The weighted distributions are widely used in many fields such as medicine, ecology and reliability, to name a few, for the development of proper statistical models. Weighted distributions are milestone for efficient modeling of statistical data and prediction when the standard distributions are not appropriate. A good deal of studies related to the weight distributions have been published in the literature. In this article, a brief review of these distributions is carried out. Implications of the differing weight models for future research as well as some possible strategies are discussed. Finally, characterizations of these distributions based on a simple relationship between two truncated moments are presented.

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Weighted Distributions: A Brief Review, Perspective and Characterizations

Overpopulation is a huge problem for the people of sub-Sahara African countries, including Nigeria. In order to tackle this problem, demographers have in the last five decades made immense efforts towards understanding the causes and mechanisms of fertility, and to explain the variations inherent in it. Several theoretical approaches that have been proposed mainly focus on cumulative or completed fertility as the main dependent variable. Moreover, it is equally important to examine variations in fertility in such entities such as timing and spacing of births. Data on birth interval offers rich and more detailed information for the analysis of reproductive behavior than does the data on number of births. And more importantly, timing of first birth after marriage which also corresponds to first-birth-interval (FBI) is strongly correlated to the number of children a woman would have by the end of her reproductive life. In this research, FBI is modeled stochastically as a survival response variable. Suitability of a class of some standard parametric models is investigated and the study comes up with a novel three-parameter inverse Gaussian model through the exponentiation of probability distribution approach. Estimations of parameters are obtained by maximizing the log-likelihood functions of the data in R software programming. Different approaches employed to discriminate between the distribution and the existing ones considered, suggests that the new model, Lehmann type II inverse Gaussian, provides a better fit for the FBI and other lifetime data. Akaike Information Criterion (AIC) which is a resemblance of coefficient of determination in linear regression, also shows a significant reduction in value as covariates are introduced into it in comparison with regressions from other distributions. This research finds that age of women at marriage, education attainment and region where women reside are significant prognostic factors that influence the chance of first birth after marriage. The thesis concludes by calling on agencies and organizations working on population control, especially in Nigeria, to initiate programs that will encourage girl-child education to higher level.

Survival Analysis of Time-to-First birth after marriage in Nigeria using three parameter inverse gaussian distribution

Length-Biased distributions are a special case of the more general form known as weighted distributions. We can exploit the conceptuality of Length-Biased distribution in the development of appropriate models for lifetime data. Its method is adjusting the original probability density function from real data and the expectation of those data. This modification can lead to correct conclusions of the models. Therefore, we introduced the Length-Biased version of the weighted Exponentiated inverted Weibull distribution in this paper. Various properties and the expressions for moments, coefficient of skewness, coefficient of kurtosis, moment generating function, hazard rate function, etc. are derived. The maximum likelihood estimates of the parameters of the proposed distribution are determined. The study results suggest that this distribution is an efficacious model in life time data analysis and other related fields.

The length-biased weighted exponentiated inverted Weibull distribution

In this paper, a potentiated lifetime model that demonstrates the bathtub-shaped hazard rate function is developed. This proposed model is referred to as Kumaraswamy modified size-biased Lehmann Type-II (Kum–MSBL–II) distribution. Various mathematical and reliability characteristics are derived and discussed. The maximum likelihood estimation method is used to estimate the model parameters and its performance is discussed by following a simulation study. Three-lifetime sets of data are utilized to illustrate the flexibility of the proposed model and the results demonstrate that the proposed model can adequately fit the real-life datasets than the competitors.

A New Generalization of Lehmann Type - II Distribution: Theory, Simulation, and Applications to Survival and Failure Rate data

Two main distributions are combining by using the logit of beta function by Jones [10]. The weighted Weibull distribution proposed by Shahbaz et al. [19] and beta distribution in order to have a better distribution (beta-weighted Weibull distribution) than each of them individually in terms of the estimate of their characteristics in their parameters. We study and provide a comprehensive treatment of the mathematical properties of the beta weighted Weibull distribution and derive expressions for its moments and moment generating function, survival rate function, hazard rate function, skewness and kurtosis, coefficient of variation and asymptotic behaviours. We also discuss maximum likelihood estimation and provide formulae for the elements of the Fisher information matrix. The new distribution is apply to a lifetime data set and clearly shows that it is much more flexible and has a better representation

Life length of components estimates with beta-weighted weibull distribution

We upgrade on the method used on weighted Weibull model proposed by Azzalini (1985) using the logit of Beta function by Jones (2004) to have Lehmann Type II weighted Weibull model with a view to obtaining a distribution that is more better than both weighted Weibull and Weibull distribution in terms of estimate of their characteristics and their parameter. The weighted Weibull distribution is proposed by slightly modifying the method of Azzalini (1985) on weighted distribution with additional shape  and scale  parameters. Some basic properties of the newly proposed distribution including moments and moment generating function, survival rate function, hazard rate function, asymptotic behaviours, and the estimation of parameters have been studied. 

	Key words: Asymptotic, hazard rate, Lehmann Type II, moments, weighted-Weibull.

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Lehmann Type II weighted Weibull distribution

A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time  has been investigated, in order to provide some insight into the performance measures in renewal process and sequence such as, the mean time between successive renewals, ; Laplace-Stiltjes transform (LST) of the mean time, ;  the Laplace-Stieltjes transform (LST) of the mean time distribution function,  ;  Laplace-Stiltjes transform (LST) of  fold convolution of distribution function, ;  the time at which the  renewal occurs,  the average number of renewals per unit time over the interval (0, t],   and expected reward, . Our quest is to analyse the distribution function of the renewal process and sequence   using the concept of discrete time Markov chain to obtain the aforementioned performance measures. Some properties of the Erlang , exponential and geometric distributions are used with the help of some existing laws, theorems and formulas of Markov chain. We concluded our study through illustrative examples that, it is not possible for an infinite number of renewals to occur in a finite period of time; Also, the expected number of renewals increases linearly with time; and from the uniqueness property, we affirmed that, the Poisson process is the only renewal process with a linear mean-value  function; and lastly, we obtained the optimal replacement policy for a manufacturing machine which showed that, the exponential property of the lifetime 

https://68.169.56.45/index.php/fjs/article/download/785/615

N. I. Badmus

Papers

Lehmann Type II weighted Weibull distribution

Application of renewal reward processes in homogeneous discrete markov chain