Length biased Gegenbauer distribution and its applications in accident statistics

This study proposes a length biased version of the Gegenbauer distribution, of which the length biased negative binomial distribution is a special case and highlights the usefulness of the proposed model through modelling certain road accident data sets. We derive several fundamental properties of the distribution, including its probability generating function, cumulative distribution function, survival function and hazard rate function, as well as formulae for both factorial and raw moments. Recurrence relations for the probabilities are also obtained. Various methods for estimating the model parameters are discussed, with particular emphasis on maximum likelihood estimation. Further, certain test procedures are developed for examining the significance of the parameters and conducted a brief simulation study to assess the efficiency of the maximum likelihood estimators.