A parametric family of bivariate Moran’s distributions has the following properties: distributions of (X, Y) and (Y, X) are identical, marginal distributions are exponential, the regression of X on Y is linear and distribution of |X - Y| is exponential. In the paper an extension of this family is presented. The family contains distributions satisfying the above conditions and which are different from Moran’s distributions for parameters.
Suppose that the random number X of particles entering a system has the distribution pn=P{X=n}, n=0,1,⋯. Due to the interference of noise, some of the arriving particles are not registered, and the number of particles observed is actually Y, where Y≤X, and Y has the distribution qn=P{Y=n}, n=0,1,⋯. The interference of noise is characterized by the conditional probabilities s(r,n)=P{Y=r|X=n}, 0≤r≤n. In this paper the author assumes that the distribution of X belongs to the class PSD of power series distributions and that the relations E(Y|X=n)=αnp and E(Y2|X=n)=αnp(1−p)+αn2p2 hold with 0
Suppose that the random number X of particles entering a system has the distribution pn=P{X=n}, n=0,1,⋯. Due to the interference of noise, some of the arriving particles are not registered, and the number of particles observed is actually Y, where Y≤X, and Y has the distribution qn=P{Y=n}, n=0,1,⋯. The interference of noise is characterized by the conditional probabilities s(r,n)=P{Y=r|X=n}, 0≤r≤n. In this paper the author considers the case in which the probabilities s(n,r) form an inflated binomial distribution for each fixed n, i.e., (1) s(0,n)=1−α+αqn and s(r,n)=α(nr)prqn−r for 1≤r≤n, where 0
Consider an incomplete multiresponse experiment with a classification of the observations into u classes. The class Si contains ti treatments with ni observations. Thus, the experiment can be described by the set {Y1,Y2,...,Yu}, where Yi are ni×ti random matrices of observations from the class Si. For the expectation of each Yi a linear model with main and nuisance parameters is proposed. The problem of (linear) estimation of the main parameters is considered using results of R. Zmyślony [in Mathematical statistics and probability theory (Wisła, 1978), 365–373, Lecture Notes in Statist., 2, Springer, New York, 1980; MR0577292] and J. K. Baksalary [Math. Operationsforsch. Statist. Ser. Statist. 15 (1984), no. 1, 3–35; MR0729609].
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