167 research outputs found
Multivariate Geometric Skew-Normal Distribution
Azzalini (1985) introduced a skew-normal distribution of which normal
distribution is a special case. Recently Kundu (2014) introduced a geometric
skew-normal distribution and showed that it has certain advantages over
Azzalini's skew-normal distribution. In this paper we discuss about the
multivariate geometric skew-normal distribution. It can be used as an
alternative to Azzalini's skew normal distribution. We discuss different
properties of the proposed distribution. It is observed that the joint
probability density function of the multivariate geometric skew normal
distribution can take variety of shapes. Several characterization results have
been established. Generation from a multivariate geometric skew normal
distribution is quite simple, hence the simulation experiments can be performed
quite easily. The maximum likelihood estimators of the unknown parameters can
be obtained quite conveniently using expectation maximization (EM) algorithm.
We perform some simulation experiments and it is observed that the performances
of the proposed EM algorithm are quite satisfactory. Further, the analyses of
two data sets have been performed, and it is observed that the proposed methods
and the model work very well
On Generalized Progressive Hybrid Censoring in presence of competing risks
The progressive Type-II hybrid censoring scheme introduced by Kundu and
Joarder (\textit{Computational Statistics and Data Analysis}, 2509-2528, 2006),
has received some attention in the last few years. One major drawback of this
censoring scheme is that very few observations (even no observation at all) may
be observed at the end of the experiment. To overcome this problem, Cho, Sun
and Lee (\textit{Statistical Methodology}, 23, 18-34, 2015) recently introduced
generalized progressive censoring which ensures to get a pre specified number
of failures. In this paper we analyze generalized progressive censored data in
presence of competing risks. For brevity we have considered only two competing
causes of failures, and it is assumed that the lifetime of the competing causes
follow one parameter exponential distributions with different scale parameters.
We obtain the maximum likelihood estimators of the unknown parameters and also
provide their exact distributions. Based on the exact distributions of the
maximum likelihood estimators exact confidence intervals can be obtained.
Asymptotic and bootstrap confidence intervals are also provided for comparison
purposes. We further consider the Bayesian analysis of the unknown parameters
under a very flexible Beta-Gamma prior. We provide the Bayes estimates and the
associated credible intervals of the unknown parameters based on the above
priors. We present extensive simulation results to see the effectiveness of the
proposed method and finally one real data set is analyzed for illustrative
purpose
Univariate and Bivariate Geometric Discrete Generalized Exponential Distributions
Marshall and Olkin (1997, Biometrika, 84, 641 - 652) introduced a very
powerful method to introduce an additional parameter to a class of continuous
distribution functions and hence it brings more flexibility to the model. They
have demonstrated their method for the exponential and Weibull classes. In the
same paper they have briefly indicated regarding its bivariate extension. The
main aim of this paper is to introduce the same method, for the first time, to
the class of discrete generalized exponential distributions both for the
univariate and bivariate cases. We investigate several properties of the
proposed univariate and bivariate classes. The univariate class has three
parameters, whereas the bivariate class has five parameters. It is observed
that depending on the parameter values the univariate class can be both zero
inflated as well as heavy tailed. We propose to use EM algorithm to estimate
the unknown parameters. Small simulation experiments have been performed to see
the effectiveness of the proposed EM algorithm, and a bivariate data set has
been analyzed and it is observed that the proposed models and the EM algorithm
work quite well in practice.Comment: arXiv admin note: text overlap with arXiv:1701.0356
Estimating the fundamental frequency using modified Newton-Raphson algorithm
In this paper, we propose a modified Newton-Raphson algorithm to estimate the
frequency parameter in the fundamental frequency model in presence of an
additive stationary error. The proposed estimator is super efficient in nature
in the sense that its asymptotic variance is less than the asymptotic variance
of the least squares estimator. With a proper step factor modification, the
proposed modified Newton-Raphson algorithm produces an estimator with the rate
, the same rate as the least squares estimator.
Numerical experiments are performed for different sample sizes, different error
variances and for different models. For illustrative purposes, two real data
sets are analyzed using the fundamental frequency model and the estimators are
obtained using the proposed algorithm. It is observed the model and the
proposed algorithm work quite well in both cases
Point and Interval Estimation of Weibull Parameters Based on Joint Progressively Censored Data
The analysis of progressively censored data has received considerable
attention in the last few years. In this paper we consider the joint
progressive censoring scheme for two populations. It is assumed that the
lifetime distribution of the items from the two populations follow Weibull
distribution with the same shape but different scale parameters. Based on the
joint progressive censoring scheme first we consider the maximum likelihood
estimators of the unknown parameters whenever they exist. We provide the
Bayesian inferences of the unknown parameters under a fairly general priors on
the shape and scale parameters. The Bayes estimators and the associated
credible intervals cannot be obtained in closed form, and we propose to use the
importance sampling technique to compute the same. Further, we consider the
problem when it is known apriori that the expected lifetime of one population
is smaller than the other. We provide the order restricted classical and
Bayesian inferences of the unknown parameters. Monte Carlo simulations are
performed to observe the performances of the different estimators and the
associated confidence and credible intervals. One real data set has been
analyzed for illustrative purpose
Interval Estimation of the Unknown Exponential Parameter Based on Time Truncated Data
In this paper we consider the statistical inference of the unknown parameter
of an exponential distribution based on the time truncated data. The time
truncated data occurs quite often in the reliability analysis for type-I or
hybrid censoring cases. All the results available today are based on the
conditional argument that at least one failure occurs during the experiment. In
this paper we provide some inferential results based on the unconditional
argument. We extend the results for some two-parameter distributions also
On a General Class of Discrete Bivariate Distributions
In this paper we develop a very general class of bivariate discrete
distributions. The basic idea is very simple. The marginals are obtained by
taking the random geometric sum of a baseline distribution function. The
proposed class of distributions is a very flexible class of distributions in
the sense the marginals can take variety of shapes. It can be multimodal as
well as heavy tailed also. It can be both over dispersed as well as under
dispersed. Moreover, the correlation can be of a wide range. We discuss
different properties of the proposes class of bivariate distributions. The
proposed distribution has some interesting physical interpretations also.
Further, we consider two specific base line distributions namely; Poisson and
negative binomial distributions for illustrative purposes. Both of them are
infinitely divisible. The maximum likelihood estimators of the unknown
parameters cannot be obtained in closed form. They can be obtained by solving
three and five dimensional non-linear optimizations problems, respectively. To
avoid that we propose to use the method of moment estimators and they can be
obtained quite conveniently. The analyses of two real data sets have been
performed to show the effectiveness of the proposed class of models. Finally,
we discuss some open problems and conclude the paper
Order Restricted Bayesian Analysis of a Simple Step Stress Model
In this article we consider a simple step stress set up under the cumulative
exposure model assumption. At each stress level the lifetime distribution of
the experimental units are assumed to follow the generalized exponential
distribution. We provide the order restricted Bayesian inference of the model
parameters by considering the fact that the expected lifetime of the
experimental units are larger in lower stress level. Analysis and the related
results are extended to different censoring schemes also. The Bayes estimates
and the associated credible intervals of the unknown parameters are constructed
using importance sampling technique. We perform extensive simulation
experiments both for the complete and censored samples to see the performances
of the proposed estimators. We analyze two simulated and one real data sets for
illustrative purposes. An optimal value of the stress changing time is obtained
by minimizing the total posterior coefficient of variations of the unknown
parameters
Chirp-like model and its parameter estimation
We propose a chirp-like signal model as an alternative to a chirp model and a
generalisation of the sinusoidal model, which is a fundamental model in the
statistical signal processing literature. It is observed that the proposed
model can be arbitrarily close to the chirp model. The propounded model is
similar to a chirp model in the sense that here also the frequency changes
linearly with time. However, the parameter estimation of a chirp-like model is
simpler compared to a chirp model. In this paper, we consider the least squares
and the sequential least squares estimation procedures and study the asymptotic
properties of these proposed estimators. These asymptotic results are
corroborated through simulation studies and analysis of four speech signal data
sets have been performed to see the effectiveness of the proposed model, and
the results are quite encouraging.Comment: 33 pages, 5 figures, 10 table
Estimation of P(X > Y ) for Weibull distribution based on hybrid censored samples
A Hybrid censoring scheme is mixture of Type-I and Type-II censoring schemes.
Based on hybrid censored samples, this paper deals with the in- ference on R =
P(X > Y ), when X and Y are two independent Weibull distributions with
different scale parameters, but having the same shape pa- rameter. The maximum
likelihood estimator (MLE), and the approximate MLE (AMLE) of R are obtained.
The asymptotic distribution of the maxi- mum likelihood estimator of R is
obtained. Based on the asymptotic distribu- tion, the confidence interval of R
can be derived. Two bootstrap confidence intervals are also proposed. We
consider the Bayesian estimate of R, and propose the corresponding credible
interval for R. Monte Carlo simulations are performed to compare the different
proposed methods. Analysis of a real data set has also been presented for
illustrative purposes
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