Inference for Mixtures

In this article, we consider the problem of estimating certain “parameters” in a mixture of probability measures. We show that a single sample is typically suitable for estimating the component measures, but not suitable for estimating the mixing measures, especially when consistency is required. To have consistent estimators of the mixing measure, several samples with increasing size are needed in general.     

Inference for response-limited time-series models

This paper Introduces a robust time series estimation procedure based on a new nonlinear time-series model called the response-limited ARMA model. Under this procedure, In contrast to location or regression, the uncorrected pseudo-observations, which produce the parameter estimates when submitted to a standard analysis, also yield satisfactory standard errors for inference purposes.     

收入基尼系数的统计推断

 基尼系数估计量的统计推断是基尼系数研究的一个重点。本文我们使用Davidson（2009）提出的近似大样本渐进分布方法,对收入基尼系数估计量进行统计推断,包括计算估计量的标准差、构造置信区间和进行假设检验。通过模拟试验,我们验证了在小样本情形下,依据该方法所做的统计推断具有较高的可靠性。在此基础上,我们对我国城镇居民的真实收入基尼系数进行了统计推断。     

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.     

Likelihood-Based Inference for Semi-Competing Risks

Consider semi-competing risks data (two times to concurrent events are studied but only one of them is right-censored by the other one) where the link between the times Y and C to non-terminal and terminal events, respectively, is modeled by a family of Archimedean copulas. Moreover, both Y and C are submitted to an independent right censoring variable D. We propose to estimate the parameter of the copula and some resulting survival functions using a pseudo maximum likelihood approach. The main advantage of this procedure is that it extends to multidimensional parameters copulas. We perform simulations to study the behavior of our estimation procedure and its impact on other related estimators and we apply our method to real data coming from a study on the Hodgkin disease.     

Some lower bounds of centered $$L_2$$-discrepancy of $$2^{s-k}$$2s-k designs and their complementary designs

The indicator function is an effective tool in studying factorial designs. This paper presents some lower bounds of centered \(L_2\)-discrepancy through indicator function. Some new lower bounds of centered \(L_2\)-discrepancy for \(2^{s-k}\) designs and their complementary designs are given. Numerical results show that our lower bounds are tight and better than the existing results.     

Likelihood-Based Inference for Weak Exogeneity in I(2) Cointegrated VAR Models

This article develops limit theory for likelihood analysis of weak exogeneity in I(2) cointegrated vector autoregressive (VAR) models incorporating deterministic terms. Conditions for weak exogeneity in I(2) VAR models are reviewed, and the asymptotic properties of conditional maximum likelihood estimators and a likelihood-based weak exogeneity test are then investigated. It is demonstrated that weak exogeneity in I(2) VAR models allows us to conduct asymptotic conditional inference based on mixed Gaussian distributions. It is then proved that a log-likelihood ratio test statistic for weak exogeneity in I(2) VAR models is asymptotically χ² distributed. The article also presents an empirical illustration of the proposed test for weak exogeneity using Japan's macroeconomic data.     

Robust Inference for Inverse Stochastic Dominance

The notion of inverse stochastic dominance is gaining increasing support in risk, inequality, and welfare analysis as a relevant criterion for ranking distributions, which is alternative to the standard stochastic dominance approach. Its implementation rests on comparisons of two distributions’ quantile functions, or of their multiple partial integrals, at fixed population proportions. This article develops a novel statistical inference model for inverse stochastic dominance that is based on the influence function approach. The proposed method allows model-free evaluations that are limitedly affected by contamination in the data. Asymptotic normality of the estimators allows to derive tests for the restrictions implied by various forms of inverse stochastic dominance. Monte Carlo experiments and an application promote the qualities of the influence function estimator when compared with alternative dominance criteria.     

Inference for clusters of extreme values

Summary. Inference for clusters of extreme values of a time series typically requires the identification of independent clusters of exceedances over a high threshold. The choice of declustering scheme often has a significant effect on estimates of cluster characteristics. We propose an automatic declustering scheme that is justified by an asymptotic result for the times between threshold exceedances. The scheme relies on the extremal index, which we show may be estimated before declustering, and supports a bootstrap procedure for assessing the variability of estimates.     

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.     

Kernel Likelihood Inference for Time Series

Abstract.  This paper develops non-parametric techniques for dynamic models whose data have unknown probability distributions. Point estimators are obtained from the maximization of a semiparametric likelihood function built on the kernel density of the disturbances. This approach can also provide Kullback–Leibler cross-validation estimates of the bandwidth of the kernel densities. Confidence regions are derived from the dual-empirical likelihood method based on non-parametric estimates of the scores. Limit theorems for martingale difference sequences support the statistical theory; moreover, simulation experiments and a real case study show the validity of the methods.     

Adaptive Inference for Multi-Stage Survey Data

Multi-level models can be used to account for clustering in data from multi-stage surveys. In some cases, the intraclass correlation may be close to zero, so that it may seem reasonable to ignore clustering and fit a single-level model. This article proposes several adaptive strategies for allowing for clustering in regression analysis of multi-stage survey data. The approach is based on testing whether the PSU-level variance component is zero. If this hypothesis is retained, then variance estimates are calculated ignoring clustering; otherwise, clustering is reflected in variance estimation. A simple simulation study is used to evaluate the various procedures.     

Large-Sample Inference for the Epsilon-Skew-t Distribution

The main object of this article is to discuss maximum likelihood inference for the epsilon-skew-t distribution. Special cases of this distribution include the epsilon-skew-Cauchy and the epsilon-skew-normal distributions. We derive the information matrix for the maximum likelihood estimators. The approach is applied to a data set presenting significant amount of skewness and heavy tails. In the application we consider the epsilon-skew-t distribution with known and unknown degrees of freedom parameter, showing great flexibility in adjusting to skew data with heavy tails.     

Inference for current leukemia free survival

Donor lymphocyte infusion (DLI) for patients who relapse following an allogeneic stem cell transplant has proved remarkably durable. Because of the potential for second remissions with DLI, the current leukemia free survival (CLFS), which is the probability that a patient has not failed the entire course of the treatment, is becoming of interest to clinical investigators. Based on either a multistate Markov model or a linear combination of Kaplan–Meier estimators, we explore regression models for the CLFS. We focus on the two sample problem and we develop confidence bands for the CLFS or for differences in CLFS as well as a Kolmogorov type hypothesis test using a re-sampling technique. We also examine the use of pseudo-values to make inference on the direct effects of covariates on the CLFS function and we develop a score test for the equality of two CLFS. We illustrate these inference methods on a bone marrow transplant dataset.     

Analysis of Variance for Bayesian Inference

This paper develops a multiway analysis of variance for non-Gaussian multivariate distributions and provides a practical simulation algorithm to estimate the corresponding components of variance. It specifically addresses variance in Bayesian predictive distributions, showing that it may be decomposed into the sum of extrinsic variance, arising from posterior uncertainty about parameters, and intrinsic variance, which would exist even if parameters were known. Depending on the application at hand, further decomposition of extrinsic or intrinsic variance (or both) may be useful. The paper shows how to produce simulation-consistent estimates of all of these components, and the method demands little additional effort or computing time beyond that already invested in the posterior simulator. It illustrates the methods using a dynamic stochastic general equilibrium model of the US economy, both before and during the global financial crisis.     

A Graphical Representation for Non-Parametric Inference

A method is described for determining the sample size required for a specified precision simultaneous confidence statement about the parameters of a multinomial population. The method is based on a simultaneous confidence interval procedure due to Goodman, and the results are compared with those obtained by separately considering each cell of the multinomial population as a binomial.     

Approximate Bayesian Inference for Survival Models

Abstract. Bayesian analysis of time‐to‐event data, usually called survival analysis, has received increasing attention in the last years. In Cox‐type models it allows to use information from the full likelihood instead of from a partial likelihood, so that the baseline hazard function and the model parameters can be jointly estimated. In general, Bayesian methods permit a full and exact posterior inference for any parameter or predictive quantity of interest. On the other side, Bayesian inference often relies on Markov chain Monte Carlo (MCMC) techniques which, from the user point of view, may appear slow at delivering answers. In this article, we show how a new inferential tool named integrated nested Laplace approximations can be adapted and applied to many survival models making Bayesian analysis both fast and accurate without having to rely on MCMC‐based inference.     

Inference for ergodic diffusions plus noise

We research an adaptive maximum‐likelihood–type estimation for an ergodic diffusion process where the observation is contaminated by noise. This methodology leads to the asymptotic independence of the estimators for the variance of observation noise, the diffusion parameter, and the drift one of the latent diffusion process. Moreover, it can lessen the computational burden compared to simultaneous maximum likelihood–type estimation. In addition to adaptive estimation, we propose a test to see if noise exists or not and analyze real data as the example such that the data contain observation noise with statistical significance.     

Inference for multivariate normal hierarchical models

This paper provides a new method and algorithm for making inferences about the parameters of a two-level multivariate normal hierarchical model. One has observed J p -dimensional vector outcomes, distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices. At level 2, the unknown mean vectors also have normal distributions, with common unknown covariance matrix A and with means depending on known covariates and on unknown regression coefficients. The algorithm samples independently from the marginal posterior distribution of A by using rejection procedures. Functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of A drawn. This estimation accounts for the uncertainty in A , unlike standard restricted maximum likelihood empirical Bayes procedures. It is based on independent draws from the exact posterior distributions, unlike Gibbs sampling. The procedure is demonstrated for profiling hospitals based on patients' responses concerning p =2 types of problems (non-surgical and surgical). The frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting.