Bathtub distributions: a review

Bathtub distributions are characterized by bathtub failure rate functions . These are possibly more realisitic models than the monotone failure rate models . A systematic account of such distributions is not available and this review aims to give such an account . We give some easily verifiable conditions to check the bathtub property of a distribution along with methods to construct such distributions . We also discuss some stochastic and reliablity mechanisms which lead to bathtub distributions. These include mixtures ( stochastic failure rate models ) , series system , stochastic differential equation models and so on. We also review inference on bathtub distributions. The paper concludes with a rather exhaustive list of bathtub distributions.     

The periodogram regression:correction and comments

Ratios of periodogram and spectral density at different harmonic frequencies are independent if the frequency zero is approached slowly enough. This is an asymptotically relevant condition for the periodogram regression to work with fractionally integrated series. In finite samples, however, this theoretical condition should be ignored as we illustrate experimentally.     

Bayesian analysis of threshold autoregressions

A nonasymptotic Bayesian approach is developed for analysis of data from threshold autoregressive processes with two regimes. Using the conditional likelihood function, the marginal posterior distribution for each of the parameters is derived along with posterior means and variances. A test for linear functions of the autoregressive coefficients is presented. The approach presented uses a posterior p-value averaged over the values of the threshold. The one-step ahead predictive distribution is derived along with the predictive mean and variance. In addition, equivalent results are derived conditional upon a value of the threshold. A numerical example is presented to illustrate the approach.     

Effect of systematic sampling on arima models

Given a general homogeneous non-stationary autoregressive integrated moving average process ARIMA(p,d,q), the corresponding model for the subseries obtained by a systematic sampling is derived. The article then shows that the sampled subseries approaches approximately to an integrated moving average process IMA(d,l), l≤(d-l), regardless of the autoregressive and moving average structures in the original series. In particular, the sampled subseries from an ARIMA (p,l,q) process approaches approximately to a simple random walk model.     

Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Obtaining prediction intervals for FARIMA processes using the sieve bootstrap

The sieve bootstrap (SB) prediction intervals for invertible autoregressive moving average (ARMA) processes are constructed using resamples of residuals obtained by fitting a finite degree autoregressive approximation to the time series. The advantage of this approach is that it does not require the knowledge of the orders, p and q, associated with the ARMA(p, q) model. Up until recently, the application of this method has been limited to ARMA processes whose autoregressive polynomials do not have fractional unit roots. The authors, in a 2012 publication, introduced a version of the SB suitable for fractionally integrated autoregressive moving average (FARIMA (p,d,q)) processes with 0<d<0.5 and established its asymptotic validity. Herein, we study the finite sample properties this new method and compare its performance against an older method introduced by Bisaglia and Grigoletto in 2001. The sieve bootstrap (SB) method is a numerically simpler alternative to the older method which requires the estimation of p, d, and q at every bootstrap step. Monte-Carlo simulation studies, carried out under the assumption of normal, mixture of normals, and exponential distributions for the innovations, show near nominal coverages for short-term and long-term SB prediction intervals under most situations. In addition, the sieve bootstrap method yields better coverage and narrower intervals compared to the Bisaglia–Grigoletto method in some situations, especially when the error distribution is a mixture of normals.     

On Non-parametric Estimation of the Survival Function with Competing Risks

The non-parametric maximum likelihood estimators (MLEs) are derived for survival functions associated with individual risks or system components in a reliability framework. Lifetimes are observed for systems that contain one or more of those components. Analogous to a competing risks model, the system is assumed to fail upon the first instance of any component failure; i.e. the system is configured in series. For any given risk or component type, the asymptotic distribution is shown to depend explicitly on the unknown survival function of the other risks, as well as the censoring distribution. Survival functions with increasing failure rate are investigated as a special case. The order restricted MLE is shown to be consistent under mild assumptions of the underlying component lifetime distributions.     

Multivariate regression analysis of panel data with binary outcomes applied to unemployment data

Summary  In panel studies binary outcome measures together with time stationary and time varying explanatory variables are collected over time on the same individual. Therefore, a regression analysis for this type of data must allow for the correlation among the outcomes of an individual. The multivariate probit model of Ashford and Sowden (1970) was the first regression model for multivariate binary responses. However, a likelihood analysis of the multivariate probit model with general correlation structure for higher dimensions is intractable due to the maximization over high dimensional integrals thus severely restricting ist applicability so far. Czado (1996) developed a Markov Chain Monte Carlo (MCMC) algorithm to overcome this difficulty. In this paper we present an application of this algorithm to unemployment data from the Panel Study of Income Dynamics involving 11 waves of the panel study. In addition we adapt Bayesian model checking techniques based on the posterior predictive distribution (see for example Gelman et al. (1996)) for the multivariate probit model. These help to identify mean and correlation specification which fit the data well. C. Czado was supported by research grant OGP0089858 of the Natural Sciences and Engineering Research Council of Canada.     

Inference for Time Series Regression Models With Weakly Dependent and Heteroscedastic Errors

Motivated by the need to assess the significance of the trend in some macroeconomic series, this article considers inference of a parameter in parametric trend functions when the errors exhibit certain degrees of nonstationarity with changing unconditional variances. We adopt the recently developed self-normalized approach to avoid the difficulty involved in the estimation of the asymptotic variance of the ordinary least-square estimator. The limiting distribution of the self-normalized quantity is nonpivotal but can be consistently approximated by using the wild bootstrap, which is not consistent in general without studentization. Numerical simulation demonstrates favorable coverage properties of the proposed method in comparison with alternative ones. The U.S. nominal wages series is analyzed to illustrate the finite sample performance. Some technical details are included in the online supplemental material.