Estimating herd-specific force of infection by using random-effects models for clustered binary data and monotone fractional polynomials

Summary.  In veterinary epidemiology, we are often confronted with hierarchical or clustered data. Typically animals are grouped within herds, and consequently we cannot ignore the possibility of animals within herds being more alike than between herds. Based on a serological survey of bovine herpes virus type 1 in cattle, we describe a method for the estimation of herd-specific rates at which susceptible animals acquire the infection at different ages. In contrast with the population-averaged force of infection, this method allows us to model the herd-specific force of infection, allowing investigation of the variability between herds. A random-effects approach is used to account for the correlation in the data, allowing us to study both population-averaged and herd-specific force of infection. In contrast, generalized estimating equations can be used when interest is only in the population-averaged force of infection. Further, a flexible predictor model is needed to describe the dependence of covariates appropriately. Fractional polynomials as proposed by Royston and Altman offer such flexibility. However, the flexibility of this model should be restricted, since only positive forces of infection have a meaningful interpretation.     

Bias reduction for high quantiles

High quantile estimation is of importance in risk management. For a heavy-tailed distribution, estimating a high quantile is done via estimating the tail index. Reducing the bias in a tail index estimator can be achieved by using either the same order or a larger order of number of the upper order statistics in comparison with the theoretical optimal one in the classical tail index estimator. For the second approach, one can either estimate all parameters simultaneously or estimate the first and second order parameters separately. Recently, the first method and the second method via external estimators for the second order parameter have been applied to reduce the bias in high quantile estimation. Theoretically, the second method obviously gives rise to a smaller order of asymptotic mean squared error than the first one. In this paper we study the second method with simultaneous estimation of all parameters for reducing bias in high quantile estimation.     

中国系统性金融风险对宏观经济的影响研究

基于2007年1月至2017年12月月度数据,本文首先选取金融机构极值风险、金融体系间的传染效应、金融市场的波动性和不稳定性、流动性和信用风险4个层面的14个代表性指标测度了系统性金融风险;然后运用分位数回归度量了单个系统性风险指标对宏观经济的影响;最后运用偏最小二乘分位数回归法构建一个系统性金融风险综合指标进一步实证分析系统性金融风险对宏观经济的影响。研究结果表明：①单个系统性金融风险指数中机构极值风险类别下的指标对宏观经济的影响最大,其中金融体系巨灾风险指数影响效果最显著;②运用偏最小二乘分位数回归构造的系统性金融风险综合指标较之单个系统性金融风险指标,能够更稳健地反映系统性金融风险对宏观经济的影响状况;③从测度效果来看,单个系统性风险指标和系统性金融风险综合指标在下尾分布（0.2分位数）的结果明显优于中间分布（0.5分位数）和上尾分布（0.8分位数）。     

基于分位回归的风险保费预测

风险保费预测是非寿险费率厘定的重要组成部分。在传统的分位回归厘定风险保费中,通常假设分位数水平是事先给定的,缺乏一定的客观性。为此,提出了一种应用分位回归厘定风险保费的新方法。基于破产概率确定保单组合的总风险保费,建立个体保单的分位回归模型,并与总风险保费建立等式关系,通过数值方法求解出分位数水平,实现对个体保单风险保费的预测。通过一组实际数据分析表明,该方法具有良好的预测效果。     

Confidence curves and improved exact confidence intervals for discrete distributions

The author describes a method for improving standard “exact” confidence intervals in discrete distributions with respect to size while retaining correct level. The binomial, negative binomial, hypergeometric, and Poisson distributions are considered explicitly. Contrary to other existing methods, the author's solution possesses a natural nesting condition: if α < α', the 1 ‐ α' confidence interval is included in the 1 ‐ α interval. Nonparametric confidence intervals for a quantile are also considered.     

Attribute Charts for Zero-Inflated Processes

The classical Shewhart c-chart and p-chart which are constructed based on the Poisson and binomial distributions are inappropriate in monitoring zero-inflated counts. They tend to underestimate the dispersion of zero-inflated counts and subsequently lead to higher false alarm rate in detecting out-of-control signals. Another drawback of these charts is that their 3-sigma control limits, evaluated based on the asymptotic normality assumption of the attribute counts, have a systematic negative bias in their coverage probability. We recommend that the zero-inflated models which account for the excess number of zeros should first be fitted to the zero-inflated Poisson and binomial counts. The Poisson parameter λ estimated from a zero-inflated Poisson model is then used to construct a one-sided c-chart with its upper control limit constructed based on the Jeffreys prior interval that provides good coverage probability for λ. Similarly, the binomial parameter p estimated from a zero-inflated binomial model is used to construct a one-sided np-chart with its upper control limit constructed based on the Jeffreys prior interval or Blyth–Still interval of the binomial proportion p. A simple two-of-two control rule is also recommended to improve further on the performance of these two proposed charts.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Solving For the parameters of a beta a distribution under two quantile constraints

It will be shown that a solution exists for the parameters of a beta distribution given any combination of a lower quantile and upper quantile constraint. A numerical procedure is developed to solve for the parameters of the beta distribution given these quantile constraints. Example solutions are provided.     

Improving extreme quantile estimation via a folding procedure

In many applications (geosciences, insurance, etc.), the peaks-over-thresholds (POT) approach is one of the most widely used methodology for extreme quantile inference. It mainly consists of approximating the distribution of exceedances above a high threshold by a generalized Pareto distribution (GPD). The number of exceedances which is used in the POT inference is often quite small and this leads typically to a high volatility of the estimates. Inspired by perfect sampling techniques used in simulation studies, we define a folding procedure that connects the lower and upper parts of a distribution. A new extreme quantile estimator motivated by this theoretical folding scheme is proposed and studied. Although the asymptotic behaviour of our new estimate is the same as the classical (non-folded) one, our folding procedure reduces significantly the mean squared error of the extreme quantile estimates for small and moderate samples. This is illustrated in the simulation study. We also apply our method to an insurance dataset.     

Quantile regression in longitudinal studies with dropouts and measurement errors

ABSTRACT

Quantile regression models, as an important tool in practice, can describe effects of risk factors on the entire conditional distribution of the response variable with its estimates robust to outliers. However, there is few discussion on quantile regression for longitudinal data with both missing responses and measurement errors, which are commonly seen in practice. We develop a weighted and bias-corrected quantile loss function for the quantile regression with longitudinal data, which allows both missingness and measurement errors. Additionally, we establish the asymptotic properties of the proposed estimator. Simulation studies demonstrate the expected performance in correcting the bias resulted from missingness and measurement errors. Finally, we investigate the Lifestyle Education for Activity and Nutrition study and confirm the effective of intervention in producing weight loss after nine month at the high quantile.     

Variance reduction for quantile estimates in simulations via nonlinear controls

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relation-ship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtain an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control     

Production Frontiers and Panel Data

Value at risk (VaR) is the standard measure of market risk used by financial institutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, the conditional autoregressive value at risk (CAViaR) model specifies the evolution of the quantile over time using an autoregressive process and estimates the parameters with regression quantiles. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the dynamic quantile test. Applications to real data provide empirical support to this methodology.     

Low dose risk estimation via simultaneous statistical inferences

Summary.  The paper develops and studies simultaneous confidence bounds that are useful for making low dose inferences in quantitative risk analysis. Application is intended for risk assessment studies where human, animal or ecological data are used to set safe low dose levels of a toxic agent, but where study information is limited to high dose levels of the agent. Methods are derived for estimating simultaneous, one-sided, upper confidence limits on risk for end points measured on a continuous scale. From the simultaneous confidence bounds, lower confidence limits on the dose that is associated with a particular risk (often referred to as a bench-mark dose ) are calculated. An important feature of the simultaneous construction is that any inferences that are based on inverting the simultaneous confidence bounds apply automatically to inverse bounds on the bench-mark dose.     

A Projection-Based Nonparametric Test of Conditional Quantile Independence

Abstract

This paper proposes a nonparametric procedure for testing conditional quantile independence using projections. Relative to existing smoothed nonparametric tests, the resulting test statistic: (i) detects the high frequency local alternatives that converge to the null hypothesis in probability at faster rate and, (ii) yields improvements in the finite sample power when a large number of variables are included under the alternative. In addition, it allows the researcher to include qualitative information and, if desired, direct the test against specific subsets of alternatives without imposing any functional form on them. We use the weighted Nadaraya-Watson (WNW) estimator of the conditional quantile function avoiding the boundary problems in estimation and testing and prove weak uniform consistency (with rate) of the WNW estimator for absolutely regular processes. The procedure is applied to a study of risk spillovers among the banks. We show that the methodology generalizes some of the recently proposed measures of systemic risk and we use the quantile framework to assess the intensity of risk spillovers among individual financial institutions.     

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number k _n of largest order statistics can be represented as a statistical tail function al, that is a functional T applied to the empirical tail quantile function Q _n. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F . For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T ( Q _n) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k _pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour     

EWMA control charts for monitoring correlated counts with finite range

In this work, we develop and study upper and lower one-sided EWMA control charts for monitoring correlated counts with finite range. Often in practice, data of that kind can be adequately described by a first-order binomial or beta-binomial autoregressive model. Especially, when there is evidence that data demonstrate extra-binomial variation, the latter model is preferable than the former. The proposed charts can be used for detecting upward or downward shifts in process mean level. Practical guidelines concerning the statistical design of the proposed charts are given, while the effect of the extra-binomial variation is investigated as well. Comparisons with existing control charting procedures are also provided. Finally, an illustrative real-data example is also given.     

Bias-reduced extreme quantile estimators of Weibull tail-distributions

In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. [2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.     

Design of accelerated life test plans under progressive Type II interval censoring with random removals

This paper investigates the design of accelerated life test (ALT) plans under progressive Type II interval censoring with random removals. Units’ lifetimes are assumed to follow a Weibull distribution, and the number of random removals at each inspection is assumed to follow a binomial distribution. The optimal ALT plans, which minimize the asymptotic variance of an estimated quantile at use condition, are determined. The expected duration of the test and the expected number of inspections on each stress level are calculated. A numerical study is conducted to investigate the properties of the derived ALT plans under different parameter values. For illustration purpose, a numerical example is also given.     

Joint modeling of correlated binary outcomes: HIV-1 and HSV-2 co-infection

Herpes Simplex Virus Type 2 (HSV-2) facilitates the sexual acquisition and transmission of HIV-1 infection and is highly prevalent in most regions experiencing severe HIV epidemics. In sub-Saharan Africa, where HIV infection is a public health burden, the prevalence of HSV-2 is substantially high. The high prevalence of HSV-2 and the association between HSV-2 infection and HIV-1 acquisition could play a significant role in the spread of HIV-1 in the region. The objective of our study was to identify risk factors for HSV-2 and HIV-1 infections among men in sub-Saharan Africa. We used a joint response model that accommodates the interdependence between the two infections in assessing their risk factors. Simulation studies show superiority of the joint response model compared to the traditional models which ignore the dependence between the two infections. We found higher odds of having HSV-2/HIV-1 among older men, in men who had multiple sexual partners, abused alcohol, or reported symptoms of sexually transmitted infections. These findings suggest that interventions that identify and control the risk factors of the two infections should be part of HIV-1 prevention programs in sub-Saharan Africa where antiretroviral therapy is not readily available.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.