Density Of Order Statistics Research Articles

The exponentiated generalized extended exponential distribution

We introduce and study a new four-parameter lifetime model named the exponentiated generalized extended exponential distribution. The proposed model has the advantage of including as special cases the exponential and exponentiated exponential distributions, among others, and its hazard function can take the classic shapes: bathtub, inverted bathtub, increasing, decreasing and constant, among others. We derive some mathematical properties of the new model such as a representation for the density function as a double mixture of Erlang densities, explicit expressions for the quantile function, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R´enyi entropy, density of order statistics and reliability. We use the maximum likelihood method to estimate the model parameters. Two applications to real data illustrate the flexibility of the proposed model.

Interval-valued time series models: Estimation based on order statistics exploring the Agriculture Marketing Service data

The current regression models for interval-valued data ignore the extreme nature of the lower and upper bounds of intervals. A new estimation approach is proposed; it considers the bounds of the interval as realizations of the max/min order statistics coming from a sample of nt random draws from the conditional density of an underlying stochastic process {Yt}. This approach is important for data sets for which the relevant information is only available in interval format, e.g., low/high prices. The interest is on the characterization of the latent process as well as in the modeling of the bounds themselves. A dynamic model is estimated for the conditional mean and conditional variance of the latent process, which is assumed to be normally distributed, and for the conditional intensity of the discrete process {nt}, which follows a negative binomial density function. Under these assumptions, together with the densities of order statistics, maximum likelihood estimates of the parameters of the model are obtained. They are needed to estimate the expected value of the bounds of the interval. This approach is implemented with the time series of livestock prices, of which only low/high prices are recorded making the price process itself a latent process. It is found that the proposed model provides an excellent fit of the intervals of low/high returns with an average coverage rate of 83%. In addition, a comparison with current models for interval-valued data is offered.

The complementary Weibull geometric distribution

In this paper, we proposed a new three-parameters lifetime distribution with unimodal, increasing and decreasing hazard rate. The new distribution, the complementary Weibull geometric (CWG), is complementary to the Weibull-geometric (WG) model proposed by Barreto-Souza et al. (The Weibull-Geometric distribution, J. Statist. Comput. Simul. 1 (2010), pp. 1–13). The CWG distribution arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its reliability and hazard rate functions, moments, density of order statistics and their moments. We provide expressions for the Rényi and Shannon entropies. The parameter estimation is based on the usual maximum likelihood approach. We obtain the observed information matrix and discuss inferences issues. We report a hazard function comparison study between the WG distribution and our complementary one. The flexibility and potentiality of the new distribution is illustrated by means of three real dataset, where we also made a comparison between Weibull, WG and CWG modelling approach.

The Weibull-geometric distribution

For the first time, we propose the Weibull-geometric (WG) distribution which generalizes the extended exponential-geometric (EG) distribution introduced by Adamidis et al. [K. Adamidis, T. Dimitrakopoulou, and S. Loukas, On a generalization of the exponential-geometric distribution, Statist. Probab. Lett. 73 (2005), pp. 259–269], the exponential-geometric distribution discussed by Adamidis and Loukas [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] and the Weibull distribution. We derive many of its standard properties. The hazard function of the EG distribution is monotone decreasing, but the hazard function of the WG distribution can take more general forms. Unlike the Weibull distribution, the new distribution is useful for modelling unimodal failure rates. We derive the cumulative distribution and hazard functions, moments, density of order statistics and their moments. We provide expressions for the Rényi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an EM algorithm [A.P. Dempster, N.M. Laird, and D.B. Rubim, Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. R. Stat. Soc. B 39 (1977), pp. 1–38; G.J. McLachlan and T. Krishnan, The EM Algorithm and Extension, Wiley, New York, 1997] is given for estimating the parameters. We obtain the observed information matrix and discuss inference issues. The flexibility and potentiality of the new distribution is illustrated by means of a real data set.

Edgeworth Expansion of Densities of Order Statistics with Fixed Rank

We give an Edgeworth expansion for densities of order statistics with fixed rank k. The Edgeworth expansion for densities of extreme values is then obtained as a special case k = 1.

Über ein nichtparametrisches Schätzproblem

This note gives invariantly optimal estimators for functions of probability densities, based on independent observations in a homogeneous space. The results are applied to the estimation of certain densities, derivatives of the density, densities of order statistics and of empirical means.

A Note on Densities of Order Statistics

(1974). A Note on Densities of Order Statistics. Mathematics Magazine: Vol. 47, No. 2, pp. 100-101.

A Note on Densities of Order Statistics

A Note on Densities of Order Statistics