General Class Of Distributions Research Articles

Why Burr Distribution: Invariance-Based Explanation

In many practical situations, we observe a special Burr probability distribution — e.g., Burr distribution describes the distribution of people by income, it describes the rainfall data, etc. In this paper, we provide a theoretical explanation for the ubiquity of Burr distributions: namely, we show that this distribution naturally follows from scale-invariance. To be more precise, the simplest distribution that can be obtained from scale invariance is the Pareto distribution — a frequent particular case of the general Burr family of distributions. Next simplest are all distributions from the Burr family. We also use the scale-invariance approach to come up with a more general class of distributions that will, hopefully, provide an even more accurate description of the corresponding phenomena.

Pricing with Samples

This paper studies the value of data for pricing purposes. Although pricing is central across industries, little is known about the minimal amount of data needed to achieve good pricing decisions. The present paper proposes a novel approach to quantify the informational content of data, through the introduction of a new class of robust data-driven policies and the development of factor-revealing dynamic programs. Studying the prototypical case of data coming in the form of samples from the willingness to pay of customers, we show that even a few samples (as few as 10) go a very long way in uncovering “good” prices. For example, quite strikingly, against a general class of distributions (monotone increasing hazard rate distributions), a single observation guarantees 64% of the performance an oracle with full knowledge of the distribution would achieve, two samples suffice to ensure 71%, and 10 samples guarantee 80% of such performance.

Reliability and Inference for Multi State Systems: The Generalized Kumaraswamy Case

Semi-Markov processes are typical tools for modeling multi state systems by allowing several distributions for sojourn times. In this work, we focus on a general class of distributions based on an arbitrary parent continuous distribution function G with Kumaraswamy as the baseline distribution and discuss some of its properties, including the advantageous property of being closed under minima. In addition, an estimate is provided for the so-called stress–strength reliability parameter, which measures the performance of a system in mechanical engineering. In this work, the sojourn times of the multi-state system are considered to follow a distribution with two shape parameters, which belongs to the proposed general class of distributions. Furthermore and for a multi-state system, we provide parameter estimates for the above general class, which are assumed to vary over the states of the system. The theoretical part of the work also includes the asymptotic theory for the proposed estimators with and without censoring as well as expressions for classical reliability characteristics. The performance and effectiveness of the proposed methodology is investigated via simulations, which show remarkable results with the help of statistical (for the parameter estimates) and graphical tools (for the reliability parameter estimate).

Generalized Rank Mapped Transmuted Distributions with Properties and Application: A Review

Generalizing probability distributions is a very common practice in the theory of statistics. Researchers have proposed several generalized classes of distributions which are very flexible and convenient to study various statistical properties of the distribution and its ability to fit the real-life data. Several methods are available in the literature to generalize new family of distributions. The Quadratic Rank Transmutation Map (QRTM) is a tool for the construction of new families of non-Gaussian distributions and to modulate a given base distribution for modifying the moments like the skewness and kurtosis with the ability to explore its tail properties and improve the adequacy of the distribution. Recently, a new family of transmutation map, defined as Cubic Rank Transmutation (CRT) has been used by several authors to develop new distributions with application to real-life data. In this article, we have done a review work on the existing generalized rank mapped transmuted probability distributions, available in the literature with various statistical properties such as the reliability, hazard rate and cumulative hazard functions, moments, mean, variance, moment-generating function, order statistics, generalized entropy and quantile function along with its applications. Some future works have also been discussed for generalized rank mapped transmuted distributions.

Test Procedures for Change Point in a General Class of Distributions

This paper is concerned with the change point analysis in a general class of distributions. The quasi-Bayes and likelihood ratio test procedures are considered to test the null hypothesis of no change point. Exact and asymptotic behaviors of the two test statistics are derived. To compare the performances of two test procedures, numerical significance levels and powers of tests are tabulated for certain selected values of the parameters. Estimation of the change point based on these two test procedures are also considered. Moreover, the epidemic change point problem is studied as an alternative model for the single change point model. A real data set with epidemic change model is analyzed by two test procedures.

Statistical inference for a general class of distributions with time-varying parameters

In this article we are interested in a general class of distributions for independent not necessarily identically distributed random variables, closed under minima, that includes a number of discrete and continuous distributions like the Geometric, Exponential, Weibull or Pareto. The main parameter involved in this class of distributions is assumed to be time varying with several possible modeling options. This is of particular interest in reliability and survival analysis for describing the time to event or failure. The maximum likelihood estimation of the parameters is addressed and the asymptotic properties of the estimators are discussed. We provide real and simulated examples and we explore the accuracy of the estimating procedure as well as the performance of classical model selection criteria in choosing the correct model among a number of competing models for the time-varying parameters of interest.

Prediction for Progressively Type-II Censored Competing Risks Data from the Half-Logistic Distribution

Point and interval predictions of the s-th order statistic in a future sample are discussed. The informative sample is assumed to be drawn from a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz and half-logistic distributions. The informative and future samples are progressively type-II censored, under competing risks model, and assumed to be obtained from the same population. A special attention is paid to the half-logistic distribution. Using six different progressive censoring schemes, numerical computations are carried out to illustrate the performance of the procedure. An illustrative example based on real data is also considered. The biases, mean squared prediction errors of the maximum likelihood predictors, coverage probabilities and average interval lengths of the Bayesian prediction intervals are computed via a simulation study.

On Estimation of Stress-Strength Reliability Using Lower Record Values from Proportional Reversed Hazard Family

In this article, a study on the stress-strength parameter based on lower record values from one parameter proportional reversed hazard family (PRHF) has been conducted. The classical and Bayesian results of Khan and Arshad (UMVU Estimation of Reliability Function and Stress-Strength Reliability from Proportional Reversed Hazard Family Based on Lower Records. American Journal of Mathematical and Management Sciences, 35(2), 171–181) and Condino et al. (Likelihood and Bayesian estimation of P(Y < X) using lower record values from a general class of distributions. Statistical Papers), when the strength and stress variables belong to different family of distributions from PRHF have been generalized. Uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE) and Bayes estimator (BE) are obtained for the powers of the parameter and reliability functions and The estimators of three parametric functions, namely, powers of parameter, and are interrelated, whereas, in the literature, researchers have handled the three estimation problems separately. Moreover, it is has been shown that the expressions for and are not required to estimate them. In this article, the technique of obtaining estimators of and is simpler as it does not require Rao-Blackwellization. Simulation studies have been performed for analyzing the behavior of the proposed estimators. An example using real data has also been considered as an illustration.

Bayesian Prediction of order statistics based on finite mixture of general class of distributions under Random Censoring

This paper focuses on the Bayesian prediction of kth ordered future observations modelled by a two-component mixture of general class of distributions. Samples under consideration are subject to random censoring. A closed form of Bayesian predictive density is obtained under a two-sample scheme. Applications to Weibull and Burr XII components are presented and comparisons with previous results are made. A numerical example is presented for special cases of the exponential and Lomax components to obtain interval prediction of first and last order statistics.

Copula based constant-stress PALT using tampered failure rate model with dependent competing risks

PurposeAccelerated life tests (ALTs) are used to make timely assessments of the lifetime distribution of highly reliable materials and components. Life test under accelerated environmental conditions may be fully accelerated or partially accelerated. In fully accelerated life testing, all the test units are run at accelerated condition, while in partially accelerated life testing, they are both run at normal and accelerated conditions. The products can fail due to one of the several possible causes of failure which need not be independent. The purpose of this paper is to design constant-stress PALT with dependent competing causes of failure using the tampered failure rate model.Design/methodology/approachGumbel–Hougaard copula is used to model and measure the dependence between the life times of competing causes of failure. The use of the copula simplifies the model specification and gives a general class of distributions with the same dependent structure and arbitrary marginal distributions.FindingsThe optimal plan consists in finding optimum allocation of test units in different chambers by minimizing the reciprocal of the determinant of Fisher Information Matrix. The confidence interval for the estimated values of the design parameters has been obtained and sensitivity analysis carried out. The results of sensitivity analysis show that the plan is robust to small deviations from the true values of baseline parameters.Originality/valueThe model formulated can help reliability engineers obtain reliability estimates quickly of high reliability products that are likely to last for several years.

Estimation and Reliability for a Special Type of Semi-Markov Process

This work deals with multi-state systems modelled by means of semi-Markov processes. The sojourn times are seen to be independent not identically distributed random variables and assumed to belong to one of two different general classes of distributions so that the first class is closed under maxima and the second one is closed under minima. We obtain maximum likelihood estimators of the parameters of interest under several estimation schemes and investigate their asymptotic properties. Plug-in type estimators are furnished for various quantities related to the system under study.

A New Class of Beta-Complementary Exponential Power Series Distributions

Abstract In this paper, we introduce a new class of beta-complementary exponential power series distributions, which is obtained by replacing the cumulative distribution function of the complementary exponential power series distributions in the logit of a beta random variable. This new class of distributions contains some new submodels, such as beta-complementary exponential geometric, beta-complementary exponential Poisson, beta-complementary exponential binomial, beta-complementary exponential logarithmic, and complementary exponential power series distributions. A general class of distributions is presented and some various properties are obtained in this paper. We characterize the beta-complementary exponential geometric distribution as one of the most applicable distributions in this class. Some mathematical properties of the beta-complementary exponential geometric distribution are reached in terms of the corresponding properties of the complementary exponential geometric and beta exponential distributions. We present expressions for the probability density function, cumulative distribution function, moment generating function, and moments. The estimation of parameters is approached by the maximum likelihood estimation procedure, and the expected information matrix is derived. The flexibility of the distribution is illustrated in application of a real data set.

Bayesian Inference Based on Multiply Type-II Censored Sample from a General Class of Distributions

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for Bayesian estimation based on an observed multiply Type-II censored sample. The problem of predicting the order statistics from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the inverse Weibull distribution is used as example. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.

Estimation and testing procedures for the reliability functions of a general class of distributions

ABSTRACTWe consider here the general class of distributions proposed by Sankaran and Gupta (2005) by zeroing in on two measures of reliability, R(t) = P(X > t) and P = P(X > Y). Thereafter, we develop point estimation for R(t) and ‘P’ and develop uniformly minimum variance unbiased estimators (UMVUES). Then we derive testing procedures for the hypotheses related to different parametric functions. Finally, we compare the results using the Monte Carlo simulation method. Using real data set, we illustrate the procedure clearly.

Bivariate Weibull-power series class of distributions

We point out that some of the results in Kundu and Gupta [3] in otherwise an excellent paper are incorrect. We propose a more general class of distributions and illustrate its use with two real data sets.

Characterizations of Certain Doubly Truncated Distribution Based on Order Statistics

In this paper, we characterize doubly truncated classes of absolutely continuous distributions by considering the conditional expectation of functions of order statistics. Specific distributions considered as a particular case of the general class of distributions are Weibull, Pareto, Power function, Rayleigh and Inverse Weibull.

Stability in electoral competition: A case for multiple votes

It is well known that the Hotelling–Downs model generically fails to admit an equilibrium when voting takes place under the plurality rule (Osborne, 1993). This paper studies the Hotelling–Downs model considering that each voter is allowed to vote for up to k candidates and demonstrates that an equilibrium exists for a non-degenerate class of distributions of voters' ideal policies – which includes all log-concave distributions – if and only ifk≥2. That is, the plurality rule (k=1) is shown to be the unique k-vote rule which generically precludes stability in electoral competition. Regarding the features of k-vote rules' equilibria, first, we show that there is no convergent equilibrium and, then, we fully characterize all divergent equilibria. We study comprehensively the simplest kind of divergent equilibria (two-location ones) and we argue that, apart from existing for quite a general class of distributions when k≥2, they have further attractive properties – among others, they are robust to free-entry and to candidates' being uncertain about voters' preferences.

Translation, contraction and dilation of dual generalized order statistics

A general class of distributions has been characterized through translation of two non-adjacent dual generalized order statistics (dgos). Moreover, the characterizing results are obtained for generalized Pareto distribution through dilation of dgos and generalized power function distribution through contraction of non-adjacent generalized order statistics.

Evidence on copula-based double-hurdle models with flexible margins

I consider copula-based double-hurdle models with flexible marginal distributions. While the copula must be specified in advance, the marginal distributions are assumed to belong to a quite general class of distributions and need not be specified by the researcher in advance. A simulation study indicates that the copula-based double-hurdle models with flexible margins perform well even when the selected copula has been misspecified. An empirical application shows that copula-based double-hurdle models with flexible margins may outperform the conventionally used classical double-hurdle model based on a bivariate normality assumption.

Recurrence Relations for Single and Product Moments of Dual Generalized Order Statistics from a General Class of Distributions

In this paper we derive some general recurrence relations between single and product moments of dual gener- alized order statistics from a general class of distributions, thus generalizing and unifying the earlier results in this direction due to several authors.