Distribution Of Order Statistics Research Articles

Distortion representations of multivariate distributions

The univariate distorted distributions were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later, they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate and proportional reversed hazard rate models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representation, especially when the marginal distributions may not be easily handled. Several examples illustrate the applications of such representations in statistical modeling. They include the study of paired (dependent) ordered data, joint residual lifetimes, order statistics and coherent systems

Asymptotic behavior of ordered random variables in mixture of two Gaussian sequences with random index

<abstract><p>When the random sample size is assumed to converge weakly and to be independent of the basic variables, the asymptotic distributions of extreme, intermediate, and central order statistics, as well as record values, for a mixture of two stationary Gaussian sequences under an equi-correlated setup are derived. Furthermore, sufficient conditions for convergence are derived in each case. An interesting fact is revealed that in several cases, the limit distributions of the aforementioned statistics are the same when the sample size is random and non-random. e.g., when one mixture component has a correlation that converges to a non-zero value.</p></abstract>

Limit Distributions of Ordered Random Variables in Mixture of Two Gaussian Sequences

In this paper, we study the limit distributions of extreme, intermediate, and central order statistics, as well as record values, of the mixture of two stationary Gaussian sequences under equicorrelated setup.

Further Properties and Estimations of Exponentiated Generalized Linear Exponential Distribution

The recent exponentiated generalized linear exponential distribution is a generalization of the generalized linear exponential distribution and the exponentiated generalized linear exponential distribution. In this paper, we study some statistical properties of this distribution such as negative moments, moments of order statistics, mean residual lifetime, and their asymptotic distributions for sample extreme order statistics. Different estimation procedures include the maximum likelihood estimation, the corrected maximum likelihood estimation, the modified maximum likelihood estimation, the maximum product of spacing estimation, and the least squares estimation are compared via a Monte Carlo simulation study in terms of their biases, mean squared errors, and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate its use for modeling a rainfall data from Orlando, Florida.

A generalization to the log-inverse Weibull distribution and its applications in cancer research

In this paper we consider a generalization of a log-transformed version of the inverse Weibull distribution. Several theoretical properties of the distribution are studied in detail including expressions for its probability density function, reliability function, hazard rate function, quantile function, characteristic function, raw moments, percentile measures, entropy measures, median, mode etc. Certain structural properties of the distribution along with expressions for reliability measures as well as the distribution and moments of order statistics are obtained. Also we discuss the maximum likelihood estimation of the parameters of the proposed distribution and illustrate the usefulness of the model through real life examples. In addition, the asymptotic behaviour of the maximum likelihood estimators are examined with the help of simulated data sets.

Study of a Modified Kumaraswamy Distribution

In this article, a structural modification of the Kumaraswamy distribution yields a new two-parameter distribution defined on (0,1), called the modified Kumaraswamy distribution. It has the advantages of being (i) original in its definition, mixing logarithmic, power and ratio functions, (ii) flexible from the modeling viewpoint, with rare functional capabilities for a bounded distribution—in particular, N-shapes are observed for both the probability density and hazard rate functions—and (iii) a solid alternative to its parental Kumaraswamy distribution in the first-order stochastic sense. Some statistical features, such as the moments and quantile function, are represented in closed form. The Lambert function and incomplete beta function are involved in this regard. The distributions of order statistics are also explored. Then, emphasis is put on the practice of the modified Kumaraswamy model in the context of data fitting. The well-known maximum likelihood approach is used to estimate the parameters, and a simulation study is conducted to examine the performance of this approach. In order to demonstrate the applicability of the suggested model, two real data sets are considered. As a notable result, for the considered data sets, statistical benchmarks indicate that the new modeling strategy outperforms the Kumaraswamy model. The transmuted Kumaraswamy, beta, unit Rayleigh, Topp–Leone and power models are also outperformed.

Generalized Inverse Power Sujatha Distribution with Applications

In this paper, we present a new lifetime distribution known as the generalized inverse power Sujatha distribution. The statistical and mathematical properties of the new distribution such as the moment and moment generating function, Renyi entropy and distribution of order statistics have been derived and discussed. Also, reliability measures like survival function, hazard function, reverse hazard rate, cumulative hazard rate and odds function are discussed. Maximum likelihood estimation technique was used to estimate the parameters. However, a 95% confidence intervals were constructed for the parameters. Finally, we applied the proposed distribution to two lifetime datasets and compare its superiority over other candidate models. Results obtained indicates that the generalized inverse power Sujatha distribution outperform the other competing models.

Extreme Value Charts & Analysis of Means Based on the New Weighted Exponential Distribution

It is assumed that the probabilistic model of the quality characteristics follows the new weighted exponential distribution. Control charts based on each subgroup’s extreme values are established. The constants in the control chart are determined by the probability distribution of the extreme value order statistics of the sub-group and the sub-group size. The proposed chart is thus referred to as an extreme values chart. A biased overall mean analysis method (ANOM for truncated population) is used for the new weighted exponential distribution. Examples based on real time data are used to explain the findings.

Inverse Lomax-Rayleigh distribution with application

In this paper, an extension of Rayleigh distribution called Inverse Lomax Rayleigh (ILR) is proposed by using the Inverse Lomax generator of [13]. Properties of ILR were derived. This includes the complete and incomplete moments, entropy, distribution of order statistics, and quantile function. A simulation study was presented to explore the properties of the estimates. This shows that they are unbiased, consistent, and efficient. An application to fatigue data shows the flexibility of ILR distribution, as it outperforms all the comparators with minimum values of all the measures.

The Iwok-Nwikpe Distribution: Statistical Properties and Its Application

In this paper, a new continuous probability distribution named Iwok-Nwikpe distribution is proposed. Some essential statistical properties of the proposed probability distribution have been derived. The graphs of the survival function, probability density function (p.d.f) and cumulative distribution function (c.d.f) were plotted at different values of the parameter. The mathematical expression for the moment generating function (mgf) was derived. Consequently, the first three crude moments were obtained; the distribution of order statistics, the second and third moments corrected for the mean have also been derived. The parameter of the Iwok-Nwikpe distribution was estimated by means of maximum likelihood technique. To establish the goodness of fit of the Iwok-Nwikpe distribution, three real data sets from engineering and medical science were fitted to the distribution. Findings of the study revealed that the Iwok-Nwikpe distribution performed better than the one parameter exponential distribution and other competing models used for the study.

The maximum order statistic of doubly truncated Inverted Gamma distribution

Dealing with phenomena is developing rapidly as a result of the evaluation of the phenomena themselves. Some phenomena benefit us from the maximum values within a certain period as a variable, for example the highest electricity load, the largest stretch of the beach, the longest period of a specific disease in the patient’s body and so on. In this paper, we present the distribution of Maximum order statistic from doubly truncated Inverted Gamma distribution (MOSDTIGD). The rth raw moments function form derives for new density. Stress-strength reliability, Shannon and relative entropies have also been discussed.

Maximum Product of Spacing Parameter Estimation of Gompertz Rayleigh Distribution and Application to Rainfall Datasets

Gompertz Rayleigh (GomR) distribution was introduced in an earlier study with few statistical properties derived and parameters estimated using only the most common traditional method, Maximum Likelihood Estimation (MLE). This paper aimed at deriving more statistical properties of the GomR distribution, estimating the three unknown parameters via a competitive method, Maximum Product of Spacing (MPS) and evaluating goodness of fit using rainfall data sets from Nigeria, Malaysia and Argentina.
 Properties of statistical distributions including distribution of smallest and largest order statistics, cumulative or integrated hazard function, odds function, rth non-central moments, moment generating function, mean, variance and entropy measures for GomR distribution were explicitly derived. The fitted data sets reveal the flexibility of GomR distribution over other distributions been compared with. Simulation study was used to evaluate the consistency, accuracy and unbiasedness of the GomR distribution parameter estimates obtained from the method of MPS. The study found that GomR distribution could not provide a better fit for Argentine rainfall data but it was the best distribution for the rainfall data sets from Nigeria and Malaysia in comparison with the distributions; Generalized Weibull Rayleigh (GWR), Exponentiated Weibull Rayleigh (EWR), Type (II) Topp Leone Generalized Inverse Rayleigh (TIITLGIR), Kumarawamy Exponential Inverse Raylrigh (KEIR), Negative Binomial Marshall-Olkin Rayleigh (NBMOR) and Exponentiated Weibull (EW). Furthermore, the estimates from MPSE were consistent as the sample size increases but not as efficient as those from MLE.

Tornumonkpe Distribution: Statistical Properties and Goodness of Fit

A new sole parameter probability distribution named the Tornumonkpe distribution has been derived in this paper. The new model is a blend of gamma (2, and gamma(3 distributions. The shape of its density for different values of the parameter has been shown. The mathematical expression for the moment generating function, the first three raw moments, the second and third moments about the mean, the distribution of order statistics, coefficient of variation and coefficient of skewness has been given. The parameter of the new distribution was estimated using the method of maximum likelihood. The goodness of fit of the Tornumonkpe distribution was established by fitting the distribution to three real life data sets. Using -2lnL, Bayesian Information Criterion (BIC), and Akaike Information Criterion(AIC) as criterial for selecting the best fitting model, it was revealed that the new distribution outperforms the one parameter exponential, Shanker and Amarendra distributions for the data sets used.

Higher Order Moments of the Order Statistics for the Rectangular, Exponential, Gamma and Weibull Distributions

In this paper we discuss the problem of Higher Order Moments for the order Statistics for the Rectangular, Exponential, Gamma and Weibull distributions by finding the order statistic distributions for the base distribution and modified distributions, the base distribution is to deduce the corresponding distribution by the polynomial modifier. These higher order moments are very much useful in most of the Data sciences and Image analysis.

Samade Probability Distribution: Its Properties and Application to Real Lifetime Data

A new two-parameter lifetime distribution has been proposed in this study. The distribution is called Samade distribution. The model is motivated by the wide use of the lifetime models derived from the mixture of gamma and exponential distributions. Its mathematical properties which include the first four moments, variance as well as coefficient of variation, reliability function, hazard function, survival function, Renyi entropy measure and distribution of order statistics have been successfully derived. The maximum likelihood estimation of its parameters and application to real life data have been discussed. Application of this model to three real datasets shown that the proposed model yields a satisfactorily better fit than other existing lifetime distributions. The comparism of goodness-of-fits were established using -2Loglikelihood, AIC and BIC.

ON THE PROPERTIES AND APPLICATIONS OF A NEW EXTENSION OF EXPONENTIATED RAYLEIGH DISTRIBUTION

Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.

Entropy and Stress-Strength Model of Lower Order Statistic of Doubly Truncated Inverted Gamma Distribution

Interest in phenomena is one of the most prominent scientific concerns of humans in general, and researchers and scientists in particular. From this standpoint came the interest to derive new probability distributions for representing the actual data taken from those phenomena more flexibly. In this paper, we present the distribution of lower order statistic from doubly truncated Inverted Gamma distribution (LOSDTIGD). The rth raw moments function form derives for new density. Stress-strength reliability, Shannon and relative entropies have also been discussed.

Generalized Quasi Lindley Distribution: Theoretical Properties, Estimation Methods, and Applications

In this paper, we introduce a new continuous distribution of two parameterscalled as a generalized Quasi Lindley distribution (GQLD). The GQLD is asum of two independent Quasi Lindley distributed random variables. Compre-hensive statistical properties of the GQLD are provided in closed forms includesmoments, reliability analysis, stochastic ordering, stress-strength reliability, andthe distribution of order statistics. The parameters of the new distribution areestimated by the maximum likelihood, maximum product of spacings, ordinaryleast squares, weighted least squares, Cramer-von-Mises, and Anderson-Darlingmethods are considered. A simulation study is conducted to investigate theeciency of the proposed estimators and applications to real data sets are pro-vided.

A Generalized Log-Weibull Distribution with Bio-Medical Applications

Here we consider a generalized log-transformed version of the Weibull distribution and investigate some of its important properties like expressions for the cumulative distribution function hazard rate function, quantile function, characteristic function, raw moments, incomplete moments, etc. The distribution and moments of order statistics are obtained along with some results on certain structural properties of the distribution. The maximum likelihood estimation of the parameters of the distribution is attempted for both complete and censored data sets and the usefulness of the distribution is illustrated with the help of real-life data sets from biomedical fields.

Two-Parameter Nwikpe (TPAN) Distribution with Application

A new two-parameter continuous distribution called the Two-Parameter Nwikpe (TPAN) distribution is derived in this paper. The new distribution is a mixture of gamma and exponential distributions. A few statistical properties of the new probability distribution have been derived. The shape of its density for different values of the parameters has also been established. The first four crude moments, the second and third moments about the mean of the new distribution were derived using the method of moment generating function. Other statistical properties derived include; the distribution of order statistics, coefficient of variation and coefficient of skewness. The parameters of the new distribution were estimated using maximum likelihood method. The flexibility of the Two-Parameter Nwikpe (TPAN) distribution was shown by fitting the distribution to three real life data sets. The goodness of fit shows that the new distribution outperforms the one parameter exponential, Shanker and Amarendra distributions for the data sets used for this study.