirical finance such as option pricing, risk management, etc. Recent advances of Markov chain Monte Carlo (MCMC) techniques made it possible to fit all kinds of stochastic volatility models of increasing complexity within Bayesian framework. In this article, we propose a new Bayesian model selection procedure based on Bayes factor and a classical thermodynamic integration technique named path sampling to select an appropriate stochastic volatility model. The performance of the developed procedure is illustrated with an application to the daily pound/dollar exchange rates data set.     

Simex approaches to measurement error in roc studies

This paper explores the estimation of the area under the ROC curve when test scores are subject to errors. The naive approach that ignores measurement errors generally yields inconsistent estimates. Finding the asymptotic bias of the naive estimator, Coffin and Sukhatme (1995, 1997) proposed bias-corrected estimators for parametric and nonparametric cases. However, the asymptotic distributions of these estimators have not been developed because of their complexity. We propose several alternative approaches, including the SIMEX procedure of Cook and Stefanski (1994). We also provide the asymptotic distributions of the SIMEX estimators for use in statistical inference. Small simulation studies illustrate that the SIMEX estimators perform reasonably well when compared to the bias-corrected estimators.     

Geometric approaches to design of experiment

Lagrangian duality theory is used to relate problems of D-optimal design of experiments to covering problems involving ellipsoids and ellipsoidal cylinders. The possibility of generalisation of the results to other optimality criteria is discussed. The covering ellipsoids find application in some robust estimation activities and there are links with the operational research problems of optimal location of facilities. The possibility of exact computation of the covering ellipsoids is investigated.     

Approaching in Distribution with Applications to Resampling of Stochastic Processes

We introduce the notion of weak approaching and conditionally weak approaching sequences of random processes. This notion generalizes the conventional weak convergence, and has been proposed for real valued random variables in Belyaev (1995). Some of the standard tools for an investigation of the behaviour of weak approaching sequences of random elements in metric spaces are developed. The spaces of smoothed and right-continuous functions with left-hand limits are considered. This technique allows us to use the resampling approach for an evaluation of distributions of continuous functionals on realizations of sum of an increasing number of independent random processes. Two numerical examples are presented for such functionals as supremum and number of level crossings.     

Two new approaches to robust estimation in time series

Two new approaches to robust time series modelling are proposed. These approaches are natural generalisations of the Yule—Walker and the least squares methods. The approaches generate further a few viable estimators. Simulation experiments are conducted to investigate the relative efficiency and the breakdown bounds of these estimators.     

A Stochastic Approach to General Nonlinear Programming Problems

Nonlinear programming problem is the general case of mathematical programming problem such that both the objective and constraint functions are nonlinear and is the most difficult case of smooth optimization problem to solve. In this article, we suggest a stochastic search method to general nonlinear programming problems which is not an iterative algorithm but it is an interior point method. The proposed method finds the near-optimal solution to the problem. The results of a few numerical studies are reported. The efficiency of the new method is compared and is found to be reasonable.     

Recursive and en-bloc approaches to signal extraction

In the literature on unobservable component models , three main statistical instruments have been used for signal extraction: fixed interval smoothing (FIS), which derives from Kalman's seminal work on optimal state-space filter theory in the time domain; Wiener-Kolmogorov-Whittle optimal signal extraction (OSE) theory, which is normally set in the frequency domain and dominates the field of classical statistics; and regularization , which was developed mainly by numerical analysts but is referred to as 'smoothing' in the statistical literature (such as smoothing splines, kernel smoothers and local regression). Although some minor recognition of the interrelationship between these methods can be discerned from the literature, no clear discussion of their equivalence has appeared. This paper exposes clearly the interrelationships between the three methods; highlights important properties of the smoothing filters used in signal extraction; and stresses the advantages of the FIS algorithms as a practical solution to signal extraction and smoothing problems. It also emphasizes the importance of the classical OSE theory as an analytical tool for obtaining a better understanding of the problem of signal extraction.     

Probabilistic approaches to current life table estimation

"Two simple current life table estimators of conditional probabilities of death result from making either a uniform or exponential distributional assumption of time at death in the age interval. Each is compared with Chiang's estimator based on the concept of fraction of the last age interval of life. Graphical and numerical results are presented to assess the magnitude and direction of differences between estimators when the true value of Chiang's fraction takes on specific values."     

Bayesian approaches to the detection of outliers in poisson samples

This paper is concerned with the detection of upper outliers in a Poisson sample.The approach is Bayesian throughout. It is supposed that a small number of observations are contaminated, that is they are generated from a Poisson sample with mean inflated by a factor §.Bayes factors for the cases when (i) § is known, (ii) it is given a proper conjugate prior or (iii) it is completely unknown are discussed. It is suggested, in contrast to classical approaches, that transforming the data to normality does not simplify the problem.     

A Note on the Stochastic Approach to Index Numbers

In a recent article, Clements and Izan (1987) used the stochastic approach to index-number theory to estimate the rate of inflation and its standard error. Selvanathan (1988) extended their approach to the prices of groups of goods and to prices within groups. In this note, I apply the within-group results to the U.K. alcohol data. Simulation results show that the estimates are unbiased, but the asymptotic standard errors understate the true sampling variability of the estimates. To overcome this problem, I applied the bootstrap technique to obtain alternate standard errors.     

Stochastic comparisons in frailty models

The frailty model in survival analysis accounts for unobserved heterogeneity between individuals by assuming that the hazard rate of an individual is the product of an individual specific quantity, called “frailty” and a baseline hazard rate. It is well known that the choice of the frailty distribution strongly affects the nonparametric estimate of the baseline hazard as well as that of the conditional probabilities. This paper reviews the basic concepts of a frailty model, presents various probability inequalities and other monotonicity results which may prove useful in choosing among alternative specifications. More specifically, our main result lies in seeing how well known stochastic orderings between distributions of two frailities translate into orderings between the corresponding survival functions. Some probabilistic aspects and implications of the models resulting from competing choices of the distributions of frailty or the baseline are compared.     

Nonparametric approaches to testing and multiple comparisons in a one-way anova

Two questions of interest involving nonparametric multiple comparisons are considered. The first question concerns whether it is appropriate to use a multiple comparison procedure as a test of the equality of k treatments, and if it is, which procedure performs best as a test. Our results show that for smaller k values some multiple comparison procedures perform well as tests. The second question concerns whether a joint ranking or a separate ranking multiple comparison procedure performs better as a test and as a device for treatment separation. We find that the joint ranking procedure does slightly better as a test, but for treatment separation the answer depends on the situation.     

Three approaches to sequential analysis and one to hidden Markov processes

Three approaches to sequential analysis are reviewed: Chernoff's development of the Wald approach, the dynamic programming analysis developed by the author some years ago and a ‘path-averaging’ approach which exploits the random-walk properties of the log-posterior under a given hypothesis. These last two approaches led to explicit determinations of the optimal decision boundary and its associated costs in the limit of a small sampling cost, for a general number of hypotheses. However, the particular interest of the path-averaging approach is that it applies also to state-estimation for a hidden Markov model, where it leads to Eq. (39), which gives an immediate indication of the effectiveness with which the different states are estimated.     

Stochastic orderings related to ranking in an nth order record process

Let X₁, X₂, …, X_m be successive observations on m objects, numbered 1,2, …, m. If X₁ belongs to the n largest observations among X₁,X₂,…,X_i, object i is called a record, i = 1,2,…,m; n?1.In investigating the influence of the ranking of the objects on the expected number of records, a hierarchy of stochastic order relations between random variables arises.It is these order relations and their relationship with known stochastic orderings that are studied in this paper.     

Stochastic geometry models in high-level vision

We survey the use of Markov models from stochastic geometry as priors in ‘high-level’ computer vision, in direct analogy with the use of discrete Markov random fields in ‘low-level’ vision. There are analogues of the Gibbs sampler, ICM and simulated annealing, and connections with existing methods in computer vision.     

New approaches to model-free dimension reduction for bivariate regression

Dimension reduction with bivariate responses, especially a mix of a continuous and categorical responses, can be of special interest. One immediate application is to regressions with censoring. In this paper, we propose two novel methods to reduce the dimension of the covariates of a bivariate regression via a model-free approach. Both methods enjoy a simple asymptotic chi-squared distribution for testing the dimension of the regression, and also allow us to test the contributions of the covariates easily without pre-specifying a parametric model. The new methods outperform the current one both in simulations and in analysis of a real data. The well-known PBC data are used to illustrate the application of our method to censored regression.