Metrika [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1435-926X - ISSN (Online) 0026-1335 Published by Springer-Verlag [2209 journals] [SJR: 0.839] [H-I: 22] |
- A necessary and sufficient condition for justifying non-parametric
likelihood with censored data- Abstract: The non-parametric likelihood L(F) for censored data, including univariate or multivariate right-censored, doubly-censored, interval-censored, or masked competing risks data, is proposed by Peto (Appl Stat 22:86–91, 1973). It does not involve censoring distributions. In the literature, several noninformative conditions are proposed to justify L(F) so that the GMLE can be consistent (see, for examples, Self and Grossman in Biometrics 42:521–530 1986, or Oller et al. in Can J Stat 32:315–326, 2004). We present the necessary and sufficient (N&S) condition so that
\(L(F)\)
is equivalent to the full likelihood under the non-parametric set-up. The statement is false under the parametric set-up. Our condition is slightly different from the noninformative conditions in the literature. We present two applications to our cancer research data that satisfy the N&S condition but has dependent censoring.
PubDate: 2014-11-01
- Abstract: The non-parametric likelihood L(F) for censored data, including univariate or multivariate right-censored, doubly-censored, interval-censored, or masked competing risks data, is proposed by Peto (Appl Stat 22:86–91, 1973). It does not involve censoring distributions. In the literature, several noninformative conditions are proposed to justify L(F) so that the GMLE can be consistent (see, for examples, Self and Grossman in Biometrics 42:521–530 1986, or Oller et al. in Can J Stat 32:315–326, 2004). We present the necessary and sufficient (N&S) condition so that
\(L(F)\)
is equivalent to the full likelihood under the non-parametric set-up. The statement is false under the parametric set-up. Our condition is slightly different from the noninformative conditions in the literature. We present two applications to our cancer research data that satisfy the N&S condition but has dependent censoring.
- U-type and column-orthogonal designs for computer experiments
- Abstract: U-type designs and orthogonal Latin hypercube designs (OLHDs) have been used extensively for performing computer experiments. Both have good spaced filling properties in one-dimension. U-type designs may not have low correlations among the main effects, quadratic effects and two-factor interactions. On the other hand, OLHDs are hard to be found due to their large number of levels for each factor. Recently, alternative classes of U-type designs with zero or low correlations among the effects of interest appear in the literature. In this paper, we present new classes of U-type or quantitative
\(3\)
-orthogonal designs for computer experiments. The proposed designs are constructed by combining known combinatorial structures and they have their main effects pairwise orthogonal, orthogonal to the mean effect, and orthogonal to both quadratic effects and two-factor interactions.
PubDate: 2014-11-01
- Abstract: U-type designs and orthogonal Latin hypercube designs (OLHDs) have been used extensively for performing computer experiments. Both have good spaced filling properties in one-dimension. U-type designs may not have low correlations among the main effects, quadratic effects and two-factor interactions. On the other hand, OLHDs are hard to be found due to their large number of levels for each factor. Recently, alternative classes of U-type designs with zero or low correlations among the effects of interest appear in the literature. In this paper, we present new classes of U-type or quantitative
\(3\)
-orthogonal designs for computer experiments. The proposed designs are constructed by combining known combinatorial structures and they have their main effects pairwise orthogonal, orthogonal to the mean effect, and orthogonal to both quadratic effects and two-factor interactions.
- Asymptotic behavior of the hazard rate in systems based on sequential
order statistics- Abstract: The limiting behavior of the hazard rate of coherent systems based on sequential order statistics is examined. Related results for the survival function of the system lifetime are also considered. For deriving the results, properties of limits involving a relevation transform are studied in detail. Then, limits of characteristics in sequential
\(k\)
-out-of-
\(n\)
systems and general coherent systems with failure-dependent components are obtained. Applications to the comparison of different systems based on their long run behavior and to limits of coefficients in a signature-based representation of the residual system lifetime are given.
PubDate: 2014-11-01
- Abstract: The limiting behavior of the hazard rate of coherent systems based on sequential order statistics is examined. Related results for the survival function of the system lifetime are also considered. For deriving the results, properties of limits involving a relevation transform are studied in detail. Then, limits of characteristics in sequential
\(k\)
-out-of-
\(n\)
systems and general coherent systems with failure-dependent components are obtained. Applications to the comparison of different systems based on their long run behavior and to limits of coefficients in a signature-based representation of the residual system lifetime are given.
- Bayesian prediction in doubly stochastic Poisson process
- Abstract: A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process
\((T_{i},Y_{i})_{i\ge 1}\)
, where
\(T_{i}\)
is the time of occurrence of the
\(i\)
th event and the mark
\(Y_{i}\)
is its characteristic (size), is supposed to be a non-homogeneous Poisson process on
\(\mathbb {R}_{+}^{2}\)
with intensity measure
\(P\times \varTheta \)
, where
\(P\)
is known, whereas
\(\varTheta \)
is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that
\(\varTheta \)
is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.
PubDate: 2014-11-01
- Abstract: A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process
\((T_{i},Y_{i})_{i\ge 1}\)
, where
\(T_{i}\)
is the time of occurrence of the
\(i\)
th event and the mark
\(Y_{i}\)
is its characteristic (size), is supposed to be a non-homogeneous Poisson process on
\(\mathbb {R}_{+}^{2}\)
with intensity measure
\(P\times \varTheta \)
, where
\(P\)
is known, whereas
\(\varTheta \)
is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that
\(\varTheta \)
is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.
- On extremes of bivariate residual lifetimes from generalized
Marshall–Olkin and time transformed exponential models- Abstract: We study here extremes of residuals of the bivariate lifetime and the residual of extremes of the two lifetimes. In the case of generalized Marshall–Olkin model and the total time transformed exponential model, we first present some sufficient conditions for the extremes of residuals to be stochastically larger than the residual of the corresponding extremes, and then investigate the stochastic order of the residual of extremes of the two lifetimes based on the majorization of the age vector of the residuals.
PubDate: 2014-11-01
- Abstract: We study here extremes of residuals of the bivariate lifetime and the residual of extremes of the two lifetimes. In the case of generalized Marshall–Olkin model and the total time transformed exponential model, we first present some sufficient conditions for the extremes of residuals to be stochastically larger than the residual of the corresponding extremes, and then investigate the stochastic order of the residual of extremes of the two lifetimes based on the majorization of the age vector of the residuals.
- Model averaging based on James–Stein estimators
- Abstract: Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.
PubDate: 2014-11-01
- Abstract: Existing model averaging methods are generally based on ordinary least squares (OLS) estimators. However, it is well known that the James–Stein (JS) estimator dominates the OLS estimator under quadratic loss, provided that the dimension of coefficient is larger than two. Thus, we focus on model averaging based on JS estimators instead of OLS estimators. We develop a weight choice method and prove its asymptotic optimality. A simulation experiment shows promising results for the proposed model average estimator.
- Blocked semifoldovers of two-level orthogonal designs
- Abstract: Follow-up experimentation is often necessary to the successful use of fractional factorial designs. When some effects are believed to be significant but cannot be estimated using an initial design, adding another fraction is often recommended. As the initial design and its foldover (or semifoldover) are usually conducted at different stages, it may be desirable to include a block factor. In this article, we study the blocking effect of such a factor on foldover and semifoldover designs. We consider two general cases for the initial designs, which can be either unblocked or blocked designs. In both cases, we explore the relationships between semifoldover of a design and its corresponding foldover design. More specifically, we obtain some theoretical results on when a semifoldover design can estimate the same two-factor interactions or main effects as the corresponding foldover. These results can be important for those who want to take advantage of the run size savings of a semifoldover without sacrificing the ability to estimate important effects.
PubDate: 2014-10-22
- Abstract: Follow-up experimentation is often necessary to the successful use of fractional factorial designs. When some effects are believed to be significant but cannot be estimated using an initial design, adding another fraction is often recommended. As the initial design and its foldover (or semifoldover) are usually conducted at different stages, it may be desirable to include a block factor. In this article, we study the blocking effect of such a factor on foldover and semifoldover designs. We consider two general cases for the initial designs, which can be either unblocked or blocked designs. In both cases, we explore the relationships between semifoldover of a design and its corresponding foldover design. More specifically, we obtain some theoretical results on when a semifoldover design can estimate the same two-factor interactions or main effects as the corresponding foldover. These results can be important for those who want to take advantage of the run size savings of a semifoldover without sacrificing the ability to estimate important effects.
- Robust minimax Stein estimation under invariant data-based loss for
spherically and elliptically symmetric distributions- 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: 2014-10-04
- 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.
- Modified maximum spacings method for generalized extreme value
distribution and applications in real data analysis- 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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.
- Optimal evaluations for the bias of trimmed means of class="a-plus-plus inline-equation id-i-eq1"> class="a-plus-plus equation-source
format-t-e-x">\(k\) th record values- 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: 2014-09-27
- 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.
- Generalized measures of information for truncated random variables
- 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: 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: 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: 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: 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: 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: 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: 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: 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: 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.