Research And Applications Of Skewed And Discrete Normal Distributions

With the advent of big data era,more and more data have serious skewness and even multimodality,traditional symmetrical distribution such as normal distribution can not be described appropriately,it is necessary to find more reasonable skew distributions to fit different types of data.In addition,it is difficult to obtain samples from continuous distribution,many observed life data are discrete in reliability research,and the diversity of discrete life data makes some classical discrete distributions such as binomial distribution and Poisson distribution not reach the expected fitting effect.Therefore,in order to better describe the data of statistical distributions,it is necessary to obtain some new probability distributions by means of skewed and discrete methods.In this paper,we choose normal distribution as the basic distribution of skewed and discrete,and propose two new kinds of skew normal distributions for the first time,which are alpha-beta-gamma skew normal distribution and alpha skew power normal distribution.A new discrete alpha skew normal distribution is obtained for the first time by the rounding discrete method.The statistical properties and parameter estimation of these three new distributions are studied in detail.The main work of this paper is divided into three parts.In the first part and the second part,the alpha-beta-gamma skew normal distribution and the alpha skew power normal distribution are proposed respectively,and their probability density functions are given.The moment,skewness,kurtosis,maximum likelihood estimator,random number generation algorithm,Fisher information matrix and some theorems of the distribution are obtained.In addition,two random simulations of parameter maximum likelihood estimation are considered,the simulation results show that the parameter estimator has good asymptotic statistical properties.Furthermore,the two new distributions are fitted to the same set of real data,and the maximum likelihood estimators of the parameters,the logarithmic likelihood values of the samples,the AIC and BIC values of each distribution are calculated.At the same time,the Kolmogrov-Smirnov test between samples and distributions are considered.The results of empirical analysis show that,compared with several existing skew distributions such as skew normal distribution,alpha skew normal distribution and alpha-beta skew normal distribution,the two new types of skew normal distribution proposed in this paper are more flexible.In addition,under AIC and BIC criteria,the fitting effect of alpha-beta-gamma skew normal distribution is better than alpha skew power normal distribution.In the third part,the related properties and parameter estimation of discrete alpha skew normal distribution are studied,including cumulative distribution function,moment,survival function,failure rate function,maximum likelihood estimation,random number generation and random simulation.Finally,the distribution is applied to a group of discrete life data,the simulation results show that the fitting effect of discrete alpha skew normal distribution is better than that of alpha skew normal distribution and discrete normal distribution.