We introduce a new family of distributions namely inverse truncated discrete Linnik G family of distributions. This family is a generalization of inverse Marshall-Olkin family of distributions, inverse family of distributions generated through truncated negative binomial distribution and inverse family of distributions generated through truncated discrete Mittag-Leffler distribution. A particular member of the family, inverse truncated negative binomial Weibull distribution is studied in detail. The shape properties of the probability density function and hazard rate, model identifiability, moments, median, mean deviation, entropy, distribution of order statistics, stochastic ordering property, mean residual life function and stress-strength properties of the new generalized inverse Weibull distribution are studied. The unknown parameters of the distribution are estimated using maximum likelihood method, product spacing method and least square method. The existence and uniqueness of the maximum likelihood estimates are proved. Simulation is carried out to illustrate the performance of maximum likelihood estimates of model parameters. An AR(1) minification model with this distribution as marginal is developed. The inverse truncated negative binomial Weibull distribution is fitted to a real data set and it is shown that the distribution is more appropriate for modeling in comparison with some other competitive models.
Earthquake in recent years has increased tremendously. This paper outlines an evaluation of Cumulative Sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) charting technique to determine if the frequency of earthquake in the world is unusual. The frequency of earthquake in the world is considered from the period 1973 to 2016. As our data is auto correlated we cannot use the regular control chart like Shewhart control chart to detect unusual earthquake frequency. An approach that has proved useful in dealing with auto correlated data is to directly model time series model such as Autoregressive Integrated Moving Average (ARIMA), and apply control charts to the residuals. The EWMA control chart and the CUSUM control chart have detected unusual frequencies of earthquake in the year 2012 and 2013 which are state of statistically out of control.
A new flexible extension of the inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties are derived. We assessed the performance of the maximum likelihood method via a simulation study. The importance of the new model is shown via three applications to real data sets. The new model is much better than other important competitive models.
The shape parameter of a symmetric probability distribution is often more difficult to estimate accurately than the location and scale parameters. In this paper, we suggest an intuitive but innovative matching quantile estimation method for this parameter. The proposed shape parameter estimate is obtained by setting its value to a level such that the central 1-1/n portion of the distribution will just cover all n observations, while the location and scale parameters are estimated using existing methods such as maximum likelihood (ML). This hybrid estimator is proved to be consistent and is illustrated by two distributions, namely Student-t and Exponential Power. Simulation studies show that the hybrid method provides reasonably accurate estimates. In the presence of extreme observations, this method provides thicker tails than the full ML method and protect inference on the location and scale parameters. This feature offered by the hybrid method is also demonstrated in the empirical study using two real data sets.
In this paper, kumaraswamy reciprocal family of distributions is introduced as a new continues model with some of approximation to other probabilistic models as reciprocal, beta, uniform, power function, exponential, negative exponential, weibull, rayleigh and pareto distribution. Some fundamental distributional properties, force of mortality, mills ratio, bowley skewness, moors kurtosis, reversed hazard function, integrated hazard function, mean residual life, probability weighted moments, bonferroni and lorenz curves, laplace-stieltjes transform of this new distribution with the maximum likelihood method of the parameter estimation are studied. Finally, four real data sets originally presented are used to illustrate the proposed estimators.
Bayesian hierarchical regression (BHR) is often used in small area estimation (SAE). BHR conditions on the samples. Therefore, when data are from a complex sample survey, neither survey sampling design nor survey weights are used. This can introduce bias and/or cause large variance. Further, if non-informative priors are used, BHR often requires the combination of multiple years of data to produce sample sizes that yield adequate precision; this can result in poor timeliness and can obscure trends. To address bias and variance, we propose a design assisted model-based approach for SAE by integrating adjusted sample weights. To address timeliness, we use historical data to define informative priors (power prior); this allows estimates to be derived from a single year of data. Using American Community Survey data for validation, we applied the proposed method to Behavioral Risk Factor Surveillance System data. We estimated the prevalence of disability for all U.S. counties. We show that our method can produce estimates that are both more timely than those arising from widely-used alternatives and are closer to ACS’ direct estimates, particularly for low-data counties. Our method can be generalized to estimate the county-level prevalence of other health related measurements.
In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors on the lifetimes of experimental units. In this paper, a step-stress model is considered in which the life-testing experiment gets terminated either at a pre-fixed time (say, Tm+1) or at a random time ensuring at least a specified number of failures (Say, y out of n). Under this model in which the data obtained are Type-II hybrid censored, the Kumaraswamy Weibull distribution is used for the underlying lifetimes. The maximum Likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model are derived. The confidence intervals of the parameters are also obtained. The hazard rate and reliability functions are estimated at usual conditions of stress. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.
This article presents a classification of disease severity for patients with cystic fibrosis (CF). CF is a genetic disease that dramatically decreases life expectancy and quality. The disease is characterized by polymicrobial infections which lead to lung remodeling and airway mucus plugging. In order to quantify disease severity of CF patients and compute a continuous severity index measure, quantile regression, rank scores, and corresponding normalized ranks are calculated for CF patients. Based on the rank scores calculated from the set of quantile regression models, a continuous severity index is computed for each CF patient and can be considered a robust estimate of CF disease severity.
The Pareto distribution is a power law probability distribution that is used to describe social scientific, geophysical, actuarial, and many other types of observable phenomena. A new weighted Pareto distribution is proposed using a logarithmic weight function. Several statistical properties of the weighted Pareto distribution are studied and derived including cumulative distribution function, location measures such as mode, median and mean, reliability measures such as reliability function, hazard and reversed hazard functions and the mean residual life, moments, shape indices such as skewness and kurtosis coefficients and order statistics. A parametric estimation is performed to obtain estimators for the distribution parameters using three different estimation methods the maximum likelihood method, the L-moments method and the method of moments. Numerical simulation is carried out to validate the robustness of the proposed distribution. The distribution is fitted to a real data set to show its importance in real life applications.
In this paper, a new version of the Poisson Lomax distributions is proposed and studied. The new density is expressed as a linear mixture of the Lomax densities. The failure rate function of the new model can be increasing-constant, increasing, U shape, decreasing and upside down-increasing. The statistical properties are derived and four applications are provided to illustrate the importance of the new density. The method of maximum likelihood is used to estimate the unknown parameters of the new density. Adequate fitting is provided by the new model.