Metrika [SJR: 0.943] [H-I: 25] [2 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1435-926X - ISSN (Online) 0026-1335 Published by Springer-Verlag [2335 journals] |
- Variability ordering of multiplicative frailty models
- Abstract: Abstract
The classical multiplicative frailty model in survival analysis accounts for unobserved heterogeneity between individuals. It is of great importance to identify how the variation of the frailty variable affects that of the overall population. This paper is mainly to present how the dispersive and the excess wealth orders between two frailty variables, translate into the corresponding orders between the resulting overall population variables. For the mean residual life and the mean inactivity time orders, we also obtain relevant analogous results in multiplicative frailty models.
PubDate: 2016-08-01
- Abstract: Abstract
The classical multiplicative frailty model in survival analysis accounts for unobserved heterogeneity between individuals. It is of great importance to identify how the variation of the frailty variable affects that of the overall population. This paper is mainly to present how the dispersive and the excess wealth orders between two frailty variables, translate into the corresponding orders between the resulting overall population variables. For the mean residual life and the mean inactivity time orders, we also obtain relevant analogous results in multiplicative frailty models.
- On multi-step MLE-process for Markov sequences
- Abstract: Abstract
We consider the problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process. The estimation is done in two or three steps. First we estimate the unknown parameter by a learning relatively short part of observations and then we use the one-step MLE idea to construct an-estimator process which is asymptotically equivalent to the MLE. To have the learning interval shorter we introduce the two-step procedure which leads to the asymptotically efficient estimator-process too. The presented results are illustrated with the help of two numerical examples.
PubDate: 2016-08-01
- Abstract: Abstract
We consider the problem of the construction of the estimator-process of the unknown finite-dimensional parameter in the case of the observations of nonlinear autoregressive process. The estimation is done in two or three steps. First we estimate the unknown parameter by a learning relatively short part of observations and then we use the one-step MLE idea to construct an-estimator process which is asymptotically equivalent to the MLE. To have the learning interval shorter we introduce the two-step procedure which leads to the asymptotically efficient estimator-process too. The presented results are illustrated with the help of two numerical examples.
- Multivariate Poisson distributions associated with Boolean models
- Abstract: Abstract
We consider a d-dimensional Boolean model
\(\varXi = (\varXi _1+X_1)\cup (\varXi _2+X_2)\cup \cdots \)
generated by a Poisson point process
\(\{X_i, i\ge 1\}\)
with intensity measure
\(\varLambda \)
and a sequence
\(\{\varXi _i, i\ge 1\}\)
of independent copies of some random compact set
\(\varXi _0\,\)
. Given compact sets
\(K_1,\ldots ,K_{\ell }\)
, we show that the discrete random vector
\((N(K_1),\ldots ,N(K_\ell ))\)
, where
\(N(K_j)\)
equals the number of shifted sets
\(\varXi _i+X_i\)
hitting
\(K_j\)
, obeys an
\(\ell \)
-variate Poisson distribution with
\(2^{\ell }-1\)
parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set
\(\varXi \)
in some unboundedly expanding window
\(W_n\)
(as
\(n \rightarrow \infty \)
) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for
\(1\le k < d\)
and more general set-valued functionals of independently marked Poisson processes.
PubDate: 2016-08-01
- Abstract: Abstract
We consider a d-dimensional Boolean model
\(\varXi = (\varXi _1+X_1)\cup (\varXi _2+X_2)\cup \cdots \)
generated by a Poisson point process
\(\{X_i, i\ge 1\}\)
with intensity measure
\(\varLambda \)
and a sequence
\(\{\varXi _i, i\ge 1\}\)
of independent copies of some random compact set
\(\varXi _0\,\)
. Given compact sets
\(K_1,\ldots ,K_{\ell }\)
, we show that the discrete random vector
\((N(K_1),\ldots ,N(K_\ell ))\)
, where
\(N(K_j)\)
equals the number of shifted sets
\(\varXi _i+X_i\)
hitting
\(K_j\)
, obeys an
\(\ell \)
-variate Poisson distribution with
\(2^{\ell }-1\)
parameters. We obtain explicit formulae for all these parameters which can be estimated consistently from an observation of the union set
\(\varXi \)
in some unboundedly expanding window
\(W_n\)
(as
\(n \rightarrow \infty \)
) provided that the Boolean model is stationary. Some of these results can be extended to unions of Poisson k-cylinders for
\(1\le k < d\)
and more general set-valued functionals of independently marked Poisson processes.
- Likelihood ratio order of parallel systems with heterogeneous Weibull
components- Abstract: Abstract
In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let
\(X_{1},\ldots ,X_{n}\)
be independent Weibull random variables with
\(X_{i}\)
having shape parameter
\(0<\alpha \le 1\)
and scale parameter
\(\lambda _{i}\)
,
\(i=1,\ldots ,n\)
, and
\(Y_{1},\ldots ,Y_{n}\)
be a random sample of size n from a Weibull distribution with shape parameter
\(0<\alpha \le 1\)
and a common scale parameter
\(\overline{\lambda }=\frac{1}{n}\sum \nolimits _{i=1}^{n}\lambda _{i}\)
, the arithmetic mean of
\(\lambda _{i}^{'}s\)
. Let
\(X_{n:n}\)
and
\(Y_{n:n}\)
denote the corresponding largest order statistics, respectively. We then prove that
\(X_{n:n}\)
is stochastically larger than
\(Y_{n:n}\)
in terms of the likelihood ratio order, and provide numerical examples to illustrate the results established here.
PubDate: 2016-08-01
- Abstract: Abstract
In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let
\(X_{1},\ldots ,X_{n}\)
be independent Weibull random variables with
\(X_{i}\)
having shape parameter
\(0<\alpha \le 1\)
and scale parameter
\(\lambda _{i}\)
,
\(i=1,\ldots ,n\)
, and
\(Y_{1},\ldots ,Y_{n}\)
be a random sample of size n from a Weibull distribution with shape parameter
\(0<\alpha \le 1\)
and a common scale parameter
\(\overline{\lambda }=\frac{1}{n}\sum \nolimits _{i=1}^{n}\lambda _{i}\)
, the arithmetic mean of
\(\lambda _{i}^{'}s\)
. Let
\(X_{n:n}\)
and
\(Y_{n:n}\)
denote the corresponding largest order statistics, respectively. We then prove that
\(X_{n:n}\)
is stochastically larger than
\(Y_{n:n}\)
in terms of the likelihood ratio order, and provide numerical examples to illustrate the results established here.
- Evaluations of expectations of order statistics and spacings based on IFR
distributions- Abstract: Abstract
We consider i.i.d. random variables
\(X_1,\ldots , X_n\)
with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on
\(X_1,\ldots , X_n\)
, expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications.
PubDate: 2016-08-01
- Abstract: Abstract
We consider i.i.d. random variables
\(X_1,\ldots , X_n\)
with a distribution function F preceding the exponential distribution function V in the convex transform order which means that F has an increasing failure rate. We determine sharp upper bounds on the expectations of order statistics and spacings based on
\(X_1,\ldots , X_n\)
, expressed in the population standard deviation units. We also specify the distributions for which all these bounds are attained. Finally, we indicate some reliability applications.
- On the records of multivariate random sequences
- Abstract: Abstract
Two types of records in multivariate sequences are considered in this paper. According to the first definition, a multivariate observation is accepted as a record if it is not dominated in at least one of the coordinates of previous record and the first observation is a record. Some basic straightforward results concerning the distributions of record times and records according to this definition are given. The development of distribution theory for these types of record and also providing examples with available analytical results still involves challenging unsolved problems. Second, we consider records of bivariate sequences according to conditionally N-ordering, introduced in Bairamov (J Multivar Anal 97:797–809, 2006). The joint distributions of record times and distributions of record values are derived. Some examples, with particular underlying distributions demonstrating the availability of obtained formulae are provided.
PubDate: 2016-08-01
- Abstract: Abstract
Two types of records in multivariate sequences are considered in this paper. According to the first definition, a multivariate observation is accepted as a record if it is not dominated in at least one of the coordinates of previous record and the first observation is a record. Some basic straightforward results concerning the distributions of record times and records according to this definition are given. The development of distribution theory for these types of record and also providing examples with available analytical results still involves challenging unsolved problems. Second, we consider records of bivariate sequences according to conditionally N-ordering, introduced in Bairamov (J Multivar Anal 97:797–809, 2006). The joint distributions of record times and distributions of record values are derived. Some examples, with particular underlying distributions demonstrating the availability of obtained formulae are provided.
- Robust Bayesian Pitman closeness
- Abstract: Abstract
In this paper, the robust Bayesian methodology has been developed in the sense of Pitman measure of closeness. To do this, the definition of Pitman posterior closeness, introduced by Ghosh and Sen (Commun Stat Theory Methods 20:3659–3678, 1991) and simultaneous closeness are integrated. First, the
\(\varGamma \)
-minimax problem is developed in the sense of Pitman’s criterion and the basic results and definitions are provided. Then, several results regarding Pitman
\(\varGamma \)
-minimax have been proved. Some examples have been presented to illustrate the application of the findings. Finally, other aspect of robust Bayesian methodology such as: Pitman stable rules and Pitman regret type estimators have been proposed.
PubDate: 2016-08-01
- Abstract: Abstract
In this paper, the robust Bayesian methodology has been developed in the sense of Pitman measure of closeness. To do this, the definition of Pitman posterior closeness, introduced by Ghosh and Sen (Commun Stat Theory Methods 20:3659–3678, 1991) and simultaneous closeness are integrated. First, the
\(\varGamma \)
-minimax problem is developed in the sense of Pitman’s criterion and the basic results and definitions are provided. Then, several results regarding Pitman
\(\varGamma \)
-minimax have been proved. Some examples have been presented to illustrate the application of the findings. Finally, other aspect of robust Bayesian methodology such as: Pitman stable rules and Pitman regret type estimators have been proposed.
- On runs of ones defined on a q -sequence of binary trials
- Abstract: Abstract
In a sequence of n binary (
\(0{-}1\)
) trials, with probability of ones varying according to a geometric rule, we consider a random variable denoting the number of runs of ones of length at least equal to a fixed threshold k,
\(1\le k\le n\)
. Closed and recursive expressions are obtained for the probability mass function, generating functions and moments of this random variable. Statistical inference problems related to the probability of ones are examined by numerical techniques. Numerics illustrate further the theoretical results.
PubDate: 2016-07-01
- Abstract: Abstract
In a sequence of n binary (
\(0{-}1\)
) trials, with probability of ones varying according to a geometric rule, we consider a random variable denoting the number of runs of ones of length at least equal to a fixed threshold k,
\(1\le k\le n\)
. Closed and recursive expressions are obtained for the probability mass function, generating functions and moments of this random variable. Statistical inference problems related to the probability of ones are examined by numerical techniques. Numerics illustrate further the theoretical results.
- Stochastic somparisons of order statistics from scaled and interdependent
random variables- Abstract: Abstract
This paper studies order statistics from random variables following the scale model. In the presence of the Archimedean copula or survival copula for the random variables, we obtain the usual stochastic order of the sample extremes and the second smallest order statistic, the dispersive order and the star order of the sample extremes.
PubDate: 2016-07-01
- Abstract: Abstract
This paper studies order statistics from random variables following the scale model. In the presence of the Archimedean copula or survival copula for the random variables, we obtain the usual stochastic order of the sample extremes and the second smallest order statistic, the dispersive order and the star order of the sample extremes.
- A mixture model of size-biased distributions
- Abstract: Abstract
In reliability and survival analysis, to model lifetime data, size-biased distributions are useful. In this paper, a mixture model of size-biased distributions is introduced and studied. Several reliability properties of this model are investigated. In addition, some implications of well-known stochastic orders and aging classes concerning the model are established. To underline the usefulness of the model, some examples of interest in reliability and statistics are given.
PubDate: 2016-07-01
- Abstract: Abstract
In reliability and survival analysis, to model lifetime data, size-biased distributions are useful. In this paper, a mixture model of size-biased distributions is introduced and studied. Several reliability properties of this model are investigated. In addition, some implications of well-known stochastic orders and aging classes concerning the model are established. To underline the usefulness of the model, some examples of interest in reliability and statistics are given.
- Single change-point detection methods for small lifetime samples
- Abstract: Abstract
In this paper, we address the problem of deciding if either n consecutive independent failure times have the same failure rate or if there exists some
\(k\in \{1,\ldots ,n\}\)
such that the common failure rate of the first k failure times is different from the common failure rate of the last
\(n-k\)
failure times, based on an exponential lifetime distribution. The statistical test we propose is based on the empirical average ratio under the assumption of exponentiality. The proposed test is compared to the one based on the Mann–Whitney statistic for which no parametric assumption on the underlying distribution is necessary. The proposed statistics are free of the unknown underlying distribution under the null hypothesis of homogeneity of the n failure times which enables the determination of critical values of the proposed tests by Monte Carlo methods for small sample sizes.
PubDate: 2016-07-01
- Abstract: Abstract
In this paper, we address the problem of deciding if either n consecutive independent failure times have the same failure rate or if there exists some
\(k\in \{1,\ldots ,n\}\)
such that the common failure rate of the first k failure times is different from the common failure rate of the last
\(n-k\)
failure times, based on an exponential lifetime distribution. The statistical test we propose is based on the empirical average ratio under the assumption of exponentiality. The proposed test is compared to the one based on the Mann–Whitney statistic for which no parametric assumption on the underlying distribution is necessary. The proposed statistics are free of the unknown underlying distribution under the null hypothesis of homogeneity of the n failure times which enables the determination of critical values of the proposed tests by Monte Carlo methods for small sample sizes.
- Asymptotic results of a nonparametric conditional cumulative distribution
estimator in the single functional index modeling for time series data
with applications- Abstract: Abstract
In this paper, we treat nonparametric estimation of the conditional cumulative distribution with a scalar response variable conditioned by a functional Hilbertian regressor. We establish asymptotic normality and uniform almost complete convergence rates of the conditional cumulative distribution estimator for dependent variables, linked semiparametrically by the single index structure. Furthermore, we provide some applications and simulations to illustrate our methodology.
PubDate: 2016-07-01
- Abstract: Abstract
In this paper, we treat nonparametric estimation of the conditional cumulative distribution with a scalar response variable conditioned by a functional Hilbertian regressor. We establish asymptotic normality and uniform almost complete convergence rates of the conditional cumulative distribution estimator for dependent variables, linked semiparametrically by the single index structure. Furthermore, we provide some applications and simulations to illustrate our methodology.
- A new method of kernel-smoothing estimation of the ROC curve
- Abstract: Abstract
The receiver operating characteristic (ROC) curve is a popular graphical tool for describing the accuracy of a diagnostic test. Based on the idea of estimating the ROC curve as a distribution function, we propose a new kernel smoothing estimator of the ROC curve which is invariant under nondecreasing data transformations. We prove that the estimator has better asymptotic mean squared error properties than some other estimators involving kernel smoothing and we present an easy method of bandwidth selection. By simulation studies, we show that for the limited sample sizes, our proposed estimator is competitive with some other nonparametric estimators of the ROC curve. We also give an example of applying the estimator to a real data set.
PubDate: 2016-07-01
- Abstract: Abstract
The receiver operating characteristic (ROC) curve is a popular graphical tool for describing the accuracy of a diagnostic test. Based on the idea of estimating the ROC curve as a distribution function, we propose a new kernel smoothing estimator of the ROC curve which is invariant under nondecreasing data transformations. We prove that the estimator has better asymptotic mean squared error properties than some other estimators involving kernel smoothing and we present an easy method of bandwidth selection. By simulation studies, we show that for the limited sample sizes, our proposed estimator is competitive with some other nonparametric estimators of the ROC curve. We also give an example of applying the estimator to a real data set.
- Robust feature screening for varying coefficient models via quantile
partial correlation- Abstract: Abstract
This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.
PubDate: 2016-06-30
- Abstract: Abstract
This article is concerned with feature screening for varying coefficient models with ultrahigh-dimensional predictors. We propose a new sure independence screening method based on quantile partial correlation (QPC-SIS), which is quite robust against outliers and heavy-tailed distributions. Then we establish the sure screening property for the QPC-SIS, and conduct simulations to examine its finite sample performance. The results of simulation study indicate that the QPC-SIS performs better than other methods like sure independent screening (SIS), sure independent ranking and screening, distance correlation-sure independent screening, conditional correlation sure independence screening and nonparametric independent screening, which shows the validity and rationality of QPC-SIS.
- AR(1) model with skew-normal innovations
- Abstract: Abstract
In this paper, we consider an autoregressive model of order one with skew-normal innovations. We propose several methods for estimating the parameters of the model and derive the limiting distributions of the estimators. Then, we study some statistical properties and the regression behavior of the proposed model. Finally, we provide a Monte Carlo simulation study for comparing performance of estimators and consider a real time series to illustrate the applicability of the proposed model.
PubDate: 2016-06-29
- Abstract: Abstract
In this paper, we consider an autoregressive model of order one with skew-normal innovations. We propose several methods for estimating the parameters of the model and derive the limiting distributions of the estimators. Then, we study some statistical properties and the regression behavior of the proposed model. Finally, we provide a Monte Carlo simulation study for comparing performance of estimators and consider a real time series to illustrate the applicability of the proposed model.
- Conditional empirical likelihood for quantile regression models
- Abstract: Abstract
In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.
PubDate: 2016-06-23
- Abstract: Abstract
In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.
- Tree based diagnostic procedures following a smooth test of
goodness-of-fit- Abstract: Abstract
This paper introduces a statistical procedure, to be applied after a goodness-of-fit test has rejected a null model, that provides diagnostic information to help the user decide on a better model. The procedure goes through a list of departures, each being tested by a local smooth test. The list is organized into a hierarchy by seeking answers to the questions “Where is the problem?” and “What is the problem there?”. This hierarchy allows to focus on finer departures as the data becomes more abundant. The procedure controls the family-wise Type 1 error rate. Simulations show that the procedure can succeed in providing useful diagnostic information.
PubDate: 2016-06-10
- Abstract: Abstract
This paper introduces a statistical procedure, to be applied after a goodness-of-fit test has rejected a null model, that provides diagnostic information to help the user decide on a better model. The procedure goes through a list of departures, each being tested by a local smooth test. The list is organized into a hierarchy by seeking answers to the questions “Where is the problem?” and “What is the problem there?”. This hierarchy allows to focus on finer departures as the data becomes more abundant. The procedure controls the family-wise Type 1 error rate. Simulations show that the procedure can succeed in providing useful diagnostic information.
- Imputation based statistical inference for partially linear quantile
regression models with missing responses- Abstract: Abstract
In this paper, we consider the confidence interval construction for partially linear quantile regression models with missing response at random. We propose an imputation based empirical likelihood method to construct confidence intervals for the parametric components and the nonparametric components, and show that the proposed empirical log-likelihood ratios are both asymptotically Chi-squared in theory. Then, the confidence region for the parametric component and the pointwise confidence interval for the nonparametric component are constructed. Some simulation studies and a real data application are carried out to assess the performance of the proposed estimation method, and simulation results indicate that the proposed method is workable.
PubDate: 2016-06-09
- Abstract: Abstract
In this paper, we consider the confidence interval construction for partially linear quantile regression models with missing response at random. We propose an imputation based empirical likelihood method to construct confidence intervals for the parametric components and the nonparametric components, and show that the proposed empirical log-likelihood ratios are both asymptotically Chi-squared in theory. Then, the confidence region for the parametric component and the pointwise confidence interval for the nonparametric component are constructed. Some simulation studies and a real data application are carried out to assess the performance of the proposed estimation method, and simulation results indicate that the proposed method is workable.
- A test of linearity in partial functional linear regression
- Abstract: Abstract
This paper investigates the hypothesis test of the parametric component in partial functional linear regression. We propose a test procedure based on the residual sums of squares under the null and alternative hypothesis, and establish the asymptotic properties of the resulting test. A simulation study shows that the proposed test procedure has good size and power with finite sample sizes. Finally, we present an illustration through fitting the Berkeley growth data with a partial functional linear regression model and testing the effect of gender on the height of kids.
PubDate: 2016-06-08
- Abstract: Abstract
This paper investigates the hypothesis test of the parametric component in partial functional linear regression. We propose a test procedure based on the residual sums of squares under the null and alternative hypothesis, and establish the asymptotic properties of the resulting test. A simulation study shows that the proposed test procedure has good size and power with finite sample sizes. Finally, we present an illustration through fitting the Berkeley growth data with a partial functional linear regression model and testing the effect of gender on the height of kids.
- Nonparametric estimation in a mixed-effect Ornstein–Uhlenbeck model
- Abstract: Abstract
Two adaptive nonparametric procedures are proposed to estimate the density of the random effects in a mixed-effect Ornstein–Uhlenbeck model. First a kernel estimator is introduced with a new bandwidth selection method developed recently by Goldenshluger and Lepski (Ann Stat 39:1608–1632, 2011). Then, we adapt an estimator from Comte et al. (Stoch Process Appl 7:2522–2551, 2013) in the framework of small time interval of observation. More precisely, we propose an estimator that uses deconvolution tools and depends on two tuning parameters to be chosen in a data-driven way. The selection of these two parameters is achieved through a two-dimensional penalized criterion. For both adaptive estimators, risk bounds are provided in terms of integrated
\(\mathbb {L}^2\)
-error. The estimators are evaluated on simulations and show good results. Finally, these nonparametric estimators are applied to neuronal data and are compared with previous parametric estimations.
PubDate: 2016-05-23
- Abstract: Abstract
Two adaptive nonparametric procedures are proposed to estimate the density of the random effects in a mixed-effect Ornstein–Uhlenbeck model. First a kernel estimator is introduced with a new bandwidth selection method developed recently by Goldenshluger and Lepski (Ann Stat 39:1608–1632, 2011). Then, we adapt an estimator from Comte et al. (Stoch Process Appl 7:2522–2551, 2013) in the framework of small time interval of observation. More precisely, we propose an estimator that uses deconvolution tools and depends on two tuning parameters to be chosen in a data-driven way. The selection of these two parameters is achieved through a two-dimensional penalized criterion. For both adaptive estimators, risk bounds are provided in terms of integrated
\(\mathbb {L}^2\)
-error. The estimators are evaluated on simulations and show good results. Finally, these nonparametric estimators are applied to neuronal data and are compared with previous parametric estimations.