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] |
- 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.
- 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: 2014-07-10
- 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 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: 2014-07-10
- 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.
- 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: 2014-07-09
- 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.
- Distributions of stopping times in some sequential estimation procedures
- Abstract: Abstract
A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure.
PubDate: 2014-07-01
- Abstract: Abstract
A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure.
- On Kullback–Leibler information of order statistics in terms of the
relative risk- Abstract: Abstract
The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391–395, 1986). In this paper, we consider a representation of the Kullback–Leibler information of the first
\(r\)
order statistics in terms of the relative risk (Park and Shin in Statistics, 2012), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback–Leibler information of the first
\(r\)
order statistics according to
\(r\)
and discuss its relation with Fisher information in order statistics.
PubDate: 2014-07-01
- Abstract: Abstract
The representation of the entropy in terms of the hazard function and its extensions have been studied by many authors including Teitler et al. (IEEE Trans Reliab 35:391–395, 1986). In this paper, we consider a representation of the Kullback–Leibler information of the first
\(r\)
order statistics in terms of the relative risk (Park and Shin in Statistics, 2012), the ratio of hazard functions, and extend it to the progressively Type II censored data. Then we study the change in Kullback–Leibler information of the first
\(r\)
order statistics according to
\(r\)
and discuss its relation with Fisher information in order statistics.
- Asymptotic properties of id-i-eq1"> format-t-e-x">\(M\) -estimators in
linear and nonlinear multivariate regression models- Abstract: Abstract
We consider the (possibly nonlinear) regression model in
\(\mathbb{R }^q\)
with shift parameter
\(\alpha \)
in
\(\mathbb{R }^q\)
and other parameters
\(\beta \)
in
\(\mathbb{R }^p\)
. Residuals are assumed to be from an unknown distribution function (d.f.). Let
\(\widehat{\phi }\)
be a smooth
\(M\)
-estimator of
\(\phi = {{\beta }\atopwithdelims (){\alpha }}\)
and
\(T(\phi )\)
a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of
\(T(\widehat{\phi })\)
and an estimator of
\(T(\phi )\)
with bias
\(\sim n^{-2}\)
requiring
\(\sim n\)
calculations. (In contrast, the jackknife and bootstrap estimators require
\(\sim n^2\)
calculations.) For a linear regression with random covariates of low skewness, if
\(T(\phi ) = \nu \beta \)
, then
\(T(\widehat{\phi })\)
has bias
\(\sim n^{-2}\)
(not
\(n^{-1}\)
) and skewness
\(\sim n^{-3}\)
(not
\(n^{-2}\)
), and the usual approximate one-sided confidence interval (CI) for
\(T(\phi )\)
has error
\(\sim n^{-1}\)
(not
\(n^{-1/2}\)
). These results extend to random covariates.
PubDate: 2014-07-01
- Abstract: Abstract
We consider the (possibly nonlinear) regression model in
\(\mathbb{R }^q\)
with shift parameter
\(\alpha \)
in
\(\mathbb{R }^q\)
and other parameters
\(\beta \)
in
\(\mathbb{R }^p\)
. Residuals are assumed to be from an unknown distribution function (d.f.). Let
\(\widehat{\phi }\)
be a smooth
\(M\)
-estimator of
\(\phi = {{\beta }\atopwithdelims (){\alpha }}\)
and
\(T(\phi )\)
a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of
\(T(\widehat{\phi })\)
and an estimator of
\(T(\phi )\)
with bias
\(\sim n^{-2}\)
requiring
\(\sim n\)
calculations. (In contrast, the jackknife and bootstrap estimators require
\(\sim n^2\)
calculations.) For a linear regression with random covariates of low skewness, if
\(T(\phi ) = \nu \beta \)
, then
\(T(\widehat{\phi })\)
has bias
\(\sim n^{-2}\)
(not
\(n^{-1}\)
) and skewness
\(\sim n^{-3}\)
(not
\(n^{-2}\)
), and the usual approximate one-sided confidence interval (CI) for
\(T(\phi )\)
has error
\(\sim n^{-1}\)
(not
\(n^{-1/2}\)
). These results extend to random covariates.