Metrika [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 1435-926X - ISSN (Online) 0026-1335 Published by Springer-Verlag [2208 journals] [SJR: 0.839] [H-I: 22] |
- 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.
- Asymptotic infimum coverage probability for interval estimation of
proportions- Abstract: Abstract
In this paper, we discuss asymptotic infimum coverage probability (ICP) of eight widely used confidence intervals for proportions, including the Agresti–Coull (A–C) interval (Am Stat 52:119–126, 1998) and the Clopper–Pearson (C–P) interval (Biometrika 26:404–413, 1934). For the A–C interval, a sharp upper bound for its asymptotic ICP is derived. It is less than nominal for the commonly applied nominal values of 0.99, 0.95 and 0.9 and is equal to zero when the nominal level is below 0.4802. The
\(1-\alpha \)
C–P interval is known to be conservative. However, we show through a brief numerical study that the C–P interval with a given average coverage probability
\(1-\gamma \)
typically has a similar or larger ICP and a smaller average expected length than the corresponding A–C interval, and its ICP approaches to
\(1-\gamma \)
when the sample size goes large. All mathematical proofs and R-codes for computation in the paper are given in Supplementary Materials.
PubDate: 2014-07-01
- Abstract: Abstract
In this paper, we discuss asymptotic infimum coverage probability (ICP) of eight widely used confidence intervals for proportions, including the Agresti–Coull (A–C) interval (Am Stat 52:119–126, 1998) and the Clopper–Pearson (C–P) interval (Biometrika 26:404–413, 1934). For the A–C interval, a sharp upper bound for its asymptotic ICP is derived. It is less than nominal for the commonly applied nominal values of 0.99, 0.95 and 0.9 and is equal to zero when the nominal level is below 0.4802. The
\(1-\alpha \)
C–P interval is known to be conservative. However, we show through a brief numerical study that the C–P interval with a given average coverage probability
\(1-\gamma \)
typically has a similar or larger ICP and a smaller average expected length than the corresponding A–C interval, and its ICP approaches to
\(1-\gamma \)
when the sample size goes large. All mathematical proofs and R-codes for computation in the paper are given in Supplementary Materials.
- An empirical likelihood inference for the coefficient difference of a
two-sample linear model with missing response data- Abstract: Abstract
In this paper, we use the empirical likelihood method to make inferences for the coefficient difference of a two-sample linear regression model with missing response data. The commonly used empirical likelihood ratio is not concave for this problem, so we append a natural and well-explained condition to the likelihood function and propose three types of restricted empirical likelihood ratios for constructing the confidence region of the parameter in question. It can be demonstrated that all three empirical likelihood ratios have, asymptotically, chi-squared distributions. Simulation studies are carried out to show the effectiveness of the proposed approaches in aspects of coverage probability and interval length. A real data set is analysed with our methods as an example.
PubDate: 2014-07-01
- Abstract: Abstract
In this paper, we use the empirical likelihood method to make inferences for the coefficient difference of a two-sample linear regression model with missing response data. The commonly used empirical likelihood ratio is not concave for this problem, so we append a natural and well-explained condition to the likelihood function and propose three types of restricted empirical likelihood ratios for constructing the confidence region of the parameter in question. It can be demonstrated that all three empirical likelihood ratios have, asymptotically, chi-squared distributions. Simulation studies are carried out to show the effectiveness of the proposed approaches in aspects of coverage probability and interval length. A real data set is analysed with our methods as an example.
- New robust tests for the parameters of the Weibull distribution for
complete and censored data- Abstract: Abstract
Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown is examined. In simulation studies the behavior in finite sample size and in contaminated data is analyzed and the new method is compared to existing ones. Here it is shown that the new tests based on likelihood depth give quite good results compared to standard methods and are robust against contamination. They are also robust in right-censored data in contrast to existing methods like the method of medians.
PubDate: 2014-07-01
- Abstract: Abstract
Using the likelihood depth, new consistent and robust tests for the parameters of the Weibull distribution are developed. Uncensored as well as type-I right-censored data are considered. Tests are given for the shape parameter and also the scale parameter of the Weibull distribution, where in each case the situation that the other parameter is known as well the situation that both parameter are unknown is examined. In simulation studies the behavior in finite sample size and in contaminated data is analyzed and the new method is compared to existing ones. Here it is shown that the new tests based on likelihood depth give quite good results compared to standard methods and are robust against contamination. They are also robust in right-censored data in contrast to existing methods like the method of medians.
- A new bounded log-linear regression model
- Abstract: Abstract
In this paper we introduce a new regression model in which the response variable is bounded by two unknown parameters. A special case is a bounded alternative to the four parameter logistic model. The four parameter model which has unbounded responses is widely used, for instance, in bioassays, nutrition, genetics, calibration and agriculture. In reality, the responses are often bounded although the bounds may be unknown, and in that situation, our model reflects the data-generating mechanism better. Complications arise for the new model, however, because the likelihood function is unbounded, and the global maximizers are not consistent estimators of unknown parameters. Although the two sample extremes, the smallest and the largest observations, are consistent estimators for the two unknown boundaries, they have a slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes in the one sample case; but even these consistent estimators do not obtain the optimal convergence rate. To obtain efficient estimation, we suggest using the local maximizers of the likelihood function, i.e., the solution to the likelihood equations. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.
PubDate: 2014-07-01
- Abstract: Abstract
In this paper we introduce a new regression model in which the response variable is bounded by two unknown parameters. A special case is a bounded alternative to the four parameter logistic model. The four parameter model which has unbounded responses is widely used, for instance, in bioassays, nutrition, genetics, calibration and agriculture. In reality, the responses are often bounded although the bounds may be unknown, and in that situation, our model reflects the data-generating mechanism better. Complications arise for the new model, however, because the likelihood function is unbounded, and the global maximizers are not consistent estimators of unknown parameters. Although the two sample extremes, the smallest and the largest observations, are consistent estimators for the two unknown boundaries, they have a slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes in the one sample case; but even these consistent estimators do not obtain the optimal convergence rate. To obtain efficient estimation, we suggest using the local maximizers of the likelihood function, i.e., the solution to the likelihood equations. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.
- Admissibility in non-regular family under squared-log error loss
- Abstract: Abstract
Consider an estimation problem under the squared-log error loss function in a one-parameter non-regular distribution when the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Some examples are given.
PubDate: 2014-06-25
- Abstract: Abstract
Consider an estimation problem under the squared-log error loss function in a one-parameter non-regular distribution when the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Some examples are given.
- Estimating covariate functions associated to multivariate risks: a level
set approach- Abstract: Abstract
The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the
\(c\)
-upper level sets
\({L(c)= \{F(x) \ge c \}}\)
, with
\(c \in (0,1)\)
, of an unknown distribution function
\(F\)
on
\(\mathbb {R}^d_+\)
. A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the
\(L_p\)
-consistency, with a convergence rate, for the regression function estimate on these level sets
\(L(c)\)
. We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed and a comparison with parametric and semi-parametric approaches is provided. Finally, a real environmental application of our risk measure is provided.
PubDate: 2014-06-14
- Abstract: Abstract
The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the
\(c\)
-upper level sets
\({L(c)= \{F(x) \ge c \}}\)
, with
\(c \in (0,1)\)
, of an unknown distribution function
\(F\)
on
\(\mathbb {R}^d_+\)
. A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the
\(L_p\)
-consistency, with a convergence rate, for the regression function estimate on these level sets
\(L(c)\)
. We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed and a comparison with parametric and semi-parametric approaches is provided. Finally, a real environmental application of our risk measure is provided.
- A characterization of the innovations of first order autoregressive models
- Abstract: Abstract
Suppose that
\(Y_t\)
follows a simple AR(1) model, that is, it can be expressed as
\(Y_t= \alpha Y_{t-1} + W_t\)
, where
\(W_t\)
is a white noise with mean equal to
\(\mu \)
and variance
\(\sigma ^2\)
. There are many examples in practice where these assumptions hold very well. Consider
\(X_t = e^{Y_t}\)
. We shall show that the autocorrelation function of
\(X_t\)
characterizes the distribution of
\(W_t\)
.
PubDate: 2014-06-07
- Abstract: Abstract
Suppose that
\(Y_t\)
follows a simple AR(1) model, that is, it can be expressed as
\(Y_t= \alpha Y_{t-1} + W_t\)
, where
\(W_t\)
is a white noise with mean equal to
\(\mu \)
and variance
\(\sigma ^2\)
. There are many examples in practice where these assumptions hold very well. Consider
\(X_t = e^{Y_t}\)
. We shall show that the autocorrelation function of
\(X_t\)
characterizes the distribution of
\(W_t\)
.