Metrika [SJR: 0.943] [H-I: 25] [5 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1435-926X - ISSN (Online) 0026-1335 Published by Springer-Verlag [2302 journals] |
- 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: 2015-05-01
- 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.
- Robust minimax Stein estimation under invariant data-based loss for
spherically and elliptically symmetric distributions- Abstract: Abstract
From an observable
\((X,U)\)
in
\(\mathbb R^p \times \mathbb R^k\)
, we consider estimation of an unknown location parameter
\(\theta \in \mathbb R^p\)
under two distributional settings: the density of
\((X,U)\)
is spherically symmetric with an unknown scale parameter
\(\sigma \)
and is ellipically symmetric with an unknown covariance matrix
\(\Sigma \)
. Evaluation of estimators of
\(\theta \)
is made under the classical invariant losses
\(\Vert d - \theta \Vert ^2 / \sigma ^2\)
and
\((d - \theta )^t \Sigma ^{-1} (d - \theta )\)
as well as two respective data based losses
\(\Vert d - \theta \Vert ^2 / \Vert U\Vert ^2\)
and
\((d - \theta )^t S^{-1} (d - \theta )\)
where
\(\Vert U\Vert ^2\)
estimates
\(\sigma ^2\)
while
\(S\)
estimates
\(\Sigma \)
. We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of
\(1 / \sigma ^2\)
) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, 2003). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that
\(X\)
is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate
\(t\)
and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.
PubDate: 2015-05-01
- Abstract: Abstract
From an observable
\((X,U)\)
in
\(\mathbb R^p \times \mathbb R^k\)
, we consider estimation of an unknown location parameter
\(\theta \in \mathbb R^p\)
under two distributional settings: the density of
\((X,U)\)
is spherically symmetric with an unknown scale parameter
\(\sigma \)
and is ellipically symmetric with an unknown covariance matrix
\(\Sigma \)
. Evaluation of estimators of
\(\theta \)
is made under the classical invariant losses
\(\Vert d - \theta \Vert ^2 / \sigma ^2\)
and
\((d - \theta )^t \Sigma ^{-1} (d - \theta )\)
as well as two respective data based losses
\(\Vert d - \theta \Vert ^2 / \Vert U\Vert ^2\)
and
\((d - \theta )^t S^{-1} (d - \theta )\)
where
\(\Vert U\Vert ^2\)
estimates
\(\sigma ^2\)
while
\(S\)
estimates
\(\Sigma \)
. We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of
\(1 / \sigma ^2\)
) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, 2003). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that
\(X\)
is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate
\(t\)
and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well.
- 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: 2015-05-01
- 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.
- 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: 2015-05-01
- 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.
- Optimal evaluations for the bias of trimmed means of $$k$$ k th record
values- Abstract: Abstract
We provide sharp upper and lower mean-variance bounds on the expectations of trimmed means of
\(k\)
th record values from general family of distributions. Also we improve these bounds in the case of non-trimmed means for parent distributions with decreasing density or decreasing failure rate. They can be viewed as bounds on the bias of approximation of expectation of the parent population by mean or trimmed mean of record values. The results are illustrated with numerical examples.
PubDate: 2015-05-01
- Abstract: Abstract
We provide sharp upper and lower mean-variance bounds on the expectations of trimmed means of
\(k\)
th record values from general family of distributions. Also we improve these bounds in the case of non-trimmed means for parent distributions with decreasing density or decreasing failure rate. They can be viewed as bounds on the bias of approximation of expectation of the parent population by mean or trimmed mean of record values. The results are illustrated with numerical examples.
- 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: 2015-05-01
- 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.
- Trimmed and winsorized semiparametric estimator for left-truncated and
right-censored regression models- Abstract: Abstract
For a linear regression model subject to left-truncation and right-censoring where the truncation and censoring points are known constants (or always observed if random), Karlsson and Laitila (Stat Probab Lett 78:2567–2571, 2008) proposed a semiparametric estimator which deals with left-truncation by trimming and right-censoring by ‘winsorizing’. The estimator was motivated by a zero moment condition where a transformed error term appears with trimmed and winsorized tails. This paper takes the semiparametric estimator further by deriving the asymptotic distribution that was not shown in Karlsson and Laitila (Stat Probab Lett 78:2567–2571, 2008) and discusses its implementation aspects in practice, albeit brief.
PubDate: 2015-05-01
- Abstract: Abstract
For a linear regression model subject to left-truncation and right-censoring where the truncation and censoring points are known constants (or always observed if random), Karlsson and Laitila (Stat Probab Lett 78:2567–2571, 2008) proposed a semiparametric estimator which deals with left-truncation by trimming and right-censoring by ‘winsorizing’. The estimator was motivated by a zero moment condition where a transformed error term appears with trimmed and winsorized tails. This paper takes the semiparametric estimator further by deriving the asymptotic distribution that was not shown in Karlsson and Laitila (Stat Probab Lett 78:2567–2571, 2008) and discusses its implementation aspects in practice, albeit brief.
- Testing order restrictions in contingency tables
- Abstract: Abstract
Though several interesting models for contingency tables are defined by a system of inequality constraints on a suitable set of marginal log-linear parameters, the specific features of the corresponding testing problems and the related procedures are not widely well known. After reviewing the most common difficulties which are intrinsic to inequality restricted testing problems, the paper concentrates on the problem of testing a set of equalities against the hypothesis that these are violated in the positive direction and also on testing the corresponding inequalities against the saturated model; we argue that valid procedures should consider these two testing problems simultaneously. By reformulating and adapting procedures appeared in the econometric literature, we propose a likelihood ratio and a multiple comparison procedure which are both based on the joint distribution of two relevant statistics; these statistics are used to divide the sample space into three regions: acceptance of the assumed equality constraints, rejection towards inequalities in the positive direction and rejection towards the unrestricted model. A simulation study indicates that the likelihood ratio based procedure perform substantially better. Our procedures are applied to the analysis of two real data sets to clarify how they work in practice.
PubDate: 2015-04-18
- Abstract: Abstract
Though several interesting models for contingency tables are defined by a system of inequality constraints on a suitable set of marginal log-linear parameters, the specific features of the corresponding testing problems and the related procedures are not widely well known. After reviewing the most common difficulties which are intrinsic to inequality restricted testing problems, the paper concentrates on the problem of testing a set of equalities against the hypothesis that these are violated in the positive direction and also on testing the corresponding inequalities against the saturated model; we argue that valid procedures should consider these two testing problems simultaneously. By reformulating and adapting procedures appeared in the econometric literature, we propose a likelihood ratio and a multiple comparison procedure which are both based on the joint distribution of two relevant statistics; these statistics are used to divide the sample space into three regions: acceptance of the assumed equality constraints, rejection towards inequalities in the positive direction and rejection towards the unrestricted model. A simulation study indicates that the likelihood ratio based procedure perform substantially better. Our procedures are applied to the analysis of two real data sets to clarify how they work in practice.
- Generalized projection discrepancy and its applications in experimental
designs- Abstract: Abstract
The objective of this paper is to study the issue of the generalized projection discrepancy along the line of Qin et al. (J Stat Plan Inference 142:1170–1177, 2012) based on generalized discrete discrepancy measure proposed by Chatterjee and Qin (J Stat Plan Inference 141:951–960, 2011). We shall study the projection properties for general asymmetric factorials and provide some analytic connections between minimum generalized projection uniformity and other optimality criteria. A new lower bound on the generalized projection discrepancy for asymmetric factorials is presented here.
PubDate: 2015-04-17
- Abstract: Abstract
The objective of this paper is to study the issue of the generalized projection discrepancy along the line of Qin et al. (J Stat Plan Inference 142:1170–1177, 2012) based on generalized discrete discrepancy measure proposed by Chatterjee and Qin (J Stat Plan Inference 141:951–960, 2011). We shall study the projection properties for general asymmetric factorials and provide some analytic connections between minimum generalized projection uniformity and other optimality criteria. A new lower bound on the generalized projection discrepancy for asymmetric factorials is presented here.
- A new variable selection approach for varying coefficient models
- Abstract: Abstract
The varying coefficient models are very important tools to explore the hidden structure between the response variable and its predictors. However, variable selection and identification of varying coefficients of the models are poorly understood. In this paper, we develop a novel method to overcome these difficulties using local polynomial smoothing and the SCAD penalty. Under some regularity conditions, we show that the proposed procedure is consistent in separating the varying coefficients from the constant ones. The resulting estimator can be as efficient as the oracle. Simulation results confirm our theories. Finally, we study the Boston housing data using the proposed method.
PubDate: 2015-04-16
- Abstract: Abstract
The varying coefficient models are very important tools to explore the hidden structure between the response variable and its predictors. However, variable selection and identification of varying coefficients of the models are poorly understood. In this paper, we develop a novel method to overcome these difficulties using local polynomial smoothing and the SCAD penalty. Under some regularity conditions, we show that the proposed procedure is consistent in separating the varying coefficients from the constant ones. The resulting estimator can be as efficient as the oracle. Simulation results confirm our theories. Finally, we study the Boston housing data using the proposed method.
- Properties of additive frailty model in survival analysis
- Abstract: Abstract
In this paper, we study a general additive frailty model along with some special cases and examples. The monotonicity of the population hazard is investigated in comparison to the baseline hazard rate. Examples are provided where the unconditional failure rate turns out to be increasing or bathtub shaped even when the baseline hazard is increasing. Association measure, for the additive case, of the correlated life times is studied with several examples.
PubDate: 2015-04-02
- Abstract: Abstract
In this paper, we study a general additive frailty model along with some special cases and examples. The monotonicity of the population hazard is investigated in comparison to the baseline hazard rate. Examples are provided where the unconditional failure rate turns out to be increasing or bathtub shaped even when the baseline hazard is increasing. Association measure, for the additive case, of the correlated life times is studied with several examples.
- On the stochastic and dependence properties of the three-state systems
- Abstract: Abstract
Suppose that a system has three states up, partial performance and down. We assume that for a random time
\(T_1\)
the system is in state up, then it moves to state partial performance for time
\(T_2\)
and then the system fails and goes to state down. We also denote the lifetime of the system by
\(T\)
, which is clearly
\(T=T_1+T_2\)
. In this paper, several stochastic comparisons are made between
\(T\)
,
\(T_1\)
and
\(T_2\)
and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided.
PubDate: 2015-04-01
- Abstract: Abstract
Suppose that a system has three states up, partial performance and down. We assume that for a random time
\(T_1\)
the system is in state up, then it moves to state partial performance for time
\(T_2\)
and then the system fails and goes to state down. We also denote the lifetime of the system by
\(T\)
, which is clearly
\(T=T_1+T_2\)
. In this paper, several stochastic comparisons are made between
\(T\)
,
\(T_1\)
and
\(T_2\)
and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided.
- 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: 2015-04-01
- 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.
- 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: 2015-04-01
- 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.
- Erratum to: On optimal designs for censored data
- PubDate: 2015-04-01
- PubDate: 2015-04-01
- Applications of the Rosenthal-type inequality for negatively superadditive
dependent random variables- Abstract: Abstract
In this paper, we give some applications of the Rosenthal-type inequality for a sequence of negatively superadditive dependent (NSD) random variables, which includes sequences of negatively associated random variables as a special case. The complete consistency for an estimator of a nonparametric regression model based on NSD errors is investigated. In addition, we extend Feller’s weak law of large numbers for independent and identically distributed random variables to the case of NSD random variables by using the Rosenthal-type inequality.
PubDate: 2015-04-01
- Abstract: Abstract
In this paper, we give some applications of the Rosenthal-type inequality for a sequence of negatively superadditive dependent (NSD) random variables, which includes sequences of negatively associated random variables as a special case. The complete consistency for an estimator of a nonparametric regression model based on NSD errors is investigated. In addition, we extend Feller’s weak law of large numbers for independent and identically distributed random variables to the case of NSD random variables by using the Rosenthal-type inequality.
- On optimal designs for censored data
- Abstract: Abstract
In time to event experiments the individuals under study are observed to experience some event of interest. If this event is not observed until the end of the experiment, censoring occurs, which is a common feature in such studies. We consider the proportional hazards model with type I and random censoring and determine locally
\(D\)
- and
\(c\)
-optimal designs for a larger class of nonlinear models with two parameters, where the experimental conditions can be selected from a finite discrete design region, as is often the case in practice. Additionally, we compute
\(D\)
-optimal designs for a three-parameter model on a continuous design region.
PubDate: 2015-04-01
- Abstract: Abstract
In time to event experiments the individuals under study are observed to experience some event of interest. If this event is not observed until the end of the experiment, censoring occurs, which is a common feature in such studies. We consider the proportional hazards model with type I and random censoring and determine locally
\(D\)
- and
\(c\)
-optimal designs for a larger class of nonlinear models with two parameters, where the experimental conditions can be selected from a finite discrete design region, as is often the case in practice. Additionally, we compute
\(D\)
-optimal designs for a three-parameter model on a continuous design region.
- 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: 2015-04-01
- 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.
- Erratum to: Improving the EBLUPs of balanced mixed-effects models
- PubDate: 2015-03-29
- PubDate: 2015-03-29
- Nonlinear wavelet density estimation with data missing at random when
covariates are present- Abstract: Abstract
In this paper, we construct the nonlinear wavelet estimator of a density with data missing at random when covariables are present, and provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. Unlike for kernel estimators, the MISE expression of the wavelet-based estimator still holds when the density function is piecewise smooth. Also, the asymptotic normality of the estimator is established.
PubDate: 2015-03-21
- Abstract: Abstract
In this paper, we construct the nonlinear wavelet estimator of a density with data missing at random when covariables are present, and provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. Unlike for kernel estimators, the MISE expression of the wavelet-based estimator still holds when the density function is piecewise smooth. Also, the asymptotic normality of the estimator is established.