 ). For a specific *, * (1: ) can be calculated from * (1: * ) = arg Taking out the constant terms from the arg function, we have * (1: * ) = arg Therefore, the optimal mean shift is calculated from * (1: * ) = * (1: * ) − (49) * (1: ) for each * = 1, …, will be calculated offline. We will use these * in the stochastic process during the evaluation.

2) Randomization of Termination Time
The termination time * is fixed in the previous section. However, ℰ (e.g. crash) may happen at any moment before time step. * needs to be randomized to reflect all the possible scenarios.
We first set the boundary of *. When * is small, it may fail to find a feasible solution to the optimization problem in Eq. (43). This is because no physically possible HV motion will lead to a crash when the test duration is very short. Let * be the minimum value of * that has a feasible solution in Eq. (43) under physical constraints. * = min * such that ∃ * (1: * − 1) that makes * * (1: * − 1) ≤ * (50) * is then randomized using a uniform distribution * ~ * ( * ) = 1 − * + 1 (51) where * = *, * + 1, …, − 1,. It should be noted that * is different from.
is the real termination time in the simulation, while * is used to randomize termination time in calculating the optimal shift * (1: * − 1).

3) Conducting the Accelerated Tests
The car-following simulation is executed with the lead HV following the modified distribution.

4) Calculation of Safety Benefit in the Real World
The real world safety benefit is calculated based on the Importance Sampling (IS) theory. By modifying the HV statistics from Eq. (17) to Eq. (18), more intense HV maneuvers are generated. Using a different distribution, however, leads to biased samples, and the key of IS is to compensate for this bias and to compute the safety performance of AVs in the real world. IS has been successfully applied to analyze critical events in reliability [50], finance [51], insurance [52], and telecommunication networks [53]. General reviews of IS can be found in [54]- [56]. To the best of our knowledge, no prior research has applied IS to AV evaluation in the car-following scenario.
We describe the core mechanism of IS as follows. First, the likelihood ratio L is defined The overall IS estimator of γ in tests is then calculated from The accuracy of the estimation is represented by the relative half-width, which is the half-width of the confidence interval relative to the probability to be estimated. With the Confidence Level at 100(1 − ) %, the relative half-width of ̂ is defined as, the half-width, is defined as where (̂1 : ) represents the standard deviation of [̂1,̂2, …,̂]. is defined by where Φ −1 is the inverse cumulative distribution function of (0,1). The Central Limit Theorem [57] implies that, as more tests are conducted, ̂ will converge to and will reduce to zero. The convergence is reached as is smaller than a constant threshold. The procedure of the accelerated test is summarized in Fig. 6.
After each test record indicator function ℰ and the termination time from Eq. (33) and Eq. (36). v) Calculate the likelihood of each test from Eq. (53). vi) Calculate the ̂ from Eq. (58). vii) Calculate the relative half-width in Eq. (59). viii) If <, output ̂. Otherwise, go back to step iii). The uniform distribution was used as the benchmark for modified distribution in the accelerated tests. The behavior of the HV was modified to be more aggressive, but the evaluation process may or may not accelerate. Three metrics, crash, injury, and conflict rates, were calculated. In each case, both accelerated and naturalistic driving simulations (non-accelerated, based on Monte Carlo) were conducted to examine the accuracy and the accelerated rates of the proposed method.

A. Simulation Results with uniform distribution
The uniform AE distribution is used as the baseline for comparison. The HV acceleration is generated from where represents the uniform distribution and parameter is chosen to be 6. The likelihood ratio of the uniform distribution method can be calculated from The model parameters used in the simulations are listed in TABLE II. An example simulation run is shown in Fig. 7. Although the lead HV acceleration oscillates up and down, overall the lead vehicle motion is not hard to follow. Therefore, crash rarely happens.
The estimated crash rate is plotted in Fig. 8. A million simulations were conducted, but because the convergence is very weak, not much can be said about the crash rate of the AV. The uniform distribution did not effectively accelerate crash evaluation because it did not consider the correlations between samples in a dynamic system.

B. Results of the Proposed Accelerated Evaluation Process
We applied the proposed Accelerated Evaluation approach on three types of events: crash, injury, and conflict events.  Fig. 9. In the accelerated simulations, the vehicle tends to accelerate and then decelerate at significant magnitudes. In the naturalistic simulations, the deceleration is much milder. Fig.  10 shows an example in an accelerated simulation that leads to a crash. The crash is caused by frequent acceleration and deceleration, and in this case, a "final blow" is a severe braking from a high speed. When the lead HV accelerates, it creates a larger. The AV then accelerates to catch up. If the HV harshly brakes at the moment when AV overshoots, it may lead to a crash. This tactic is frequently observed in accelerated simulations, but not in the current Euro-NCAP test protocols or ISO standard tests. In other words, the proposed accelerated evaluation method generates high risk maneuvers, some of which could be considered in future government certification process. The crash and conflict events are binary events with indication function defined in Eq. (33). When a crash happens injury may or may not occur. We model the injury event as a probabilistic function. Here we focus on injuries with the Maximum Abbreviated Injury Score equal or larger than 2 (MAIS2+), representing moderate-to-fatal injuries. The probability of injury is directly related to the relative velocity The probability of moderate-to-fatal injuries for the AV passengers is estimated by a nonlinear model which was proposed by Kusano and Gabler [58] with 0 = −6.068, 1 = 0.1, and 2 = −0.6234. The injury rate (P inj (Δ )) is calculated from E (P inj (Δ )) = Ê * (P inj (Δ )) where ( ) is the likelihood in the n th simulation. The estimation of crash, injury, and conflict rate are shown in Fig. 11. It is shown that the estimated crash rates calculated in the accelerated tests converge to those in the naturalistic driving tests, which demonstrates that the Accelerated Evaluation is unbiased.

Crash Injury Conflict
Estimation a) Estimation of the crash rate b) Estimation of the injury rate c) Estimation of the conflict rate Convergence rate d) Convergence of crash rate estimation e) Convergence of injury rate estimation f) Convergence of conflict rate estimation Fig. 11. Estimation and convergence rate of crash, injury, and conflict events Both accelerated and naturalistic simulations were conducted until the crash, injury, conflict rates converged with 80 % confidence level and = 0.2. The speed of convergences are shown in d), e), and f) in Fig. 11. The accelerated tests converged after simulations, while the naturalistic method needed simulations. TABLE III summarizes the performance of the Accelerated Evaluation in estimating the three metrics of the AV. The accelerated rate is defined as /. In the crash and injury cases, the proposed method accelerates the evaluation by five orders of magnitude. In the conflict case, the acceleration rate is lower at three hundred times. We think this is because the Importance Sampling theory provides larger accelerated rate when the target events are rarer.

V. CONCLUSION
An accelerated evaluation method is proposed to assess the safety performance of Automated Vehicles (AV) in the car-following scenario. The statistics of the lead Human Controlled Vehicle (HV) motion is modified so that the interactions between the HV and AV were significantly intensified. A stochastic optimization method was used to minimize the evaluation duration. Importance Sampling theory is then used to estimate the safety performance of AV in the naturalistic driving condition using test results in the accelerated tests.
Simulation results show that the accelerated tests can reduce the evaluation time of crash, injury, or conflict events between 300 and 100,000 times. In other words, simulating for 1,000 miles can expose the AV with challenging scenarios that take 300 thousand to 100 million miles in the real-world to encounter. This technique thus has the potential to dramatically reduce the development and validation time of AVs and can be used by both the government in AV assessment and auto companies to improve their products.Text: the somali government saturday described as `` barbaric '' the desecration of an italian cemetery in the bullet-scarred capital of mogadishu, saying it had sent an army general to negotiate the withdrawal of gunmen from the area.

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