In fact, it provides a better fit to these data than the other competitive models. The applicability of the truncated Lomax inverse Lomax model is illustrated through two real lifetime data sets and its goodness-of-fit is compared with that of the recent models. Simulation outcomes show that there is a great agreement between theoretical and empirical studies. Plus, it is a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. A simulation study framework is established to assess the accuracy of estimates through some measures. Lomax distribution is a special case of q-exponential distribution, generalized Pareto distribution, beta prime distribution and F distribution. Also, the approximate confidence interval of parameters is constructed. Maximum likelihood estimators of the population parameters are derived. The new distribution is capable of monotonically increasing, decreasing, reversed J-shaped and upside-down shaped hazard rates. Expansions for quantile function, moment generating function, probability weighted moments, ordinary moments, incomplete moments, inverse moments, conditional moments, and Rényi entropy measure are investigated. The density of the new model can be represented as a linear combination of the inverse Lomax distribution. power Lomax distribution with application to manufacturing Amjad D. The new modified distribution is called the truncated Lomax inverse Lomax distribution. The new three-parameter model is provided as a member of the truncated Lomax-G procedure. A practical procedure of estimating safety is proposed that identifies the longest suitable threshold separation for each case based on the trends in the estimation results.This paper introduces a modified form of the inverse Lomax distribution which offers more flexibility for modeling lifetime data. The most important outcome of the presented study is confirmation that traffic conflicts claimed based on sufficiently small threshold separation (such as Time to Collision) allow unbiased estimation of the expected number of crashes during the conflicts observation period. A new three-parameter discrete distribution has been improved named as a Discrete MarshallOlkin Lomax (DMOL) distribution. Unlike the benchmark methods, the proposed method produces real estimates in each case. The SPE method is found more accurate and efficient than the other two methods. The performance of the MLE, PWM, and SPE methods are compared. Then, a new Single Parameter Estimation (SPE) method is proposed and evaluated with extensive Monte Carlo experiments. The existing Maximum Likelihood Estimate (MLE) method and the Probability-Weighted Moments (PWM) method of estimating the probability of crash and the expected number of crashes based on the proposed theory are presented. Harris used Lomax distribution for income and wealth data.
The published concepts and study results that support the derived model are provided in the paper. Lomax suggested an important model for lifetime analysis, known as Lomax or Pareto Type II.It is a heavy-tail probability distribution often used in business, economics, and actuarial modeling. Then, a model applicable to heterogeneous conditions is derived and the model’s relevance, useful properties, and limitations are discussed. The pre-crash process leading to a conflict or a crash as the result of a failure is discussed as this conceptualization is the basis for proposing a simple model of the probability of a crash at the moment when a conflict is still progressing.
This paper justifies the Lomax distribution for counterfactual modeling of the probability of crash given a traffic conflict.