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A mixed bivariate distribution with exponential and geometric marginals. (English) Zbl 1072.62003
Let $$X_1, X_2, \dots$$ be $$i.i.d.$$ (independent and identically distributed) exponential random variables with parameter $$\beta > 0$$ and the probability density function ($$p.d.f.$$) defined as $$f(x) = \beta\cdot e^{-\beta x}, x > 0.$$ Let $$N$$ be a geometric random variable with p.d.f. $$h(n) = P(N = n) = p(1 - p)^{n - 1}, n = 1, 2,\dots,$$ independent of every $$X_i$$ random variable and, since the exponential distribution is closed under geometric compound, the random sum $$X =\sum^{N}_{i=1} X_i$$ has an exponential distribution with the parameter $$p\cdot\beta$$.
The aim of this paper is to study the joint distribution of $$X$$ and $$N$$, and to show that the resulting class $$(X, N)$$ of mixed bivariate distributions, referred hereafter as BEG (Bivariate distributions with Exponential and Geometric marginals), carries remarkable properties and supports useful applications in stochastic modelling. Section 2 of the paper defines the mixed BEG class of distributions and derives their basic properties: relationship between BEG distribution and the Laplace distribution, survival function computation, convergence in distribution. Section 3 presents certain conditional distributions that are useful in the practical implementation of goodness of fit analyses connected with the BEG model. Section 4 gives three representations of BEG distributions involving random and deterministic sums of i.i.d. random vectors.
The following results generalize similar properties of univariate exponential and geometric distributions, and justify the applicability of the BEG distributions as stochastic models: infinite divisibility, stability with respect to geometric summation, and compound Poisson representation. Section 5 considers maximum likelihood estimation of the BEG parameters, showing that the obtained maximum likelihood estimators are consistent, asymptotically normal, and efficient ones. Section 6 illustrates the modelling potential of the proposed BEG distributions on a currency exchange data set, $$N$$ representing the number of consecutive positive daily $$log$$-returns of currency exchange rates.
The numerical findings prove a very good fit in all five studied cases, entailing that the BEG model captures not only the marginal distributions of the data but also the conditional distributions very well, concluding the remarkable fit of the BEG model to the considered currency data. The modelling potential of the investigated laws may be also successfully applied in various fields such as water resources, climate research, and finance.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62F10 Point estimation 60E05 Probability distributions: general theory 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010) 91B32 Resource and cost allocation (including fair division, apportionment, etc.)
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