33,838 research outputs found
Beta Linear Failure Rate Geometric Distribution with Applications
This paper introduces the beta linear failure rate geometric (BLFRG) distribution, which contains a number of distributions including the exponentiated linear failure rate geometric, linear failure rate geometric, linear failure rate, exponential geometric, Rayleigh geometric, Rayleigh and exponential distributions as special cases. The model further generalizes the linear failure rate distribution. A comprehensive investigation of the model properties including moments, conditional moments, deviations, Lorenz and Bonferroni curves and entropy are presented. Estimates of model parameters are given. Real data examples are presented to illustrate the usefulness and applicability of the distribution
Intersection Numbers of Geodesic Arcs
For a compact surface with constant negative curvature (for
some ) and genus , we show that the tails of the distribution
of (where is the
intersection number of the closed geodesics and denotes the
geometric length) are estimated by a decreasing exponential function. As a
consequence, we find the asymptotic normalized average of the intersection
numbers of pairs of closed geodesics on . In addition, we prove that the
size of the sets of geodesics whose -self-intersection number is not close
to is also estimated by a decreasing exponential
function. And, as a corollary of the latter, we obtain a result of S. Lalley
which states that most of the closed geodesics on with
have roughly
self-intersections, when is large
Limit theorems for Bajraktarevi\'c and Cauchy quotient means of independent identically distributed random variables
We derive strong law of large numbers and central limit theorems for
Bajraktarevi\'c, Gini and exponential- (also called Beta-type) and logarithmic
Cauchy quotient means of independent identically distributed (i.i.d.) random
variables. The exponential- and logarithmic Cauchy quotient means of a sequence
of i.i.d. random variables behave asymptotically normal with the usual square
root scaling just like the geometric means of the given random variables.
Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d.
random variables behave asymptotically in a rather different way: in order to
get a non-trivial normal limit distribution a time dependent centering is
needed.Comment: 25 page
Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions
The geometric mean is shown to be an appropriate statistic for the scale of a
heavy-tailed coupled Gaussian distribution or equivalently the Student's t
distribution. The coupled Gaussian is a member of a family of distributions
parameterized by the nonlinear statistical coupling which is the reciprocal of
the degree of freedom and is proportional to fluctuations in the inverse scale
of the Gaussian. Existing estimators of the scale of the coupled Gaussian have
relied on estimates of the full distribution, and they suffer from problems
related to outliers in heavy-tailed distributions. In this paper, the scale of
a coupled Gaussian is proven to be equal to the product of the generalized mean
and the square root of the coupling. From our numerical computations of the
scales of coupled Gaussians using the generalized mean of random samples, it is
indicated that only samples from a Cauchy distribution (with coupling parameter
one) form an unbiased estimate with diminishing variance for large samples.
Nevertheless, we also prove that the scale is a function of the geometric mean,
the coupling term and a harmonic number. Numerical experiments show that this
estimator is unbiased with diminishing variance for large samples for a broad
range of coupling values.Comment: 17 pages, 5 figure
The Weibull-Geometric distribution
In this paper we introduce, for the first time, the Weibull-Geometric
distribution which generalizes the exponential-geometric distribution proposed
by Adamidis and Loukas (1998). The hazard function of the last distribution is
monotone decreasing but the hazard function of the new distribution can take
more general forms. Unlike the Weibull distribution, the proposed distribution
is useful for modeling unimodal failure rates. We derive the cumulative
distribution and hazard functions, the density of the order statistics and
calculate expressions for its moments and for the moments of the order
statistics. We give expressions for the R\'enyi and Shannon entropies. The
maximum likelihood estimation procedure is discussed and an algorithm EM
(Dempster et al., 1977; McLachlan and Krishnan, 1997) is provided for
estimating the parameters. We obtain the information matrix and discuss
inference. Applications to real data sets are given to show the flexibility and
potentiality of the proposed distribution
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