Mathematical and Statistical Properties for a New IDNAI Distribution[a]

<ns3:p>Over the past decade, the statistical and reliability literature has seen an enormous rise in the number of various probability distributions. Many of them are designed to integrate, enhance, or add to traditional models. This rise shows that people are becoming more aware of empirical integrity, flexibility, and adaptability, especially when it comes to modeling income distributions, dependability difficulties, and longevity statistics. In this paper, more general and flexible distributions are provided by employing a technique that generates the T−R {Y} family distributions as well as a brief overview for this kind of family. This technique may be applied to agree with certain data distribution types, including very left, right, thin or heavy tailed distribution, or to develop new distributions that are very applicable and flexible. In this context, produces a distinct member of a new sub-family called the IDNAI distribution, which is referred to the Lomax–Rayleigh {Exponential} distribution. The most important statistical and mathematical characteristics of the new family distribution are discussed. In order to estimate the parameters of the new class distribution, two estimation techniques will be additionally presented: the maximum likelihood method (MLE) and the least square method (LS). In order to analyze the effectiveness of suggested estimation techniques, simulation tools have been constructed. The results of the comparison indicated that the maximum likelihood method performed better than the least squares approach in terms of bias and mean squared error criteria. However, the application of a real dataset showed that, depending on several kinds of information criteria, the new class distribution fit the data more effectively than the specific classical and moderate distributions.</ns3:p>