Exact Sampling Research Articles

Exact Power and Sample Size Calculations for the Two One-Sided Tests of Equivalence.

Equivalent testing has been strongly recommended for demonstrating the comparability of treatment effects in a wide variety of research fields including medical studies. Although the essential properties of the favorable two one-sided tests of equivalence have been addressed in the literature, the associated power and sample size calculations were illustrated mainly for selecting the most appropriate approximate method. Moreover, conventional power analysis does not consider the allocation restrictions and cost issues of different sample size choices. To extend the practical usefulness of the two one-sided tests procedure, this article describes exact approaches to sample size determinations under various allocation and cost considerations. Because the presented features are not generally available in common software packages, both R and SAS computer codes are presented to implement the suggested power and sample size computations for planning equivalence studies. The exact power function of the TOST procedure is employed to compute optimal sample sizes under four design schemes allowing for different allocation and cost concerns. The proposed power and sample size methodology should be useful for medical sciences to plan equivalence studies.

Phylogeography by diffusion on a sphere: whole world phylogeography.

BackgroundTechniques for reconstructing geographical history along a phylogeny can answer many questions of interest about the geographical origins of species. Bayesian models based on the assumption that taxa move through a diffusion process have found many applications. However, these methods rely on diffusion processes on a plane, and do not take the spherical nature of our planet in account. Performing an analysis that covers the whole world thus does not take in account the distortions caused by projections like the Mercator projection.ResultsIn this paper, we introduce a Bayesian phylogeographical method based on diffusion on a sphere. When the area where taxa are sampled from is small, a sphere can be approximated by a plane and the model results in the same inferences as with models using diffusion on a plane. For taxa sampled from the whole world, we obtain substantial differences. We present an efficient algorithm for performing inference in a Markov Chain Monte Carlo (MCMC) algorithm, and show applications to small and large samples areas. We compare results between planar and spherical diffusion in a simulation study and apply the method by inferring the origin of Hepatitis B based on sequences sampled from Eurasia and Africa.ConclusionsWe describe a framework for performing phylogeographical inference, which is suitable when the distortion introduced by map projections is large, but works well on a smaller scale as well. The framework allows sampling tips from regions, which is useful when the exact sample location is unknown, and placing prior information on locations of clades in the tree. The method is implemented in the GEO_SPHERE package in BEAST 2, which is open source licensed under LGPL and allows joint tree and geography inference under a wide range of models.

Estimation of a Modified Capability Index for Non-Normal Distributions

Abstract Process capability indices (PCIs), which are very important in quality control have been one of a numerical measure index in manufacturing processes. Index Cpk is the most popular one used in the manufacturing industry. It is applied under the assumption that the processes are normally distributed. In real-world applications, non-normal processes may occur in industries, and the index CNpk has been proposed for non-normal processes in which its exact sampling distribution is mathematically intractable. Quality practitioners commonly use the existing NCPPM (non-conformities in parts per million) table of Cpk to obtain process yields. However, the table could not be applied directly via the value of index CNpk. For the consistency of NCPPM mapping, we propose procedures to obtain the modified index C*Npk and its approximately unbiased estimator C˜*Npk for three non-normal distributions, involving Log-normal, Gamma, and Weibull distributions. The values of modified index C*Npk could be used to inquire the existing popular NCPPM table of Cpk. In addition, four bootstrap methods were used to construct the lower confidence bounds of the index C*Npk, which are useful to the practitioners for making reliable decisions regarding process performance based on process yield.

Influence of Demographic Variables on Saving Behaviour of Rural Households – A Study with Reference to Sriperumpudur, Chennai

Saving is really a good habit particularly for the common layman. With the understanding of the above, a study was conducted in Sriperumpudur near Chennai about the saving behaviour of households and the influence of demographics over them. Objectives: The study was conducted by the authors to know the influence of demographic variables on a whole in saving behaviour and specifically, to know the level of influence of demographic factors in savings, to identify the level of saving of the people from their income and to identify the nature of investors. Analysis: The descriptive research design is adopted by the researchers in this study. They have adopted the non-probabilistic convenience sampling method for the collection of sample and the exact sample size of the study is 457. The respondents are approached with the systematic self-administered questionnaire for data collection. The researcher has used the chi-square test and the cluster analysis for the data analysis and further execution of the study. The study was concentrated on Sriperumpudur town near to Chennai only. Findings: The study has major findings that demographic variables like gender, age, marital status, number of earning members in the family, annual income have influenced annual savings. It is also found that there is an association between different heterogeneous groups of households' investment in financial market and their annual savings. But it has been indentified that there is no association between education and annual savings. Hence it is concluded that the demographic variables are playing vital role in influencing the saving behaviour of the individuals. Applications: The study has contributed for the improvements in the area where saving behaviour can be investigated from the dimension of influence of demographic variables. There are different applications that are made to identify the demographical behaviour of the respondents towards their investment choice. This study can also extensively be applied in the case of classification of demographic and psychographic variables. Improvements: Different educational qualifications of the respondents found in this study and they are considered as School Education, UG, PG, and Professional Education. They have different qualifications and aims in the determination of the investment avenues but at the same time the qualification does not influence them to choose the investment avenue. But all other demographic variables have an influence over the choice of investment avenues and change the behaviour of the investors towards the investment process. It is also found that there are certain developments in the behaviour of the individuals regarding their investment choice by their demographics.

Multilevel sequential Monte Carlo samplers

In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice, one often has to solve the associated PDE numerically, using, for instance finite element methods which depend on the step-size level hL. In addition, the expectation cannot be computed analytically and one often resorts to Monte Carlo methods. In the context of this problem, it is known that the introduction of the multilevel Monte Carlo (MLMC) method can reduce the amount of computational effort to estimate expectations, for a given level of error. This is achieved via a telescoping identity associated to a Monte Carlo approximation of a sequence of probability distributions with discretization levels ∞>h0>h1⋯>hL. In many practical problems of interest, one cannot achieve an i.i.d. sampling of the associated sequence and a sequential Monte Carlo (SMC) version of the MLMC method is introduced to deal with this problem. It is shown that under appropriate assumptions, the attractive property of a reduction of the amount of computational effort to estimate expectations, for a given level of error, can be maintained within the SMC context. That is, relative to exact sampling and Monte Carlo for the distribution at the finest level hL. The approach is numerically illustrated on a Bayesian inverse problem.

Quantitative study on exact reconstruction sampling condition by verifying solution uniqueness in limited-view CT

In limited-view computed tomography reconstruction, iterative image reconstruction with sparsity-exploiting methods, such as total variation (TV) minimization, inspired by compressive sensing, potentially claims large reductions in sampling requirements. However, a quantitative notion of this claim is non-trivial because of the ill-defined reduction in sampling achieved by the sparsity-exploiting method. In this paper, exact reconstruction sampling condition for limited-view problem is studied by verifying the uniqueness of solution in TV minimization model. Uniqueness is tested by solving a convex optimization problem derived from the sufficient and necessary condition of solution uniqueness. Through this method, the sufficient sampling number of exact reconstruction is quantified for any fixed phantom and settled geometrical parameter in the limited-view problem. This paper provides a reference to quantify the sampling condition. Three phantoms are tested to study the sampling condition of limited view exact reconstruction in this paper. The experiment results show the quantified sampling number and indicate that an object would be accurately reconstructed as the scanning range becomes narrower by increasing sampling number. The increased samplings compensate for the deficiency of the projection angle. However, the lower bound of the scanning range corresponding to three different phantoms are presented, in which an exact reconstruction cannot be obtained once the projection angular is narrowed to this extent no matter how to increase sampling.

Exact sampling of diffusions with a discontinuity in the drift

AbstractWe introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time, and are based on the principle of retrospective rejection sampling. A simple illustration is provided.

Low complexity state space representation and algorithms for closed queueing networks exact sampling

We consider exact sampling from the stationary distribution of a closed queueing network with finite capacities. In a recent work a compact representation of sets of states was proposed that enables exact sampling from the stationary distribution without considering all initial conditions in the coupling from the past (CFTP) scheme. This representation reduces the complexity of the one-step transition in the CFTP algorithm to O(KM2), where K is the number of queues and M the total number of customers; while the cardinality of the state space is exponential in the number of queues. In this paper, we extend these previous results to the multiserver case. The main focus and the contribution of this paper is on the algorithmic and the implementation issues. We propose a new representation, that leads to one-step transition complexity of the CFTP algorithm that is in O(KM). We provide a detailed description of our matrix-based implementation. Matlab toolbox Clones (CLOsed queueing Networks Exact Sampling) can be downloaded at http://www.di.ens.fr/~rovetta/Clones.

Determination of the optimal sample size for a clinical trial accounting for the population size.

The problem of choosing a sample size for a clinical trial is a very common one. In some settings, such as rare diseases or other small populations, the large sample sizes usually associated with the standard frequentist approach may be infeasible, suggesting that the sample size chosen should reflect the size of the population under consideration. Incorporation of the population size is possible in a decision‐theoretic approach either explicitly by assuming that the population size is fixed and known, or implicitly through geometric discounting of the gain from future patients reflecting the expected population size. This paper develops such approaches. Building on previous work, an asymptotic expression is derived for the sample size for single and two‐arm clinical trials in the general case of a clinical trial with a primary endpoint with a distribution of one parameter exponential family form that optimizes a utility function that quantifies the cost and gain per patient as a continuous function of this parameter. It is shown that as the size of the population, N, or expected size, N∗ in the case of geometric discounting, becomes large, the optimal trial size is O(N1/2) or O(N∗1/2). The sample size obtained from the asymptotic expression is also compared with the exact optimal sample size in examples with responses with Bernoulli and Poisson distributions, showing that the asymptotic approximations can also be reasonable in relatively small sample sizes.

Theoretical Guarantees for Approximate Sampling from Smooth and Log-Concave Densities

Summary Sampling from various kinds of distribution is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful non-asymptotic guarantees for the approximate sampling procedures, especially in high dimensional problems. The paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on Rp, for some integer p > 0. We establish non-asymptotic bounds for the error of approximating the target distribution by the distribution obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous time diffusion processes, which may be of interest beyond the framework of log-concave densities that are considered in the present work.

A Balanced Tissue Composition Reveals New Metabolic and Gene Expression Markers in Prostate Cancer.

Molecular analysis of patient tissue samples is essential to characterize the in vivo variability in human cancers which are not accessible in cell-lines or animal models. This applies particularly to studies of tumor metabolism. The challenge is, however, the complex mixture of various tissue types within each sample, such as benign epithelium, stroma and cancer tissue, which can introduce systematic biases when cancers are compared to normal samples. In this study we apply a simple strategy to remove such biases using sample selections where the average content of stroma tissue is balanced between the sample groups. The strategy is applied to a prostate cancer patient cohort where data from MR spectroscopy and gene expression have been collected from and integrated on the exact same tissue samples. We reveal in vivo changes in cancer-relevant metabolic pathways which are otherwise hidden in the data due to tissue confounding. In particular, lowered levels of putrescine are connected to increased expression of SRM, reduced levels of citrate are attributed to upregulation of genes promoting fatty acid synthesis, and increased succinate levels coincide with reduced expression of SUCLA2 and SDHD. In addition, the strategy also highlights important metabolic differences between the stroma, epithelium and prostate cancer. These results show that important in vivo metabolic features of cancer can be revealed from patient data only if the heterogeneous tissue composition is properly accounted for in the analysis.

Exact sample size determination for the ratio of two incidence rates under the Poisson distribution

We present sample size determination using exact approaches for testing the ratio of incidence rates from a two-arm non-inferiority trial. The first approach is the exact conditional approach by treating the total number of events from two groups fixed, and the other is the exact unconditional approach based on maximization. Exact approaches guarantee the type I error rate which is often not satisfied in the commonly used asymptotic approaches on the basis of the corresponding limiting distributions of test statistics. We provide tables for sample size determination in commonly used cases in practice, and an example is used to show the application of these exact approaches. Sample size comparison using different exact approaches concludes that the exact unconditional approach based on the Wald-type test statistic has good performance in balanced studies.

Fine-Scale Vertical Stratification and Guild Composition of Saproxylic Beetles in Lowland and Montane Forests: Similar Patterns despite Low Faunal Overlap.

ObjectiveThe finer scale patterns of arthropod vertical stratification in forests are rarely studied and poorly understood. Further, there are no studies investigating whether and how altitude affects arthropod vertical stratification in temperate forests. We therefore investigated the fine-scale vertical stratification of diversity and guild structure of saproxylic beetles in temperate lowland and montane forests and compared the resulting patterns between the two habitats.MethodsThe beetles were sampled with flight intercept traps arranged into vertical transects (sampling heights 0.4, 1.2, 7, 14, and 21 m). A triplet of such transects was installed in each of the five sites in the lowland and in the mountains; 75 traps were used in each forest type.Results381 species were collected in the lowlands and 236 species in the mountains. Only 105 species (21%) were found at both habitats; in the montane forest as well as in the lowlands, the species richness peaked at 1.2 m, and the change in assemblage composition was most rapid near the ground. The assemblages clearly differed between the understorey (0.4 m, 1.2 m) and the canopy (7 m, 14 m, 21 m) and between the two sampling heights within the understorey, but less within the canopy. The stratification was better pronounced in the lowland, where canopy assemblages were richer than those near the forest floor (0.4 m). In the mountains the samples from 14 and 21 m were more species poor than those from the lower heights. The guild structure was similar in both habitats.ConclusionsThe main patterns of vertical stratification and guild composition were strikingly similar between the montane and the lowland forest despite the low overlap of their faunas. The assemblages of saproxylic beetles were most stratified near ground. The comparisons of species richness between canopy and understorey may thus give contrasting results depending on the exact sampling height in the understorey.

Exact Sampling with Integer Linear Programs and Random Perturbations

We consider the problem of sampling from a discrete probability distribution specified by a graphical model. Exact samples can, in principle, be obtained by computing the mode of the original model perturbed with an exponentially many i.i.d. random variables. We propose a novel algorithm that views this as a combinatorial optimization problem and searches for the extreme state using a standard integer linear programming (ILP) solver, appropriately extended to account for the random perturbation. Our technique, GumbelMIP, leverages linear programming (LP) relaxations to evaluate the qualityof samples and prune large portions of the search space, and can thus scale to large tree-width models beyond the reach of current exact inference methods. Further, when the optimization problem is not solved to optimality, our method yields a novel approximate sampling technique. We empirically demonstrate that our approach parallelizes well, our exact sampler scales better than alternative approaches, and our approximate sampler yields better quality samples than a Gibbs sampler and a low-dimensional perturbation method.

An Euler–Poisson scheme for Lévy driven stochastic differential equations

Abstract We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of Ferreiro-Castilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that the mean-square error converges with rate O(n-1/2). The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE. Classical examples, such as stable processes, subclasses of spectrally one-sided Lévy processes, and new families, such as meromorphic Lévy processes (Kuznetsov et al. (2012), are examples for which our algorithm provides an interesting alternative to existing methods, due to its straightforward implementation and its robustness with respect to the jump structure of the driving Lévy process.

A Short Note on the Exact Stochastic Simulation Scheme of the Hull-White Model and Its Implementation

In this short note we derive an exact simulation scheme for the joint distribution of (r(t),N(t)), where r denotes the short rate following a Hull-White model and $N$ denotes the numeraire.To sample the correct joint distribution of (r(t),N(t)) our scheme requires a two-factor Brownian driver. In other words: the time-discrete Hull-White model is a two factor model.We investigate the performance of this scheme compared to the classical schemes for correlation dependent products like the LIBOR in arrears and Bermudan swaptions. We show that the exact sampling of the joint distribution is important in some applications, e.g., the valuation of Bermudan options using American Monte-Carlo method. Traditional approximation of the numeraire may generate biases if the model uses larger simulation time-steps.Large time-step simulation of short rate models and the American Monte-Carlo method is of interest in the calculation of portfolio effects (xVAs), where usually many risk factors have to be simulated and computational resources need to be saved.

Fast and accurate Monte Carlo sampling of first-passage times from Wiener diffusion models.

We present a new, fast approach for drawing boundary crossing samples from Wiener diffusion models. Diffusion models are widely applied to model choices and reaction times in two-choice decisions. Samples from these models can be used to simulate the choices and reaction times they predict. These samples, in turn, can be utilized to adjust the models’ parameters to match observed behavior from humans and other animals. Usually, such samples are drawn by simulating a stochastic differential equation in discrete time steps, which is slow and leads to biases in the reaction time estimates. Our method, instead, facilitates known expressions for first-passage time densities, which results in unbiased, exact samples and a hundred to thousand-fold speed increase in typical situations. In its most basic form it is restricted to diffusion models with symmetric boundaries and non-leaky accumulation, but our approach can be extended to also handle asymmetric boundaries or to approximate leaky accumulation.

Genome-scale estimate of the metabolic turnover of E. Coli from the energy balance analysis

In this article the notion of metabolic turnover is revisited in the light of recent results of out-of-equilibrium thermodynamics. By means of Monte Carlo methods we perform an exact sampling of the enzymatic fluxes in a genome scale metabolic network of E. Coli in stationary growth conditions from which we infer the metabolites turnover times. However the latter are inferred from net fluxes, and we argue that this approximation is not valid for enzymes working nearby thermodynamic equilibrium. We recalculate turnover times from total fluxes by performing an energy balance analysis of the network and recurring to the fluctuation theorem. We find in many cases values one of order of magnitude lower, implying a faster picture of intermediate metabolism.

Missing Data, Imputation, and Endogeneity

Basmann (Basmann, R.L., 1957, A generalized classical method of linear estimation of coefficients in a structural equation. Econometrica 25, 77-83; Basmann, R.L., 1959, The computation of generalized classical estimates of coefficients in a structural equation. Econometrica 27, 72-81) introduced two-stage least squares (2SLS). In subsequent work, Basmann (Basmann, R.L., F.L. Brown, W.S. Dawes and G.K. Schoepfle, 1971, Exact finite sample density functions of GCL estimators of structural coefficients in a leading exactly identifiable case. Journal of the American Statistical Association 66, 122-126) investigated its finite sample performance. Here, we build on this tradition focusing on the issue of 2SLS estimation of a structural model when data on the endogenous covariate is missing for some observations. Many such imputation techniques have been proposed in the literature. However, there is little guidance available for choosing among existing techniques, particularly when the covariate being imputed is endogenous. Moreover, because the finite sample bias of 2SLS is not monotonically decreasing in the degree of measurement accuracy, the most accurate imputation method is not necessarily the method that minimizes the bias of 2SLS. Instead, we explore imputation methods designed to increase the first-stage strength of the instrument(s), even if such methods entail lower imputation accuracy. We do so via simulations as well as with an application related to the medium-run effects of birth weight.

Structural Inference from Reduced Forms with Many Instruments

This paper develops exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates. Particular attention is given to models with large numbers of instruments, wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments and reduced forms. Comparisons are made with methods based on unrestricted reduced forms, and numerical computations showing finite sample performance of the tests are reported. Some new results are obtained on inequalities between noncentral chi-squared distributions with different degrees of freedom that assist in analytic power comparisons.