Exact Statistical Inference Research Articles

Exact Inference with Approximate Computation for Differentially Private Data via Perturbations

This paper discusses how two classes of approximate computation algorithms can be adapted, in a modular fashion, to achieve exact statistical inference from differentially private data products. Considered are approximate Bayesian computation for Bayesian inference, and Monte Carlo Expectation-Maximization for likelihood inference. Up to Monte Carlo error, inference from these algorithms is exact with respect to the joint specification of both the analyst's original data model, and the curator's differential privacy mechanism. Highlighted is a duality between approximate computation on exact data, and exact computation on approximate data, which can be leveraged by a well-designed computational procedure for statistical inference.

Exact statistical inference for the Wasserstein distance by selective inference

In this paper, we study statistical inference for the Wasserstein distance, which has attracted much attention and has been applied to various machine learning tasks. Several studies have been proposed in the literature, but almost all of them are based on asymptotic approximation and do not have finite-sample validity. In this study, we propose an exact (non-asymptotic) inference method for the Wasserstein distance inspired by the concept of conditional selective inference (SI). To our knowledge, this is the first method that can provide a valid confidence interval (CI) for the Wasserstein distance with finite-sample coverage guarantee, which can be applied not only to one-dimensional problems but also to multi-dimensional problems. We evaluate the performance of the proposed method on both synthetic and real-world datasets.

Products, Sums and Quotients of Upper Truncated Pareto Random Variables with an Application in Hydrology

Some hydrological quantities such as the rainfall intensity, which is defined as the quotient of the rainfall depth to the rainfall duration, are based on functions of random variables. At this point, the probability distribution of that quantity arises. Then one may take this distribution into account for the exact statistical inference without referring to a simulation study. There are a lot of works on the exact distributions of functions of random variables in the literature. One case is for the Pareto distributed random variables. Pareto distribution and its upper truncated version have many applications in hydrological modelling. In this paper, the exact distributions of the product, sum and quotient of two independently distributed upper truncated Pareto random variables are obtained. Although the probability density functions of the product and quotient are obtained in elementary mathematical functions, that for the sum is obtained in terms of a special function. Some characteristics of these functions such as moments and percentiles can be easily obtained. The distributions of the quotient and the sum are applied on a rainfall data set from hydraulic efficiency research of green roofs. The parameters are estimated by the method of maximum likelihood. The theoretical results of this paper may also be useful to other practitioners of the upper truncated Pareto distribution.

Lattice Paths for Persistent Diagrams.

Persistent homology has undergone significant development in recent years. However, one outstanding challenge is to build a coherent statistical inference procedure on persistent diagrams. In this paper, we first present a new lattice path representation for persistent diagrams. We then develop a new exact statistical inference procedure for lattice paths via combinatorial enumerations. The lattice path method is applied to the topological characterization of the protein structures of the COVID-19 virus. We demonstrate that there are topological changes during the conformational change of spike proteins.

Designing historical control studies with survival endpoints using exact statistical inference

Historical control trials compare an experimental treatment with a previously conducted control treatment. By assigning all recruited samples to the experimental arm, historical control trials can better identify promising treatments in early phase trials compared with randomized control trials. Existing designs of historical control trials with survival endpoints are based on asymptotic normal distribution. However, it remains unclear whether the asymptotic distribution of the test statistic is close enough to the true distribution given relatively small sample sizes in early phase trials. In this article, we address this question by introducing an exact design approach for exponentially distributed survival endpoints, and compare it with an asymptotic design in both real examples and simulation examples. Simulation results show that the asymptotic test could lead to bias in the sample size estimation. We conclude the proposed exact design should be used in the design of historical control trials.

ON THE PRIVACY AND UTILITY PROPERTIES OF TRIPLE MATRIX-MASKING.

Privacy protection is an important requirement in many statistical studies. A recently proposed data collection method, triple matrix-masking, retains exact summary statistics without exposing the raw data at any point in the process. In this paper, we provide theoretical formulation and proofs showing that a modified version of the procedure is strong collection obfuscating: no party in the data collection process is able to gain knowledge of the individual level data, even with some partially masked data information in addition to the publicly published data. This provides a theoretical foundation for the usage of such a procedure to collect masked data that allows exact statistical inference for linear models, while preserving a well-defined notion of privacy protection for each individual participant in the study. This paper fits into a line of work tackling the problem of how to create useful synthetic data without having a trustworthy data aggregator. We achieve this by splitting the trust between two parties, the “masking service provider” and the “data collector.”

Efficient neutrino oscillation parameter inference using Gaussian processes

The unified approach of Feldman and Cousins allows for exact statistical inference of small signals that commonly arise in high energy physics. It has gained widespread use, for instance, in measurements of neutrino oscillation parameters in long-baseline experiments. However, the approach relies on the Neyman construction of the classical confidence interval and is computationally intensive as it is typically done in a grid-based fashion over the entire parameter space. In this letter, we propose an efficient algorithm for the Feldman-Cousins approach using Gaussian processes to construct confidence intervals iteratively. We show that in the neutrino oscillation context, one can obtain confidence intervals 5 times faster in one dimension and 10 times faster in two dimensions, while maintaining an accuracy above 99.5%.

Computing the exact distribution of the Bartlett's test statistic by numerical inversion of its characteristic function

Application of the exact statistical inference frequently leads to non-standard probability distributions of the considered estimators or test statistics. The exact distributions of many estimators and test statistics can be specified by their characteristic functions, as is the case for the null distribution of the Bartlett's test statistic. However, analytical inversion of the characteristic function, if possible, frequently leads to complicated expressions for computing the distribution function and the corresponding quantiles. An efficient alternative is the well-known method based on numerical inversion of the characteristic functions, which is, however, ignored in popular statistical software packages. In this paper, we present the explicit characteristic function of the corrected Bartlett's test statistic together with the computationally fast and efficient implementation of the approach based on numerical inversion of this characteristic function, suggested for evaluating the exact null distribution used for testing homogeneity of variances in several normal populations, with possibly unequal sample sizes.

Exact Topological Inference for Paired Brain Networks via Persistent Homology.

We present a novel framework for characterizing paired brain networks using techniques in hyper-networks, sparse learning and persistent homology. The framework is general enough for dealing with any type of paired images such as twins, multimodal and longitudinal images. The exact nonparametric statistical inference procedure is derived on testing monotonic graph theory features that do not rely on time consuming permutation tests. The proposed method computes the exact probability in quadratic time while the permutation tests require exponential time. As illustrations, we apply the method to simulated networks and a twin fMRI study. In case of the latter, we determine the statistical significance of the heritability index of the large-scale reward network where every voxel is a network node.

Exact inference on Gaussian graphical models of arbitrary topology using path-sums

We present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from the closed-form resummation of infinite families of terms of the walk-sum representation of the covariance matrix. We prove that the path-sum formulation always exists for models whose covariance matrix is positive definite: i.e. it is valid for both walk-summable and non-walk-summable graphical models of arbitrary topology. We show that for graphical models on trees the path-sum formulation is equivalent to Gaussian belief propagation. We also recover, as a corollary, an existing result that uses determinants to calculate the covariance matrix. We show that the path-sum formulation formulation is valid for arbitrary partitions of the inverse covariance matrix. We give detailed examples demonstrating our results.

Approximate Inference for Observation-Driven Time Series Models with Intractable Likelihoods

In this article, we consider approximate Bayesian parameter inference for observation-driven time series models. Such statistical models appear in a wide variety of applications, including econometrics and applied mathematics. This article considers the scenario where the likelihood function cannot be evaluated pointwise; in such cases, one cannot perform exact statistical inference, including parameter estimation, which often requires advanced computational algorithms, such as Markov Chain Monte Carlo (MCMC). We introduce a new approximation based upon Approximate Bayesian Computation (ABC). Under some conditions, we show that as n → ∞, with n the length of the time series, the ABC posterior has, almost surely, a Maximum A Posteriori (MAP) estimator of the parameters that is often different from the true parameter. However, a noisy ABC MAP, which perturbs the original data, asymptotically converges to the true parameter, almost surely. In order to draw statistical inference, for the ABC approximation adopted, standard MCMC algorithms can have acceptance probabilities that fall at an exponential rate in n and slightly more advanced algorithms can mix poorly. We develop a new and improved MCMC kernel, which is based upon an exact approximation of a marginal algorithm, whose cost per iteration is random, but the expected cost, for good performance, is shown to be O ( n 2 ) per iteration. We implement our new MCMC kernel for parameter inference from models in econometrics.

Standard error of inverse prediction for dose–response relationship: approximate and exact statistical inference

This paper develops a new metric, the standard error of inverse prediction (SEIP), for a dose-response relationship (calibration curve) when dose is estimated from response via inverse regression. SEIP can be viewed as a generalization of the coefficient of variation to regression problem when x is predicted using y-value. We employ nonstandard statistical methods to treat the inverse prediction, which has an infinite mean and variance due to the presence of a normally distributed variable in the denominator. We develop confidence intervals and hypothesis testing for SEIP on the basis of the normal approximation and using the exact statistical inference based on the noncentral t-distribution. We derive the power functions for both approaches and test them via statistical simulations. The theoretical SEIP, as the ratio of the regression standard error to the slope, is viewed as reciprocal of the signal-to-noise ratio, a popular measure of signal processing. The SEIP, as a figure of merit for inverse prediction, can be used for comparison of calibration curves with different dependent variables and slopes. We illustrate our theory with electron paramagnetic resonance tooth dosimetry for a rapid estimation of the radiation dose received in the event of nuclear terrorism.

Exact nonparametric inference for component lifetime distribution based on lifetime data from systems with known signatures

In this paper, we develop exact nonparametric statistical inference for some characteristics of the component lifetime distribution based on the lifetimes of coherent systems with known signatures. Distribution-free confidence limits for population quantiles of component lifetime distributions are derived. Computational formulas as well as a procedure for choosing suitable confidence limits are presented. Construction of tolerance limits for the component lifetime distribution is also described. Finally, some examples are presented to illustrate the methods of inference developed here.

Some exact tests for manifest properties of latent trait models

Item response theory is one of the modern test theories with applications in educational and psychological testing. Recent developments made it possible to characterize some desired properties in terms of a collection of manifest ones, so that hypothesis tests on these traits can, in principle, be performed. But the existing test methodology is based on asymptotic approximation, which is impractical in most applications since the required sample sizes are often unrealistically huge. To overcome this problem, a class of tests is proposed for making exact statistical inference about four manifest properties: covariances given the sum are non-positive (CSN), manifest monotonicity (MM), conditional association (CA), and vanishing conditional dependence (VCD). One major advantage is that these exact tests do not require large sample sizes. As a result, tests for CSN and MM can be routinely performed in empirical studies. For testing CA and VCD, the exact methods are still impractical in most applications, due to the unusually large number of parameters to be tested. However, exact methods are still derived for them as an exploration toward practicality. Some numerical examples with applications of the exact tests for CSN and MM are provided.

Some Exact Tests for Manifest Properties of Latent Trait Models

Item response theory is one of the modern test theories with applications in educational and psychological testing. Recent developments made it possible to characterize some desired properties in terms of a collection of manifest ones, so that hypothesis tests on these traits can, in principle, be performed. But the existing test methodology is based on asymptotic approximation, which is impractical in most applications since the required sample sizes are often unrealistically huge. To overcome this problem, a class of tests is proposed for making exact statistical inference about four manifest properties: covariances given the sum are non-positive (CSN), manifest monotonicity (MM), conditional association (CA), and vanishing conditional dependence (VCD). One major advantage is that these exact tests do not require large sample sizes. As a result, tests for CSN and MM can be routinely performed in empirical studies. For testing CA and VCD, the exact methods are still impractical in most applications, due to the unusually large number of parameters to be tested. However, exact methods are still derived for them as an exploration toward practicality. Some numerical examples with applications of the exact tests for CSN and MM are provided.

Bayesian variable selection and model averaging in the arbitrage pricing theory model

Empirical tests of the arbitrage pricing theory using measured variables rely on the accuracy of standard inferential theory in approximating the distribution of the estimated risk premiums and factor betas. The techniques employed thus far perform factor selection and model inference sequentially. Recent advances in Bayesian variable selection are adapted to an approximate factor model to investigate the role of measured economic variables in the pricing of securities. In finite samples, exact statistical inference is carried out using posterior distributions of functions of risk premiums and factor betas. The role of the panel dimensions in posterior inference is investigated. New empirical evidence is found of time-varying risk premiums with higher and more volatile expected compensation for bearing systematic risk during contraction phases. In addition, investors are rewarded for exposure to “Economic” risk.

Estimation of population size in closed animal populations from mark-resighting surveys

The problem of estimating the size of a closed population based on the results of a certain type of mark-resighting sampling design is considered. The sampling design is similar to the commonly used multiple capture–recapture design. However, with some animal populations this procedure is more economical and easier to use. Sampling is done by first tagging a number of randomly selected animals with visible markers and later randomly sighting them on several occasions and counting the number of tagged animals. This type of sampling procedure is being used in several wildlife population monitoring studies. Some approximate statistical inference procedures which are based on the binomial approximation are available in the literature. Exact statistical inference involves dealing with the hypergeometric distribution. In this paper, classical and Bayesian inference procedures related to the exact sampling distribution and several other approximate procedures are given. Applications of these procedures for a real data set and for a couple of simulated data sets are provided. A limited simulation study is also given to demonstrate the performance of the proposed procedure.

Viral infections and chemical exposures as risk factors for hepatocellular carcinoma in Vietnam

A case-control study investigating risk factors for hepatocellular carcinoma (HCC) was conducted in Hanoi, in the north of Vietnam, between 1989 and 1992. Male cases of HCC (152) diagnosed in 2 hospitals were included. Hospital controls (241) admitted mainly to abdominal surgery departments were frequency-matched to cases for sex, age, hospital and place of residence (Hanoi, province). Odds ratios adjusted for matching variables and other potential confounders were estimated using unconditional logistic regression, or exact non-parametric statistical inference when numbers were small. Positivity for hepatitis B surface antigen (HBsAg) was the main risk factor for HCC in this sample. Five subjects (3 cases, 2 controls) had been infected by hepatitis C virus (HCV), and none of them were carriers of HBsAg, giving an OR of 38 associated with HCV infection among HBsAG-negative subjects. Alcohol drinking was associated with HCC and interacted with HBsAg positivity. Agricultural use of organophosphorous pesticides (30 liters/year or more) and military service in the south of Vietnam for 10 years or more were also associated with an increased risk of HCC. This study confirms the major role played by HBV infection and its association with HCC in south-east Asia. It also suggests how other factors such as alcohol consumption or exposure to chemicals may interact with HBV infection.

Control-Variate Models of Common Random Numbers for Multiple Comparisons with the Best

Using common random numbers (CRN) in simulation experiment design is known to reduce the variance of estimators of differences in system performance. However, when more than two systems are compared, exact simultaneous statistical inference in conjunction with CRN is typically impossible. We introduce control-variate models of CRN that permit exact statistical inference, specifically multiple comparisons with the best. These models explain the effect of CRN via a linear regression of the simulation output on “control variates” that are functions of the simulation inputs. We establish theoretically, and illustrate empirically, that the control-variate models lead to sharper statistical inference in the sense that the probability of detecting differences in systems' performance is increased.

Exact Statistical Inference for Group Sequential Trials

This paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference. The existing group sequential methods for such trials are based on the large-sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses. We demonstrate through extensive numerical studies that, for small and even moderate-sized trials, these approximate procedures may lead to tests with supranominal size (mainly when unpooled estimators of variance are used) and confidence intervals with under-nominal coverage probability. We then study exact methods for group sequential testing, repeated interval estimation, and interval estimation following sequential testing. The new procedures can accommodate any treatment allocation rules. An example using real data is provided.