Metrika [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1435-926X - ISSN (Online) 0026-1335 Published by Springer-Verlag [2210 journals] [SJR: 0.839] [H-I: 22] |
- Modified maximum spacings method for generalized extreme value
distribution and applications in real data analysis- Abstract: Abstract
This paper analyzes weekly closing price data of the S&P 500 stock index and electrical insulation element lifetimes data based on generalized extreme value distribution. A new estimation method, modified maximum spacings (MSP) method, is proposed and obtained by using interior penalty function algorithm. The standard error of the proposed method is calculated through Bootstrap method. The asymptotic properties of the modified MSP estimators are discussed. Some simulations are performed, which show that the proposed method is not only available for the whole shape parameter space, but is also of high efficiency. The benchmark risk index, value at risk (VaR), is evaluated according to the proposed method, and the confidence interval of VaR is also calculated through Bootstrap method. Finally, the results are compared with those derived by empirical calculation and some existing methods.
PubDate: 2014-10-01
- Abstract: Abstract
This paper analyzes weekly closing price data of the S&P 500 stock index and electrical insulation element lifetimes data based on generalized extreme value distribution. A new estimation method, modified maximum spacings (MSP) method, is proposed and obtained by using interior penalty function algorithm. The standard error of the proposed method is calculated through Bootstrap method. The asymptotic properties of the modified MSP estimators are discussed. Some simulations are performed, which show that the proposed method is not only available for the whole shape parameter space, but is also of high efficiency. The benchmark risk index, value at risk (VaR), is evaluated according to the proposed method, and the confidence interval of VaR is also calculated through Bootstrap method. Finally, the results are compared with those derived by empirical calculation and some existing methods.
- On sooner and later waiting time distributions associated with simple
patterns in a sequence of bivariate trials- Abstract: Abstract
In this article, we study sooner/later waiting time problems for simple patterns in a sequence of bivariate trials. The double generating functions of the sooner/later waiting times for the simple patterns are expressed in terms of the double generating functions of the numbers of occurrences of the simple patterns. Effective computational tools are developed for the evaluation of the waiting time distributions along with some examples. The results presented here provide perspectives on the waiting time problems arising from bivariate trials and extend a framework for studying the exact distributions of patterns. Finally, some examples are given in order to illustrate how our theoretical results are employed for the investigation of the waiting time problems for simple patterns.
PubDate: 2014-10-01
- Abstract: Abstract
In this article, we study sooner/later waiting time problems for simple patterns in a sequence of bivariate trials. The double generating functions of the sooner/later waiting times for the simple patterns are expressed in terms of the double generating functions of the numbers of occurrences of the simple patterns. Effective computational tools are developed for the evaluation of the waiting time distributions along with some examples. The results presented here provide perspectives on the waiting time problems arising from bivariate trials and extend a framework for studying the exact distributions of patterns. Finally, some examples are given in order to illustrate how our theoretical results are employed for the investigation of the waiting time problems for simple patterns.
- Asymptotic behaviour of near-maxima of Gaussian sequences
- Abstract: Abstract
Let
\((X_1,X_2,\ldots ,X_n)\)
be a Gaussian random vector with a common correlation coefficient
\(\rho _n,\,0\le \rho _n<1\)
, and let
\(M_n= \max (X_1,\ldots , X_n),\,n\ge 1\)
. For any given
\(a>0\)
, define
\(T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)\)
and
\(S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1\)
. In this paper, we obtain the limit distributions of
\((K_n(a))\)
and
\((S_n(a))\)
, under the assumption that
\(\rho _n\rightarrow \rho \)
as
\(n\rightarrow \infty ,\)
for some
\(\rho \in [0,1)\)
.
PubDate: 2014-10-01
- Abstract: Abstract
Let
\((X_1,X_2,\ldots ,X_n)\)
be a Gaussian random vector with a common correlation coefficient
\(\rho _n,\,0\le \rho _n<1\)
, and let
\(M_n= \max (X_1,\ldots , X_n),\,n\ge 1\)
. For any given
\(a>0\)
, define
\(T_n(a)= \left\{ j,\,1\le j\le n,\,X_j\in (M_n-a,\,M_n]\right\} ,\,K_n(a)= \#T_n(a)\)
and
\(S_n(a)=\sum \nolimits _{j\in T_n(a)}X_j,\,n\ge 1\)
. In this paper, we obtain the limit distributions of
\((K_n(a))\)
and
\((S_n(a))\)
, under the assumption that
\(\rho _n\rightarrow \rho \)
as
\(n\rightarrow \infty ,\)
for some
\(\rho \in [0,1)\)
.
- Empirical likelihood for high-dimensional linear regression models
- Abstract: Abstract
High-dimensional data are becoming prevalent, and many new methodologies and accompanying theories for high-dimensional data analysis have emerged in response. Empirical likelihood, as a classical nonparametric method of statistical inference, has proved to possess many good features. In this paper, our focus is to investigate the asymptotic behavior of empirical likelihood for regression coefficients in high-dimensional linear models. We give regularity conditions under which the standard normal calibration of empirical likelihood is valid in high dimensions. Both random and fixed designs are considered. Simulation studies are conducted to check the finite sample performance.
PubDate: 2014-10-01
- Abstract: Abstract
High-dimensional data are becoming prevalent, and many new methodologies and accompanying theories for high-dimensional data analysis have emerged in response. Empirical likelihood, as a classical nonparametric method of statistical inference, has proved to possess many good features. In this paper, our focus is to investigate the asymptotic behavior of empirical likelihood for regression coefficients in high-dimensional linear models. We give regularity conditions under which the standard normal calibration of empirical likelihood is valid in high dimensions. Both random and fixed designs are considered. Simulation studies are conducted to check the finite sample performance.
- Characterizations of bivariate distributions using concomitants of record
values- Abstract: Abstract
In this paper, we consider a family of bivariate distributions which is a generalization of the Morgenstern family of bivariate distributions. We have derived some properties of concomitants of record values which characterize this generalized class of distributions. The role of concomitants of record values in the unique determination of the parent bivariate distribution has been established. We have also derived properties of concomitants of record values which characterize each of the following families viz Morgenstern family, bivariate Pareto family and a generalized Gumbel’s family of bivariate distributions. Some applications of the characterization results are discussed and important conclusions based on the characterization results are drawn.
PubDate: 2014-10-01
- Abstract: Abstract
In this paper, we consider a family of bivariate distributions which is a generalization of the Morgenstern family of bivariate distributions. We have derived some properties of concomitants of record values which characterize this generalized class of distributions. The role of concomitants of record values in the unique determination of the parent bivariate distribution has been established. We have also derived properties of concomitants of record values which characterize each of the following families viz Morgenstern family, bivariate Pareto family and a generalized Gumbel’s family of bivariate distributions. Some applications of the characterization results are discussed and important conclusions based on the characterization results are drawn.
- Second order longitudinal dynamic models with covariates: estimation and
forecasting- Abstract: Abstract
In this paper, we propose an extension to the first-order branching process with immigration in the presence of fixed covariates and unobservable random effects. The extension permits the possibility that individuals from the second generation of the process may contribute to the total number of offsprings at time
\(t\)
by producing offsprings of their own. We will study the basic properties of the second order process and discuss a generalized quasilikelihood (GQL) estimation of the mean and variance parameters and the generalized method of moments estimation of the correlation parameters. We will discuss the asymptotic distribution of the GQL estimator by first deriving the influence curve of the estimator. For the fixed effects model we shall derive a forecasting function and the variance of the forecast error. The performance of the proposed estimators and forecasts will be examined through a simulation study.
PubDate: 2014-10-01
- Abstract: Abstract
In this paper, we propose an extension to the first-order branching process with immigration in the presence of fixed covariates and unobservable random effects. The extension permits the possibility that individuals from the second generation of the process may contribute to the total number of offsprings at time
\(t\)
by producing offsprings of their own. We will study the basic properties of the second order process and discuss a generalized quasilikelihood (GQL) estimation of the mean and variance parameters and the generalized method of moments estimation of the correlation parameters. We will discuss the asymptotic distribution of the GQL estimator by first deriving the influence curve of the estimator. For the fixed effects model we shall derive a forecasting function and the variance of the forecast error. The performance of the proposed estimators and forecasts will be examined through a simulation study.
- Generalized measures of information for truncated random variables
- Abstract: Abstract
In the present work we focus on the generalization of two types of measures of information namely divergence-type and entropy-type. Kullback–Leibler discrimination measure and Shannon entropy have been considered in this context for truncated random variables. We propose a generalized discrimination measure between two residual and past lifetime distributions along a similar line of Varma’s entropy. Some properties of this measure are studied and a characterization of the proportional (reversed) hazards model is given. Furthermore, Shannon entropy is generalized on the basis of Varma’s entropy for past lifetime distribution. These results generalize and enhance the related existing results that are developed based on Kullback–Leibler information and Shannon entropy.
PubDate: 2014-08-29
- Abstract: Abstract
In the present work we focus on the generalization of two types of measures of information namely divergence-type and entropy-type. Kullback–Leibler discrimination measure and Shannon entropy have been considered in this context for truncated random variables. We propose a generalized discrimination measure between two residual and past lifetime distributions along a similar line of Varma’s entropy. Some properties of this measure are studied and a characterization of the proportional (reversed) hazards model is given. Furthermore, Shannon entropy is generalized on the basis of Varma’s entropy for past lifetime distribution. These results generalize and enhance the related existing results that are developed based on Kullback–Leibler information and Shannon entropy.
- A note on relationships between some univariate stochastic orders and the
corresponding joint stochastic orders- Abstract: Abstract
In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, 1991). These characterizations gave rise to new stochastic comparisons, commonly called joint stochastic orders, which are equivalent to the original ones under assumption of independence, but are different whenever the variables to be compared are dependent. In this note we provide sufficient conditions on the survival copula describing the dependence among the compared variables such that the standard stochastic orders imply the corresponding joint stochastic orders, and viceversa. Also, simple conditions for joint stochastic orders between the components of random vectors defined through multivariate frailty models are provided.
PubDate: 2014-08-23
- Abstract: Abstract
In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, 1991). These characterizations gave rise to new stochastic comparisons, commonly called joint stochastic orders, which are equivalent to the original ones under assumption of independence, but are different whenever the variables to be compared are dependent. In this note we provide sufficient conditions on the survival copula describing the dependence among the compared variables such that the standard stochastic orders imply the corresponding joint stochastic orders, and viceversa. Also, simple conditions for joint stochastic orders between the components of random vectors defined through multivariate frailty models are provided.
- Limit results for concomitants of order statistics
- Abstract: Abstract
In this paper, we discuss the concomitants of order statistics. We study asymptotic properties of the concomitants of largest order statistics and we pay special attention to strong limit results. The strong limit results of this work are derived by applying the Borel–Cantelli lemma and some of its recent generalizations. The theoretical results of this paper are illustrated with examples.
PubDate: 2014-08-21
- Abstract: Abstract
In this paper, we discuss the concomitants of order statistics. We study asymptotic properties of the concomitants of largest order statistics and we pay special attention to strong limit results. The strong limit results of this work are derived by applying the Borel–Cantelli lemma and some of its recent generalizations. The theoretical results of this paper are illustrated with examples.
- Inference for types and structured families of commutative orthogonal
block structures- Abstract: Abstract
Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.
PubDate: 2014-08-19
- Abstract: Abstract
Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.
- Construction and selection of the optimal balanced blocked definitive
screening design- Abstract: Abstract
The definitive screening (DS) design was proposed recently. This new class of three-level designs provides efficient estimates of main effects that are unaliased with any second-order effects. For practical use, we further study the optimal scheme for blocking DS designs. We propose a construction method and utilize the blocked count function to select the optimal balanced blocked definitive screening (BBDS) design in terms of generalized minimum aberration. The proposed BBDS design not only inherits properties of the original DS design but also guarantees that main effects are unconfounded by block effects. Besides that, it has minimum run size and is a saturated design for estimating the mean, all block effects, all main effects, and all quadratic effects.
PubDate: 2014-08-14
- Abstract: Abstract
The definitive screening (DS) design was proposed recently. This new class of three-level designs provides efficient estimates of main effects that are unaliased with any second-order effects. For practical use, we further study the optimal scheme for blocking DS designs. We propose a construction method and utilize the blocked count function to select the optimal balanced blocked definitive screening (BBDS) design in terms of generalized minimum aberration. The proposed BBDS design not only inherits properties of the original DS design but also guarantees that main effects are unconfounded by block effects. Besides that, it has minimum run size and is a saturated design for estimating the mean, all block effects, all main effects, and all quadratic effects.
- Circular block bootstrap for coefficients of autocovariance function of
almost periodically correlated time series- Abstract: Abstract
In the paper the consistency of the circular block bootstrap for the coefficients of the autocovariance function of almost periodically correlated time series is proved. The pointwise and the simultaneous bootstrap equal-tailed confidence intervals for these coefficients are constructed. Application of the results to detect the second-order significant frequencies is provided. The simulation and real data examples are also presented.
PubDate: 2014-08-09
- Abstract: Abstract
In the paper the consistency of the circular block bootstrap for the coefficients of the autocovariance function of almost periodically correlated time series is proved. The pointwise and the simultaneous bootstrap equal-tailed confidence intervals for these coefficients are constructed. Application of the results to detect the second-order significant frequencies is provided. The simulation and real data examples are also presented.
- Erratum to: On optimal designs for censored data
- PubDate: 2014-08-09
- PubDate: 2014-08-09
- Testing for the bivariate Poisson distribution
- Abstract: Abstract
This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramér–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.
PubDate: 2014-08-01
- Abstract: Abstract
This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramér–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.
- Some results on constructing general minimum lower order confounding
format-t-e-x">\(2^{n-m}\) designs for
format-t-e-x">\(n\le 2^{n-m-2}\)- Abstract: Abstract
Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC
\(2^{n-m}\)
designs with
\(N/4+1\le n\le N-1\)
were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where
\(N=2^{n-m}\)
is run number and
\(n\)
is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC
\(2^{n-m}\)
designs respectively with the three parameter cases of
\(n\le N-1\)
: (i)
\(m\le 4\)
, (ii)
\(m\ge 5\)
and
\(n=(2^m-1)u+r\)
for
\(u>0\)
and
\(r=0,1,2\)
, and (iii)
\(m\ge 5\)
and
\(n=(2^m-1)u+r\)
for
\(u\ge 0\)
and
\(r=2^m-3,2^m-2\)
.
PubDate: 2014-08-01
- Abstract: Abstract
Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC
\(2^{n-m}\)
designs with
\(N/4+1\le n\le N-1\)
were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where
\(N=2^{n-m}\)
is run number and
\(n\)
is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC
\(2^{n-m}\)
designs respectively with the three parameter cases of
\(n\le N-1\)
: (i)
\(m\le 4\)
, (ii)
\(m\ge 5\)
and
\(n=(2^m-1)u+r\)
for
\(u>0\)
and
\(r=0,1,2\)
, and (iii)
\(m\ge 5\)
and
\(n=(2^m-1)u+r\)
for
\(u\ge 0\)
and
\(r=2^m-3,2^m-2\)
.
- Shrinkage estimation for the mean of the inverse Gaussian population
- Abstract: Abstract
We consider improved estimation strategies for a two-parameter inverse Gaussian distribution and use a shrinkage technique for the estimation of the mean parameter. In this context, two new shrinkage estimators are suggested and demonstrated to dominate the classical estimator under the quadratic risk with realistic conditions. Furthermore, based on our shrinkage strategy, a new estimator is proposed for the common mean of several inverse Gaussian distributions, which uniformly dominates the Graybill–Deal type unbiased estimator. The performance of the suggested estimators is examined by using simulated data and our shrinkage strategies are shown to work well. The estimation methods and results are illustrated by two empirical examples.
PubDate: 2014-08-01
- Abstract: Abstract
We consider improved estimation strategies for a two-parameter inverse Gaussian distribution and use a shrinkage technique for the estimation of the mean parameter. In this context, two new shrinkage estimators are suggested and demonstrated to dominate the classical estimator under the quadratic risk with realistic conditions. Furthermore, based on our shrinkage strategy, a new estimator is proposed for the common mean of several inverse Gaussian distributions, which uniformly dominates the Graybill–Deal type unbiased estimator. The performance of the suggested estimators is examined by using simulated data and our shrinkage strategies are shown to work well. The estimation methods and results are illustrated by two empirical examples.
- Testing equality of shape parameters in several inverse Gaussian
populations- Abstract: Abstract
Due to the strikingly resemblance to the normal theory and inference methods, the inverse Gaussian (IG) distribution is commonly applied to model positive and right-skewed data. As the shape parameter in the IG distribution is greatly related to other important quantities such as the mean, skewness, kurtosis and the coefficient of variation, it plays an important role in distribution theory. This paper focuses on testing the equality of shape parameters in several inverse Gaussian distributions. Three tests are suggested: the exact generalized inference-based test, the asymptotic test and a test that is based on parametric bootstrap approximation. Simulation studies are undertaken to examine the performances of the these methods, and three real data examples are analyzed for illustration.
PubDate: 2014-08-01
- Abstract: Abstract
Due to the strikingly resemblance to the normal theory and inference methods, the inverse Gaussian (IG) distribution is commonly applied to model positive and right-skewed data. As the shape parameter in the IG distribution is greatly related to other important quantities such as the mean, skewness, kurtosis and the coefficient of variation, it plays an important role in distribution theory. This paper focuses on testing the equality of shape parameters in several inverse Gaussian distributions. Three tests are suggested: the exact generalized inference-based test, the asymptotic test and a test that is based on parametric bootstrap approximation. Simulation studies are undertaken to examine the performances of the these methods, and three real data examples are analyzed for illustration.
- Optimal and robust designs for trigonometric regression models
- Abstract: Abstract
This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.
PubDate: 2014-08-01
- Abstract: Abstract
This article presents discussions on the optimal and robust designs for trigonometric regression models under different optimality criteria. First, we investigate the classical Q-optimal designs for estimating the response function in a full trigonometric regression model with a given order. The equivalencies of Q-, A-, and G-optimal designs for trigonometric regression in general are also articulated. Second, we study minimax designs and their implementation in the case of trigonometric approximation under Q-, A-, and D-optimality. Then, We indicate the existence of the symmetric designs that are D-optimal minimax designs for general trigonometric regression models, and prove the existence of the symmetric designs that are Q- or A-optimal minimax designs for two particular trigonometric regression models under certain conditions.
- On the maxima of heterogeneous gamma variables with different shape and
scale parameters- Abstract: Abstract
In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let
\(X_{1}\)
and
\(X_{2}\)
be two independent gamma random variables with
\(X_{i}\)
having shape parameter
\(r_{i}>0\)
and scale parameter
\(\lambda _{i}\)
,
\(i=1,2\)
, and let
\(X^{*}_{1}\)
and
\(X^{*}_{2}\)
be another set of independent gamma random variables with
\(X^{*}_{i}\)
having shape parameter
\(r_{i}^{*}>0\)
and scale parameter
\(\lambda _{i}^{*}\)
,
\(i=1,2\)
. Denote by
\(X_{2:2}\)
and
\(X^{*}_{2:2}\)
the corresponding maxima, respectively. It is proved that, among others, if
\((r_{1},r_{2})\)
majorize
\((r_{1}^{*},r_{2}^{*})\)
and
\((\lambda _{1},\lambda _{2})\)
weakly majorize
\((\lambda _{1}^{*},\lambda _{2}^{*})\)
, then
\(X_{2:2}\)
is stochastically larger that
\(X^{*}_{2:2}\)
in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.
PubDate: 2014-08-01
- Abstract: Abstract
In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let
\(X_{1}\)
and
\(X_{2}\)
be two independent gamma random variables with
\(X_{i}\)
having shape parameter
\(r_{i}>0\)
and scale parameter
\(\lambda _{i}\)
,
\(i=1,2\)
, and let
\(X^{*}_{1}\)
and
\(X^{*}_{2}\)
be another set of independent gamma random variables with
\(X^{*}_{i}\)
having shape parameter
\(r_{i}^{*}>0\)
and scale parameter
\(\lambda _{i}^{*}\)
,
\(i=1,2\)
. Denote by
\(X_{2:2}\)
and
\(X^{*}_{2:2}\)
the corresponding maxima, respectively. It is proved that, among others, if
\((r_{1},r_{2})\)
majorize
\((r_{1}^{*},r_{2}^{*})\)
and
\((\lambda _{1},\lambda _{2})\)
weakly majorize
\((\lambda _{1}^{*},\lambda _{2}^{*})\)
, then
\(X_{2:2}\)
is stochastically larger that
\(X^{*}_{2:2}\)
in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.
- Asymptotic properties of the number of near minimum-concomitant
observations in the case of progressive type-II censoring- Abstract: Abstract
In this paper, we study the number of near minimum-concomitant observations for Progressively Type-II Censored Order Statistics (PCOS). We first define the concomitants of PCOS and the number of near minimum-concomitant observations. We then investigate distributional and asymptotic properties of these random variables. Finally, we propose simulation techniques for generating the concomitants of PCOS.
PubDate: 2014-07-11
- Abstract: Abstract
In this paper, we study the number of near minimum-concomitant observations for Progressively Type-II Censored Order Statistics (PCOS). We first define the concomitants of PCOS and the number of near minimum-concomitant observations. We then investigate distributional and asymptotic properties of these random variables. Finally, we propose simulation techniques for generating the concomitants of PCOS.