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The multiplicative censoring model introduced in Vardi [Biometrika 76 (1989) 751-761] is an incomplete data problem whereby two independent samples from the lifetime distribution G, Xm = (X1, . . . , Xm) and Zn = (Z1, . . . , Zn), are observed subject to a form of coarsening. Specifically, sample Xm is fully observed while Yn = (Y1, . . . , Yn) is observed instead of Zn, where Yi = UiZi and (U1, . . . , Un) is an independent sample from the standard uniform distribution. Vardi [Biometrika 76 (1989) 751-761] showed that this model unifies several important statistical problems, such as the deconvolution of an exponential random variable, estimation under a decreasing density constraint and an estimation problem in renewal processes. In this paper, we establish the large-sample properties of kernel density estimators under the multiplicative censoring model. We first construct a strong approximation for the process √ k(Ĝ − G), whereĜ is a solution of the nonparametric score equation based on (Xm, Yn), and k = m + n is the total sample size. Using this strong approximation and a result on the global modulus of continuity, we establish conditions for the strong uniform consistency of kernel density estimators. We also make use of this strong approximation to study the weak convergence and integrated squared error properties of these estimators. We conclude by extending our results to the setting of length-biased sampling.

Objectives Every day, an average of 3,400 deaths and tens of millions of injuries occur as a result of traffic accidents. This study aims to model and validate road traffic injury (RTI) times series, specifically considering gender. Study design Time trend studies of monthly road traffic injuries (RTI) in Iran from March 2005 to February 2016, as well as those of males and females from March 2009 to February 2016 were performed. Methods The seasonal auto-regressive integrated moving average method (SARIMA) was employed to predict RTI time series. The final model was selected from various SARIMA models based on the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). To examine whether the residuals were white noise, the Ljung-Box (LB) test and residuals plots were used for un-correlation, and the zero mean and stationarity, respectively. Additionally, smoothing methods were utilized to validate the SARIMA models for fitting and out-of-range prediction of the time series models under study. The sample auto-correlation function (ACF) and the partial autocorrelation function (PACF) with 20 lags were employed to determine the order of models and to ascertain if the residuals of the model were uncorrelated. Results Based on the obtained results, SARIMA (2,1,0)(0,1,1) 12 , SARIMA (0,1,1)(0,1,1) 12 , SARIMA (1,1,1)(0,0,1) 12 , and SARIMA (2,0,0)(1,0,0) 12 were chosen for the time series including incidence rates of total road traffic injuries (IRTI), IRTI of males, females, and males-to-females, respectively. The AIC values were -87.57, 413.38, -732.91, and -85.32, respectively. The LB test for the residuals of the time series models of (0.539) IRTI, (0.3) IRTI of males, (0.23) females, and (0.237) males-to-females indicated that residuals were uncorrelated. Furthermore, prediction values for the next 24 months (2016 to 2018) showed no decline in the incidence rate of male and female traffic injuries. Results of the predictions using exponential smoothing methods indicated out-of-range prediction validity of the SARIMA models. Conclusion This study exemplified the high efficiency of SARIMA models in predicting road traffic injuries (RTIs). Based on observations, the IRTI mean in Iran was 35.57 in 2016. The predicted values of the IRTI for 2016–2018 by the SARIMA model showed no decreasing trend. During the studied period, the observed values of IRTI for males were two to three times the female values. Thus, prediction of RTI can provide a useful tool for traffic safety policymaking by simulating interrupted time series when applying new traffic enforcement interventions and regulations in the future. Additionally, IRTI analysis of males and females showed that men had a non-increasing trend but higher incidence of traffic injuries, whereas the IRTI for wom...

Background Road traffic law enforcement was implemented on 1 st April 2011 (the first intervention) and traffic ticket fines have been increased on 1 st March 2016 (the second intervention) in Iran. The aim of the current study was to evaluate the effects of the law enforcement on reduction in the incidence rate of road traffic fatality (IRRTF), the incidence rate of road traffic injuries (IRRTI) and the incidence rate of rural road traffic offenses (IRRRTO) in Iran. Methods Interrupted time series analysis was conducted to evaluate the impact of law enforcement and increased traffic tickets fines. Monthly data of fatality on urban, rural and local rural roads, injuries with respect to gender and traffic offenses namely speeding, illegal overtaking and tailgating were investigated separately for the period 2009-2016. Results Results showed a reduction in the incidence rate of total road traffic fatality (IRTRTF), the incidence rate of rural road traffic fatality (IRRRTF) and the incidence rate of urban road traffic fatality (IRURTF) by-21.44% (-39.3 to-3.59, 95% CI),-21.25% (-31.32 to-11.88, 95% CI) and-26.75% (-37.49 to-16, 95% CI) through the first intervention which resulted in 0.383, 0.255 and 0.222 decline in casualties per 100 000 population, respectively. Conversely, no reduction was found in the incidence rate of local rural road traffic fatality (IRLRRTF) and the IRRTI. Second intervention was found to only affect the IRURTF with-26.75% (-37.49 to-16, 95% CI) which led to 0.222 casualties per 100 000 population. In addition, a reduction effect was observed on the incidence rate of illegal overtaking (IRIO) and the incidence rate of speeding (IRS) with-42.8% (-57.39 to-28.22, 95% CI) and-10.54% (-21.05 to-0.03, 95% CI which implied a decrease of 415.85 and 1003.8 in monthly traffic offenses per 100 000 vehicles), respectively.

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