The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some ...newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.
The Mittag-Leffler stochastic process is an important tool in practical applications. In this paper, we first focus on some claims of Burr type-XII as a superstatistical stationary distribution ...(Sánchez, 2019). Then, we present the capabilities of the Mittag-Leffler distribution and introduce truncated Mittag-Leffler distributions. Finally, in the oil price analysis of time series real data, we show that the truncated Mittag-Leffler distribution performs better than other nominated distributions, especially the Burr distribution.
•Focus on some claims for Burr type-XII (Sánchez, 2019).•The capabilities of truncated Mittag-Leffler distribution.•Oil price analysis of time series real data.•Time series real data based on 100 days moving average.
A new three-parameter exponential-type family of distributions which can be used in modeling survival data, reliability problems and fatigue life studies is introduced. Its failure rate function can ...be constant, decreasing, increasing, upside-down bathtub or bathtub-shaped depending on its parameters. It includes as special sub-models the exponential distribution, the generalized exponential distribution Gupta, R.D., Kundu, D., 1999. Generalized exponential distributions. Australian and New Zealand Journal of Statistics 41, 173–188 and the extended exponential distribution Nadarajah, S., Haghighi, F., 2011. An extension of the exponential distribution. Statistics 45, 543–558. A comprehensive account of the mathematical properties of the new family of distributions is provided. Maximum likelihood estimation of the unknown parameters of the new model for complete sample as well as for censored sample is discussed. Estimation of the stress–strength parameter is also considered. Two empirical applications of the new model to real data are presented for illustrative purposes.
M. I. Yadrenko discovered that the expectation of the minimum number N
1
of independent and identically distributed uniform random variables on (0, 1) that have to be added to exceed 1 is e. For any ...threshold a > 0, K. G. Russell found the distribution, mean, and variance of the minimum number N
a
of independent and identically distributed uniform random summands required to exceed a. Here we calculate the distribution and moments of N
a
when the summands obey the negative exponential and Lévy distributions. The Lévy distribution has infinite mean. We compare these results with the results of Yadrenko and Russell for uniform random summands to see how the expected first-passage time
E
(
N
a
)
,
a
>
0
, and other moments of N
a
depend on the distribution of the summand.
In this paper, we consider two main families of bivariate distributions with exponential marginals for a couple of random variables (X1,X2). More specifically, we derive closed-form expressions for ...the distribution of the sum S=X1+X2, the TVaR of S and the contributions of each risk under the TVaR-based allocation rule. The first family considered is a subset of the class of bivariate combinations of exponentials, more precisely, bivariate combinations of exponentials with exponential marginals. We show that several well-known bivariate exponential distributions are special cases of this family. The second family we investigate is a subset of the class of bivariate mixed Erlang distributions, namely bivariate mixed Erlang distributions with exponential marginals. For this second class of distributions, we propose a method based on the compound geometric representation of the exponential distribution to construct bivariate mixed Erlang distributions with exponential marginals. Notably, we show that this method not only leads to Moran–Downton’s bivariate exponential distribution, but also to a generalization of this bivariate distribution. Moreover, we also propose a method to construct bivariate mixed Erlang distributions with exponential marginals from any absolutely continuous bivariate distributions with exponential marginals. Inspired from Lee and Lin (2012), we show that the resulting bivariate distribution approximates the initial bivariate distribution and we highlight the advantages of such an approximation.
A generalization of the exponential distribution is presented. The generalization always has its mode at zero and yet allows for increasing, decreasing and constant hazard rates. It can be used as an ...alternative to the gamma, Weibull and exponentiated exponential distributions. A comprehensive account of the mathematical properties of the generalization is presented. A real data example is discussed to illustrate its applicability.
A finite-step procedure is proposed for maximum likelihood estimation of the three-parameter asymmetric Laplace distribution. Its performance is compared with the iterative method most commonly used ...for this distribution. The new procedure is much faster and reliably identifies samples for which maximum likelihood estimates lie on the boundary of the parameter space, making it a good choice for simulation studies and simulation-based methodologies.
This article puts forth a novel category of probability distributions obtained from the Topp–Leone distribution, the inverse-
exponential distribution, and the power functions. To obtain this new ...family, we used the original cumulative distribution functions. After introducing this new family, we gave the motivations that led us to this end and the basis of the new family obtained, followed by the mathematical properties related to the family. Then, we presented the statistic order, the quantile function, the series expansion, the moments, and the entropy (Shannon, Reiny, and Tsallis), and we estimated the parameters by the maximum likelihood method. Finally, using real data, we presented numerical results through data analysis with a comparison of rival models.
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