1
1726-4057
Iranian Statistical Society
127
60: Probability theory and stochastic processes
The Local Limit Theorem: A Historical Perspective
McDonald
D. R.
1
11
2005
4
2
73
86
22
10
2011
12
09
2015
The local limit theorem describes how the density of a
sum of random variables follows the normal curve. However the local
limit theorem is often seen as a curiosity of no particular importance
when compared with the central limit theorem. Nevertheless the local
limit theorem came first and is in fact associated with the foundation
of probability theory by Blaise Pascal and Pierre de Fermat and was
originally formalized by Jakob Bernoulli, Abraham DeMoivre and
Pierre-Simon Laplace.
Here we describe the historical roots of the local limit theorem.
We describe how it was supplanted by the central limit theorem in
applications. Then we review the revival started by B. V. Gnedenko
and we describe modern developments.
128
60: Probability theory and stochastic processes
Estimation Based on an Appropriate Choice of Loss Function
Ganjali
M.
Shafie
K.
1
11
2005
4
2
87
95
22
10
2011
12
09
2015
Some examples of absurd uniformly minimum variance
unbiased estimators are discussed. Two reasons, argued in the literature,
for having such estimators are lack of enough information in the
available data and property of unbiasedness. In this paper, accepting
both of these views, we show that an appropriate choice of loss function
using a general concept of unbiasedness leads to risk unbiased,
admissible and reasonable estimators. For this we extend the Rao-
Blackwell theorem using a new way of defining unbiased estimator.
129
60: Probability theory and stochastic processes
Wavelet Based Estimation of the Derivatives of a Density for m-Dependent Random Variables
Chaubey
Yogendra P.
Doosti
Hassan
1
11
2005
4
2
97
105
22
10
2011
12
09
2015
Here, we propose a method of estimation of the derivatives
of probability density based wavelets methods for a sequence
of m−dependent random variables with a common one-dimensional
probability density function and obtain an upper bound on Lp-losses
for the such estimators.
130
60: Probability theory and stochastic processes
A Note on the Strong Law of Large Numbers
Fakoor
V.
Azarnoosh
H. A.
1
11
2005
4
2
107
111
22
10
2011
12
09
2015
Petrov (1996) proved the connection between general
moment conditions and the applicability of the strong law of large
numbers to a sequence of pairwise independent and identically distributed
random variables. This note examines this connection to a
sequence of pairwise negative quadrant dependent (NQD) and identically
distributed random variables. As a consequence of the main theorem
(Theorem 2.1), we arrive at an improvement of Marcinkiewicz–
Zygmund theorem for pairwise NQD random variables.
131
60: Probability theory and stochastic processes
Bayesian Estimation for the Pareto Income Distribution under Asymmetric LINEX Loss Function
Ertefaie
Ashkan
Parsian
Ahmad
1
11
2005
4
2
113
133
22
10
2011
12
09
2015
The use of the Pareto distribution as a model for various
socio-economic phenomena dates back to the late nineteenth century.
In this paper, after some necessary preliminary results we deal with
Bayes estimation of some of the parameters of interest under an asymmetric
LINEX loss function, using suitable choice of priors when the
scale parameter is known and unknown. Results of a Monte Carlo
simulation study conducted to evaluate the performances of these estimators
compared to the MLE’s and MME’s in terms of estimated
risks under LINEX loss function.