This study introduces a new discrete distribution which is a weighted version of Poisson-Lindley distribution. The weighted distribution is obtained using the negative binomial weight function and can be fitted to count data with over-dispersion. The p.m.f., p.g.f. and simulation procedure of the new weighted distribution, namely weighted negative binomial Poisson-Lindley (WNBPL), are provided. The maximum likelihood method for parameter estimation is also presented. The WNBPL distribution is fitted to several insurance datasets, and is compared to the Poisson and negative binomial distributions in terms of several statistical tests.

In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve their fit. In this paper, we explore a further extension of the quasi-Lindley model. Maximum likelihood, least square error, Anderson-Darling algorithm, and expectation-maximization algorithm are four techniques for estimating the parameters under study. All techniques provide accurate and reliable estimates of the parameters. However, the mean square error of the expectation maximization approach was lower. The usefulness of the proposed model was demonstrated by analyzing a dynamical systems data set, and the analysis shows that it outperforms the other models in all statistical models considered.

In this paper, under the symmetric entropy and the scale squared error loss functions, we consider the maximum likelihood (ML) estimation and Bayesian estimation of the Shannon entropy and Rényi entropy of the two-parameter inverse Weibull distribution. In the ML estimation, the dichotomy is used to solve likelihood equation. In addition, the approximation confidence interval is given by the Delta method. Because the form of estimation results is more complex in the Bayesian estimation, the Lindley approximation method is used to achieve the numerical calculation. Finally, Monte Carlo simulations and a real data set are used to illustrate the results derived. By comparing the mean square error between the estimated value and the real value, it can be found that the performance of ML estimation of Shannon entropy is better than that of Bayesian estimation, and there is no significant difference between the performance of ML estimation of Rényi entropy and that of Bayesian estimation.

Torenia Concolor Lindley var. formosama Yamazaki ethanolic extract (TCEE) is reported to have anti-inflammatory and anti-obesity properties. However, the effects of TCEE and its underlying mechanisms in the activation of endothelial nitric oxide synthase (eNOS) have not yet been investigated. Increasing the endothelium-derived nitric oxide (NO) production has been known to be beneficial against the development of cardiovascular diseases. In this study, we investigated the effect of TCEE on eNOS activation and NO-related endothelial function and inflammation by using in vitro system. In endothelial cells (ECs), TCEE increased NO production in a concentration-dependent a manner without affecting the expression of eNOS. In addition, TCEE increased the phosphorylation of eNOS at serine 635 residue (Ser635) and Ser1179, Akt at Ser473, calmodulin kinase II (CaMKII) at threonine residue 286 (Thr286) and AMP-activated protein kinase (AMPK) at Thr172. Moreover, TCEE-induced NO production, and EC proliferation, migration and tube formation were diminished by pretreatment with LY294002 (an Akt inhibitor), KN62 (a CaMKII inhibitor) and compound C (an AMPK inhibitor). Additionally, TCEE attenuated the tumor necrosis factor-α-induced inflammatory response as evidenced by the expression of adhesion molecules in ECs and monocyte adhesion onto ECs. These inflammatory effects of TCEE were abolished by L-NG-nitroarginine methyl ester (a NOS inhibitor). Collectively, our findings suggest that TCEE confers protection from endothelial dysfunction by activating the Akt/CaMKII/AMPK/eNOS/NO signaling pathway.