On quantum statistical inference

Summary. Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context.     

Belief functions and statistical inference

The Dempster Shafer theory of belief functions is a method of quantifying uncertainty that generalizes probability theory. We review the theory of belief functions in the context of statistical inference. We mainly focus on a particular belief function based on the likelihood function and its application to problems with partial prior information. We also consider connections to upper and lower probabilities and Bayesian robustness.     

A note on statistical inference for differences of covariances

This article investigates statistical inferences about differences of covariances matrices when the response has more than two values. The subspace constructed by differences of covariance matrices is related to the sufficient dimension subspace and the central space. The asymptotic distribution of test statistic for structural dimension is outlined.     

The dilation phenomenon in robust Bayesian inference

Upper and lower probabilities may become uniformly less precise after conditioning. We call this dilation. We review some results about dilation, present some examples and explore the effect of Bayesian updating. Also, we show a connection between dilation and nonconglomerability. Finally, we consider the implications of this phenomenon.     

Linear statistical inference for global and local minimum variance portfolios

Traditional portfolio optimization has often been criticized for not taking estimation risk into account. Estimation risk is mainly driven by the parameter uncertainty regarding the expected asset returns rather than their variances and covariances. The global minimum variance portfolio has been advocated by many authors as an appropriate alternative to the tangential portfolio. This is because there are no expectations which have to be estimated and thus the impact of estimation errors can be substantially reduced. However, in many practical situations an investor is not willing to choose the global minimum variance portfolio but he wants to minimize the variance of the portfolio return under specific constraints for the portfolio weights. Such a portfolio is called local minimum variance portfolio. Small-sample hypothesis tests for global and local minimum variance portfolios are derived and the exact distributions of the estimated portfolio weights are calculated in the present work. The first two moments of the estimator for the expected portfolio returns are also provided and the presented instruments are illustrated by an empirical study.     

Quantile regression for robust inference on varying coefficient partially nonlinear models

In this paper, we propose a robust statistical inference approach for the varying coefficient partially nonlinear models based on quantile regression. A three-stage estimation procedure is developed to estimate the parameter and coefficient functions involved in the model. Under some mild regularity conditions, the asymptotic properties of the resulted estimators are established. Some simulation studies are conducted to evaluate the finite performance as well as the robustness of our proposed quantile regression method versus the well known profile least squares estimation procedure. Moreover, the Boston housing price data is given to further illustrate the application of the new method.     

Conditional statistical inference and quantification of relevance

Summary. We argue that it can be fruitful to take a predictive view on notions such as the precision of a point estimator and the confidence of an interval estimator in frequentist inference. This predictive approach has implications for conditional inference, because it immediately allows a quantification of the concept of relevance for conditional inference. Conditioning on an ancillary statistic makes inference more relevant in this sense, provided that the ancillary is a precision index. Not all ancillary statistics satisfy this demand. We discuss the problem of choice between alternative ancillary statistics. The approach also has implications for the best choice of variance estimator, taking account of correlations with the squared error of estimation itself. The theory is illustrated by numerous examples, many of which are classical.     

Nonconglomerative coherent conditional probabilities in statistical inference

Noteworthy connections among conglomerability, countable additivity and coherence are discussed in detail, reaching the conclusion that nonconglomerable conditional probabilities must not be doomed and play a significant role in statistical inference. Extended and updated version of a contributed paper presented at the International Conference on “Information Processing and Management of Uncertainty in knowledge-based systems”, IPMU 2004, Perugia, Italy.     

Union–intersection principle and constrained statistical inference

Most statistical models arising in real life applications as well as in interdisciplinary research are complex in their designs, sampling plans, and associated probability laws, which in turn are often constrained by inequality, order, functional, shape or other restraints. Optimality of conventional likelihood ratio based statistical inference may not be tenable here, although the use of restricted or quasi-likelihood has spurred in such environments. S.N. Roy's ingenious union–intersection principle provides an alternative avenue, often having some computational advantages, increased scope of adaptability, and flexibility beyond conventional likelihood paradigms. This scenario is appraised here with some illustrative examples, and with some interesting problems of inference on stochastic ordering (dominance) in parametric as well as beyond parametric setups.     

Order restricted statistical inference for scale parameters based on sequential order statistics

In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of-n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework.     

Views on conditional and marginal methods of statistical inference

Conditional and marginal likelihood analysis has a long history of development. Some recent methods using exact and approximate density and distribution functions lead to more sharply defined likelihoods and to accurate observed levels of significance for a wide range of problems including nonnormal regression and exponential linear models. These developments will be surveyed.     

Alternating event processes during lifetimes: population dynamics and statistical inference

In the literature studying recurrent event data, a large amount of work has been focused on univariate recurrent event processes where the occurrence of each event is treated as a single point in time. There are many applications, however, in which univariate recurrent events are insufficient to characterize the feature of the process because patients experience nontrivial durations associated with each event. This results in an alternating event process where the disease status of a patient alternates between exacerbations and remissions. In this paper, we consider the dynamics of a chronic disease and its associated exacerbation-remission process over two time scales: calendar time and time-since-onset. In particular, over calendar time, we explore population dynamics and the relationship between incidence, prevalence and duration for such alternating event processes. We provide nonparametric estimation techniques for characteristic quantities of the process. In some settings, exacerbation processes are observed from an onset time until death; to account for the relationship between the survival and alternating event processes, nonparametric approaches are developed for estimating exacerbation process over lifetime. By understanding the population dynamics and within-process structure, the paper provide a new and general way to study alternating event processes.     

Bayesian statistical inference of the loglogistic model with interval-censored lifetime data

Interval-censored data arise when a failure time say, T cannot be observed directly but can only be determined to lie in an interval obtained from a series of inspection times. The frequentist approach for analysing interval-censored data has been developed for some time now. It is very common due to unavailability of software in the field of biological, medical and reliability studies to simplify the interval censoring structure of the data into that of a more standard right censoring situation by imputing the midpoints of the censoring intervals. In this research paper, we apply the Bayesian approach by employing Lindley's 1980, and Tierney and Kadane 1986 numerical approximation procedures when the survival data under consideration are interval-censored. The Bayesian approach to interval-censored data has barely been discussed in literature. The essence of this study is to explore and promote the Bayesian methods when the survival data been analysed are is interval-censored. We have considered only a parametric approach by assuming that the survival data follow a loglogistic distribution model. We illustrate the proposed methods with two real data sets. A simulation study is also carried out to compare the performances of the methods.     

Bayesian statistical inference for start-up demonstration tests with rejection of units upon observing d failures

This paper is concerned with Bayesian estimation and prediction in the context of start-up demonstration tests in which rejection of a unit is possible when a pre-specified number of failures is observed prior to obtaining the number of consecutive successes required for acceptance of the unit. A method for implementing Bayesian inference on the probability of success is developed for use when the test result of each start-up is not reported or even recorded, and only the number of trials until termination of the testing is available. Some errors in the related literature on the Bayesian analysis of start-up demonstration tests are corrected. The method developed in this paper is a Markov chain Monte Carlo (MCMC) method incorporating data augmentation, and it additionally enables Bayesian posterior inference on the number of failures given the number of start-up trials until termination to be made, along with Bayesian predictive inferences on the number of start-up trials and the number of failures until termination for any future run of the start-up demonstration test. An illustrative example is also included.     

An overlook on statistical inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition

ABSTRACT

In this paper, we are interested in nonparametric inference issues for stochastic damping hamiltonian systems under the fluctuation-dissipation condition. This condition relates the magnitude of the dissipative term and the magnitude of the random term. The precise balance between the drift term which removes energy in average and the stochastic term provided by the fluctuation-dissipation relation insures that the canonical measure is preserved by the dynamics. In this framework, it is possible to give an explicit construction of a Lyapunov function and thus to prove exponential ergodicity. Then, we consider various estimation procedures and provide also a numerical section, where simulations are conducted.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

Some inequalities in elementary special functions with applications to nonparametric statistical inference

Some inequalities are established in P₁(r, s) and P₁(r+1, s), where P₁(r, s) is the confidence coefficient of Wilks’ (1962) outer confidence interval (X_(r) X_(s)) for the quantile interval (ξ_p1, ξ_p2). An inequality concerning incomplete beta functions is also presented and it is shown to be an improved version of one of Koti's (1989) inequalities.     

Some aspects of statistical inference on the lagrange (generalized) poisson distribution

A generalization of the Poisson distribution was defined by Consul and Jain (Ann. Math. Statist., 41, (1970)) and was obtained as a particular family of Lagrange distributions by Consul and Shenton (SIAM. J. Appl. Math., 23, (1972)). The distribution is subsequently named the generalized Poisson distribution (GPD). This GPD reduces to the Poisson distribution for ? = 0. When the data have a one-way layout structure, the asymptotically locally optimal Neyman's C(d) test is constructed and compared with the conditional test on the hypothesis H_o? = 0. Within the framework of the generalized linear models an appropriate link function is given, and the asymptotic distributions of the estimated parameters are derived.     

On the statistical inference of a machine-generated autoregressive AR(1) model

We have obtained the asymptotic bias and the limiting distribution for the Yule–Walker estimator of the autoregressive parameter under a considerably weaker assumption than that of independence in the noise sequence. Among other things, these suggest robustness of the classical results and throw some light on the use of simulations based on pseudorandom numbers in verifying these results.