9,783 research outputs found
On the Truncated Pareto Distribution with applications
The Pareto probability distribution is widely applied in different fields
such us finance, physics, hydrology, geology and astronomy. This note deals
with an application of the Pareto distribution to astrophysics and more
precisely to the statistical analysis of mass of stars and of diameters of
asteroids. In particular a comparison between the usual Pareto distribution and
its truncated version is presented. Finally a possible physical mechanism that
produces Pareto tails for the distribution of the masses of stars is suggested.Comment: 10 pages 6 figure
A note on maximum likelihood estimation of a Pareto mixture
In this paper we study Maximum Likelihood Estimation of the parameters of a Pareto mixture. Application of standard techniques to a mixture of Pareto is problematic. For this reason we develop two alternative algorithms. The first one is the Simulated Annealing and the second one is based on Cross-Entropy minimization. The Pareto distribution is a commonly used model for heavy-tailed data. It is a two-parameter distribution whose shape parameter determines the degree of heaviness of the tail, so that it can be adapted to data with different features. This work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Losses below an unknown threshold are discarded, so that the observed data are truncated. The thresholds used in the two business lines are unknown. Thus, under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto where all the parameters have to be estimated.
Characterizations of Continuous Distributions by Truncated Moment
A probability distribution can be characterized through various methods. In this paper, some new characterizations of continuous distribution by truncated moment have been established. We have considered standard normal distribution, Studentâs t, exponentiated exponential, power function, Pareto, and Weibull distributions and characterized them by truncated moment
Modelling Censored Losses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme Value Distributions
In risk analysis, a global fit that appropriately captures the body and the
tail of the distribution of losses is essential. Modelling the whole range of
the losses using a standard distribution is usually very hard and often
impossible due to the specific characteristics of the body and the tail of the
loss distribution. A possible solution is to combine two distributions in a
splicing model: a light-tailed distribution for the body which covers light and
moderate losses, and a heavy-tailed distribution for the tail to capture large
losses. We propose a splicing model with a mixed Erlang (ME) distribution for
the body and a Pareto distribution for the tail. This combines the flexibility
of the ME distribution with the ability of the Pareto distribution to model
extreme values. We extend our splicing approach for censored and/or truncated
data. Relevant examples of such data can be found in financial risk analysis.
We illustrate the flexibility of this splicing model using practical examples
from risk measurement
Zero-inflated truncated generalized Pareto distribution for the analysis of radio audience data
Extreme value data with a high clump-at-zero occur in many domains. Moreover,
it might happen that the observed data are either truncated below a given
threshold and/or might not be reliable enough below that threshold because of
the recording devices. These situations occur, in particular, with radio
audience data measured using personal meters that record environmental noise
every minute, that is then matched to one of the several radio programs. There
are therefore genuine zeros for respondents not listening to the radio, but
also zeros corresponding to real listeners for whom the match between the
recorded noise and the radio program could not be achieved. Since radio
audiences are important for radio broadcasters in order, for example, to
determine advertisement price policies, possibly according to the type of
audience at different time points, it is essential to be able to explain not
only the probability of listening to a radio but also the average time spent
listening to the radio by means of the characteristics of the listeners. In
this paper we propose a generalized linear model for zero-inflated truncated
Pareto distribution (ZITPo) that we use to fit audience radio data. Because it
is based on the generalized Pareto distribution, the ZITPo model has nice
properties such as model invariance to the choice of the threshold and from
which a natural residual measure can be derived to assess the model fit to the
data. From a general formulation of the most popular models for zero-inflated
data, we derive our model by considering successively the truncated case, the
generalized Pareto distribution and then the inclusion of covariates to explain
the nonzero proportion of listeners and their average listening time. By means
of simulations, we study the performance of the maximum likelihood estimator
(and derived inference) and use the model to fully analyze the audience data of
a radio station in a certain area of Switzerland.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS358 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stochastic Lotka-Volterra Systems of Competing Auto-Catalytic Agents Lead Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market Returns, Clustered Volatility, Booms and Craches
We give a microscopic representation of the stock-market in which the
microscopic agents are the individual traders and their capital. Their basic
dynamics consists in the auto-catalysis of the individual capital and in the
global competition/cooperation between the agents mediated by the total wealth
invested in the stock (which we identify with the stock-index). We show that
such systems lead generically to (truncated) Pareto power-law distribution of
the individual wealth. This, in turn, leads to intermittent market (short time)
returns parametrized by a (truncated) Levy distribution. We relate the
truncation in the Levy distribution to the (truncation in the Pareto Power Law
i.e. to the) fact that at each moment no trader can own more than the current
total wealth invested in the stock. In the cases where the system is dominated
by the largest traders, the dynamics looks similar to noisy low-dimensional
chaos. By introducing traders memory and/or feedback between individual and
collective wealth fluctuations (the later identified with the stock returns),
one obtains clustered "volatility" as well as market booms and crashes. The
basic feedback loop consists in: - computing the market price of the stock as
the sum of the individual wealths invested in the stock by the traders and -
determining the time variation of the individual trader wealth as his/her
previous wealth multiplied by the stock return (i.e. the variation of the stock
price).Comment: 13 Pages, no figure
Cubic Rank Transmuted Modified Burr III Pareto Distribution: Development, Properties, Characterizations and Applications
In this paper, a flexible lifetime distribution called Cubic rank transmuted modified Burr III-Pareto (CRTMBIII-P) is developed on the basis of the cubic ranking transmutation map. The density function of CRTMBIII-P is arc, exponential, left-skewed, right-skewed and symmetrical shaped. Descriptive measures such as moments, incomplete moments, inequality measures, residual life function and reliability measures are theoretically established. The CRTMBIII-P distribution is characterized via ratio of truncated moments. Parameters of the CRTMBIII-P distribution are estimated using maximum likelihood method. The simulation study for the performance of the maximum likelihood estimates (MLEs) of the parameters of the CRTMBIII-P distribution is carried out. The potentiality of CRTMBIII-P distribution is demonstrated via its application to the real data sets: tensile strength of carbon fibers and strengths of glass fibers. Goodness of fit of this distribution through different methods is studied
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