Probability Generating Function Based Jeffrey's Divergence for Statistical Inference

Statistical inference procedures based on transforms such as characteristic function and probability generating function have been examined by many researchers because they are much simpler than probability density functions. Here, a probability generating function based Jeffrey's divergence measure is proposed for parameter estimation and goodness-of-fit test. Being a member of the M-estimators, the proposed estimator is consistent. Also, the proposed goodness-of-fit test has good statistical power. The proposed divergence measure shows improved performance over existing probability generating function based measures. Real data examples are given to illustrate the proposed parameter estimation method and goodness-of-fit test.     

A Statistical Inference Course Based on p-Values

Introductory statistical inference texts and courses treat the point estimation, hypothesis testing, and interval estimation problems separately, with primary emphasis on large-sample approximations. Here, I present an alternative approach to teaching this course, built around p-values, emphasizing provably valid inference for all sample sizes. Details about computation and marginalization are also provided, with several illustrative examples, along with a course outline. Supplementary materials for this article are available online.     

Statistical Inference for Expectile‐based Risk Measures

Expectiles were introduced by Newey and Powell in 1987 in the context of linear regression models. Recently, Bellini et al. revealed that expectiles can also be seen as reasonable law‐invariant risk measures. In this article, we show that the corresponding statistical functionals are continuous w.r.t. the 1‐weak topology and suitably functionally differentiable. By means of these regularity results, we can derive several properties such as consistency, asymptotic normality, bootstrap consistency and qualitative robustness of the corresponding estimators in nonparametric and parametric statistical models.     

Influence Measures Based on Cressie-Read Divergence Measures in Multivariate Linear Model

We define a new family of influence measures based on the divergence measures, in the multivariate general linear model. Influence measures are obtained by quantifying the divergence between the sample distribution of an estimate obtained with all the observations and the sample distribution of the same estimate obtained without any observation. This approach is applied to best linear unbiased estimates of estimable functions. Therefore, these diagnostics can be applied to every statistical multivariate technique that can be formulated like this kind of model. Some examples are considered to clarify the applicability of the introduced diagnostics.     

Statistical Inference Based on Progressively Type II Censored Data from Weibull Model

In this article, we consider the problem of estimating the shape and scale parameters and predicting the unobserved removed data based on a progressive type II censored sample from the Weibull distribution. Maximum likelihood and Bayesian approaches are used to estimate the scale and shape parameters. The sampling-based method is used to draw Monte Carlo (MC) samples and it has been used to estimate the model parameters and also to predict the removed units in multiple stages of the censored sample. Two real datasets are presented and analyzed for illustrative purposes and Monte carlo simulations are performed to study the behavior of the proposed methods.     

Statistical Inference Based on Pooled Data: A Moment-Based Estimating Equation Approach

We consider statistical inference on parameters of a distribution when only pooled data are observed. A moment-based estimating equation approach is proposed to deal with situations where likelihood functions based on pooled data are difficult to work with. We outline the method to obtain estimates and test statistics of the parameters of interest in the general setting. We demonstrate the approach on the family of distributions generated by the Box-Cox transformation model, and, in the process, construct tests for goodness of fit based on the pooled data.     

Statistical Inference on a Stochastic Epidemic Model

In this work, we develop statistical inference for the parameters of a discrete-time stochastic SIR epidemic model. We use a Markov chain for describing the dynamic behavior of the epidemic. Specifically, we propose estimators for the contact and removal rates based on the maximum likelihood and martingale methods, and establish their asymptotic distributions. The obtained results are applied in the statistical analysis of the basic reproduction number, a quantity that is useful in establishing vaccination policies. In order to evaluate the population size for which the results are useful, a numerical study is carried out. Finally, a comparison of the maximum likelihood and martingale estimators is conducted by means of Monte Carlo simulations.     

Statistical Inference for Renewal Processes

We consider non‐parametric estimation for interarrival times density of a renewal process. For continuous time observation, a projection estimator in the orthonormal Laguerre basis is built. Nonstandard decompositions lead to bounds on the mean integrated squared error (MISE), from which rates of convergence on Sobolev–Laguerre spaces are deduced, when the length of the observation interval gets large. The more realistic setting of discrete time observation is more difficult to handle. A first strategy consists in neglecting the discretization error. A more precise strategy aims at taking into account the convolution structure of the data. Under a simplifying ‘dead‐zone’ condition, the corresponding MISE is given for any sampling step. In the three cases, an automatic model selection procedure is described and gives the best MISE, up to a logarithmic term. The results are illustrated through a simulation study.     

The Limited Role of Formal Statistical Inference in Scientific Inference

ABSTRACT

Such is the grip of formal methods of statistical inference—that is, frequentist methods for generalizing from sample to population in enumerative studies—in the drawing of scientific inferences that the two are routinely deemed equivalent in the social, management, and biomedical sciences. This, despite the fact that legitimate employment of said methods is difficult to implement on practical grounds alone. But supposing the adoption of these procedures were simple does not get us far; crucially, methods of formal statistical inference are ill-suited to the analysis of much scientific data. Even findings from the claimed gold standard for examination by the latter, randomized controlled trials, can be problematic.

Scientific inference is a far broader concept than statistical inference. Its authority derives from the accumulation, over an extensive period of time, of both theoretical and empirical knowledge that has won the (provisional) acceptance of the scholarly community. A major focus of scientific inference can be viewed as the pursuit of significant sameness, meaning replicable and empirically generalizable results among phenomena. Regrettably, the obsession with users of statistical inference to report significant differences in data sets actively thwarts cumulative knowledge development.

The manifold problems surrounding the implementation and usefulness of formal methods of statistical inference in advancing science do not speak well of much teaching in methods/statistics classes. Serious reflection on statistics' role in producing viable knowledge is needed. Commendably, the American Statistical Association is committed to addressing this challenge, as further witnessed in this special online, open access issue of The American Statistician.     

Statistical Inference for Damaged Poisson Distribution

Abstract

For non-negative integer-valued random variables, the concept of “damaged” observations was introduced, for the first time, by Rao and Rubin [Rao, C. R., Rubin, H. (1964). On a characterization of the Poisson distribution. Sankhya 26:295–298] in 1964 on a paper concerning the characterization of Poisson distribution. In 1965, Rao [Rao, C. R. (1965). On discrete distribution arising out of methods of ascertainment. Sankhya Ser. A. 27:311–324] discusses some results related with inferences for parameters of a Poisson Model when it has occurred partial destruction of observations. A random variable is said to be damaged if it is unobservable, due to a damage mechanism which randomly reduces its magnitude. In subsequent years, considerable attention has been given to characterizations of distributions of such random variables that satisfy the “Rao–Rubin” condition. This article presents some inference aspects of a damaged Poisson distribution, under reasonable assumption that, when an observation on the random variable is made, it is also possible to determine whether or not some damage has occurred. In other words, we do not know how many items are damaged, but we can identify the existence of damage. Particularly it is illustrated the situation in which it is possible to identify the occurrence of some damage although it is not possible to determine the amount of items damaged. Maximum likelihood estimators of the underlying parameters and their asymptotic covariance matrix are obtained. Convergence of the estimates of parameters to the asymptotic values are studied through Monte Carlo simulations.     

The p-value Function and Statistical Inference

ABSTRACT

This article has two objectives. The first and narrower is to formalize the p-value function, which records all possible p-values, each corresponding to a value for whatever the scalar parameter of interest is for the problem at hand, and to show how this p-value function directly provides full inference information for any corresponding user or scientist. The p-value function provides familiar inference objects: significance levels, confidence intervals, critical values for fixed-level tests, and the power function at all values of the parameter of interest. It thus gives an immediate accurate and visual summary of inference information for the parameter of interest. We show that the p-value function of the key scalar interest parameter records the statistical position of the observed data relative to that parameter, and we then describe an accurate approximation to that p-value function which is readily constructed.     

网络访问固定样本调查的统计推断研究

如何解决网络访问固定样本调查的统计推断问题,是大数据背景下网络调查面临的严重挑战。针对此问题,提出将网络访问固定样本的调查样本与概率样本结合,利用倾向得分逆加权和加权组调整构造伪权数来估计目标总体,进一步采用基于有放回概率抽样的Vwr方法、基于广义回归估计的Vgreg方法与Jackknife方法来估计方差,并比较不同方法估计的效果。研究表明:无论概率样本的样本量较大还是较小,本研究所提出的总体均值估计方法效果较好,并且在方差估计中Jackknife方法的估计效果最好。     

Multiple Perspectives on Inference for Two Simple Statistical Scenarios

ABSTRACT

When data analysts operate within different statistical frameworks (e.g., frequentist versus Bayesian, emphasis on estimation versus emphasis on testing), how does this impact the qualitative conclusions that are drawn for real data? To study this question empirically we selected from the literature two simple scenarios—involving a comparison of two proportions and a Pearson correlation—and asked four teams of statisticians to provide a concise analysis and a qualitative interpretation of the outcome. The results showed considerable overall agreement; nevertheless, this agreement did not appear to diminish the intensity of the subsequent debate over which statistical framework is more appropriate to address the questions at hand.     

有限总体基尼系数的统计推断

进行基尼系数的统计推断时,已有研究一般都设定总体是无限的。研究在有限总体随机抽样的现实背景下,如何利用样本数据对总体基尼系数进行估计及其评价。从总体基尼系数的内涵和定义出发,介绍了非参数估计和参数估计的方法,构造了相应的估计量,基于蒙特卡罗的模拟结果,论证和揭示了不同情况下估计量的性质,并论述了方法的适用性以及在实际应用中需要注意的问题。     

Bayesian Statistical Inference for Weighted Exponential Distribution

Exponential distribution has an extensive application in reliability. Introducing shape parameter to this distribution have produced various distribution functions. In their study in 2009, Gupta and Kundu brought another distribution function using Azzalini's method, which is applicable in reliability and named as weighted exponential (WE) distribution. The parameters of this distribution function have been recently estimated by the above two authors in classical statistics. In this paper, Bayesian estimates of the parameters are derived. To achieve this purpose we use Lindley's approximation method for the integrals that cannot be solved in closed form. Furthermore, a Gibbs sampling procedure is used to draw Markov chain Monte Carlo samples from the posterior distribution indirectly and then the Bayes estimates of parameters are derived. The estimation of reliability and hazard functions are also discussed. At the end of the paper, some comparisons between classical and Bayesian estimation methods are studied by using Monte Carlo simulation study. The simulation study incorporates complete and Type-II censored samples.     

Statistical Inference for Transformation Inhomogeneous Point Processes

Abstract.  Statistical inference for exponential inhomogeneous Markov point processes by transformation is discussed. It is argued that the inhomogeneity parameter can be estimated, using a partial likelihood based on an inhomogeneous Poisson point process. The inhomogeneity parameter can thereby be estimated without taking the interaction into account, which simplifies the statistical analysis considerably. Data analysis and simulation experiments support the results.