Due to the advance computer technology, the use of probability distributions has been raised up to solve the real life problems. These applications are found in reliability engineering, computer sciences, economics, psychology, survival analysis, and some others. This study offers a new probability model called Marshall–Olkin Extended Gumbel Type-II (MOEGT-II) which can model various shapes of the failure rate function. The proposed distribution is capable to model increasing, decreasing, reverse J-shaped, and upside down bathtub shapes of the failure rate function. Various statistical properties of the proposed distribution are derived such as alternate expressions for the density and distribution function, special cases of MOEGT-II distribution, quantile function, Lorenz curve, and Bonferroni curve. Estimation of the unknown parameters is carried out by the method of maximum likelihood. A simulation study is conducted using three different iterative methods with different samples of sizes n. The usefulness and potentiality of the MOEGT-II distribution have been shown using three real life data sets. The MOEGT-II distribution has been demonstrated as better fit than Exponentiated Gumbel Type-II (EGT-II), Marshall–Olkin Gumbel Type-II (MOGT-II), Gumbel Type-II (GT-II), Marshall–Olkin–Frechet (MOF), Frechet (F), Burr III, Log Logistic (LL), Beta Inverse Weibull (BIW), and Kumaraswamy Inverse Weibull (KIW) distributions.

Water is important to live because of its impacts on food supply and the natural environment for all living things. Approximately 0.3 percent of water resources are usable in the world. Groundwater is the principal source of drinking water but the air, sea, land, rivers, lakes, ocean, and wells are also the essential sources of water. Many statistical models have been used to analyze the water data set so that to make predictions in a better way with limited resources of water. In this paper, a new approach is offered to model the data of water purification using vinyl chloride data. Various statistical properties of the proposed model have been derived. Maximum likelihood estimation is used to estimate the model parameters. Monte Carlo simulations are used to show the consistency of parameters. Using water purification using vinyl chloride, the suggested model application is studied and compared with that of other existing models such as EGF, GF, and F. The results showed that our model provides a much better fit while modeling this data set rather than EGF, GF, and F distributions.

Developing countries lack studies investigated the socioeconomic and parental role on students’ learning skills. This study is helpful to detect bottlenecks in the foundational learning skills (reading skills and numeracy skills) in the education system of Pakistan. Reading skills of children are found better who had no functional disabilities. Mothers with higher education had a significant positive contribution toward children learning skills. Children deprived of books for reading in appropriate language had a negative impact on their reading skills. Rich children had predominantly higher possibilities of good learning skills than poor children. Parents who had not attended children’s school to discuss child progress had a significantly negative effect on children’s numeracy skills. Overall parental involvement in some forms had insignificantly improved children reading and numeracy skills in Punjab, Pakistan.

In this paper we proposed Bayes estimators for complete sample of the Modified InverseRayleigh (MIR) parameters which was introduced by Khan (2014). Different approximationmethods with squared error loss function (SELF) have been used to develop the bayesestimators for the unknown parameters. The proposed estimators are compared with the correspondingmaximum likelihood estimators by simulation study on the basis of mean squareerror (MSE). To illustrate the usefulness and goodness of fit of Modified Inverse Rayleighdistribution we considered two real data sets.

A class of goodness of fit tests for Marshal-Olkin Extended Rayleigh distribution with estimated parameters is proposed. The tests are based on the empirical distribution function. For determination of asymptotic percentage points, Kolomogorov-Sminrov, Cramer-von-Mises, Anderson-Darling,Watson, and Liao-Shimokawa test statistic are used. This article uses Monte Carlo simulations to obtain asymptotic percentage points for Marshal-Olkin extended Rayleigh distribution. Moreover, power of the goodness of fit test statistics is investigated for this lifetime model against several alternatives.