In this paper, estimation of the location parameter of a half-normal distribution is considered. ... more In this paper, estimation of the location parameter of a half-normal distribution is considered. Some unbiased as well as biased estimators are derived. Admissibility and minimaxity of Pitman estimator are proved. A complete class of estimators among multiples of the maximum likelihood estimator is obtained. We develop a one-sided asymptotic confidence interval for the location parameter. Numerical comparisons of the percentage risk improvements over maximum likelihood estimator of various estimators are carried out.
Consider two independent normal populations with a common variance and ordered means. For this mo... more Consider two independent normal populations with a common variance and ordered means. For this model, we study the problem of estimating a common variance and a common precision with respect to a general class of scale invariant loss functions. A general minimaxity result is established for estimating the common variance. It is shown that the best affine equivariant estimator and the restricted maximum likelihood estimator are inadmissible. In this direction, we derive a Stein-type improved estimator. We further derive a smooth estimator which improves upon the best affine equivariant estimator. In particular, various scale invariant loss functions are considered and several improved estimators are presented. Furthermore, a simulation study is performed to find the performance of the improved estimators developed in this paper. Similar results are obtained for the problem of estimating a common precision for the stated model under a general class of scale invariant loss functions.
ABSTRACT Let be populations with having an inverse Gaussian distribution with unknown mean and un... more ABSTRACT Let be populations with having an inverse Gaussian distribution with unknown mean and unknown scale-like parameter , respectively. We study the problem of classification of an observation when prior information suggests some orderings on parameters. When the means are equal but unknown, we derive plug-in Bayes classification rules based on the maximum likelihood estimator (MLE), Graybill-Deal type estimator and shrinkage estimator of the common mean. When all parameters are unknown and unequal, we also derive likelihood ratio-based classification rules. For more than two populations, we suggest ordered rules when s follow an ordering. When the means are unequal, we also derive rules assuming ordering among either s or s. Extensive simulations are carried out to compare the proposed rules with respect to expected probabilities of correct classification. Applications of these classification rules are described using real data sets.
International journal of applied mathematics and statistics, 2008
... Subscription or Fee Access. Admissibility of the Pitman Estimator of Ordered Location Paramet... more ... Subscription or Fee Access. Admissibility of the Pitman Estimator of Ordered Location Parameters. Somesh Kumar, Ajay Kumar, Tanuja Srivastava. Abstract. Abstract available in paper due to mathematical expressions. Full Text: PDF.
Estimation of the scale parameter of the scale mixture of a location–scale family under the scale... more Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.
Communications in Statistics - Simulation and Computation, 2015
ABSTRACT A Langevin distribution with two parameters (mean direction and concentration parameter)... more ABSTRACT A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this article, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Jeffreys’ priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behavior of Bayes estimators is carried out with the help of simulations.
A Langevin distribution with two parameters (mean direction and concentration parameter) has been... more A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this paper, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Je ffreys priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behaviour of Bayes estimators is carried out with the help of simulations.
The main purpose of this work is to identify the determinants for Attainment in Schooling in a ru... more The main purpose of this work is to identify the determinants for Attainment in Schooling in a rural setting. The Ordinary Least Square Method (O.L.S.) of the regression analysis has been the main plank of the present paper. The estimates of multiple regression analysis have been tested for multicollinearity, heteroscedasticity and serial correlation.
In this paper, the problem of estimating the mean of a two parameter lognormal distribution is co... more In this paper, the problem of estimating the mean of a two parameter lognormal distribution is considered under the assumption that one of its parameter is restricted a priori. Several classical and Bayes estimators are proposed. Some of them are shown to be inadmissible using Rao-Blackwellization. Finally, risk performance of all estimators is compared numerically.
In many real life situations, prior information about the parameters is available, such as the or... more In many real life situations, prior information about the parameters is available, such as the ordering of the parameters. Incorporating this prior information about the order restrictions on parameters leads to more efficient estimators. In the present communication, we investigate estimation of the ordered scale parameters of two shifted exponential distributions with unknown location parameters under a class of bowl-shaped loss functions. We have proved that the best affine equivariant estimator (BAEE) is inadmissible. Various non smooth and smooth estimators has been obtained which improve upon the BAEE. In particular we have derived the improved estimators for some well known loss functions. Finally numerical comparison is carried out to compare the risk performance of the proposed estimators.
Abstract In the present article, we have studied the estimation of the reciprocal of scale parame... more Abstract In the present article, we have studied the estimation of the reciprocal of scale parameter , that is, hazard rate of a two parameter exponential distribution based on a doubly censored sample. This estimation problem has been investigated under a general class of bowl-shaped scale invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of scale equivariant estimator is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for three special loss functions. A simulation study is conducted to compare the risk performance of the proposed estimators. Finally, we analyze a real data set.
There are various situations where prior information is available on parameters in the form of or... more There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: <inline-formula> <tex-math notation="LaTeX">$(i)$ </tex-math></inline-formula> i.i.d. sampling, <inline-formula> <tex-math notation="LaTeX">$(ii)$ </tex-math></inline-formula> record values, <inline-formula> <tex-math notation="LaTeX">$(iii)$ </tex-math></inline-formula> Type-II censoring and <inline-formula> <tex-math notation="LaTeX">$(iv)$ </tex-math></inline-formula> progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.
In this paper, estimation of the location parameter of a half-normal distribution is considered. ... more In this paper, estimation of the location parameter of a half-normal distribution is considered. Some unbiased as well as biased estimators are derived. Admissibility and minimaxity of Pitman estimator are proved. A complete class of estimators among multiples of the maximum likelihood estimator is obtained. We develop a one-sided asymptotic confidence interval for the location parameter. Numerical comparisons of the percentage risk improvements over maximum likelihood estimator of various estimators are carried out.
Consider two independent normal populations with a common variance and ordered means. For this mo... more Consider two independent normal populations with a common variance and ordered means. For this model, we study the problem of estimating a common variance and a common precision with respect to a general class of scale invariant loss functions. A general minimaxity result is established for estimating the common variance. It is shown that the best affine equivariant estimator and the restricted maximum likelihood estimator are inadmissible. In this direction, we derive a Stein-type improved estimator. We further derive a smooth estimator which improves upon the best affine equivariant estimator. In particular, various scale invariant loss functions are considered and several improved estimators are presented. Furthermore, a simulation study is performed to find the performance of the improved estimators developed in this paper. Similar results are obtained for the problem of estimating a common precision for the stated model under a general class of scale invariant loss functions.
ABSTRACT Let be populations with having an inverse Gaussian distribution with unknown mean and un... more ABSTRACT Let be populations with having an inverse Gaussian distribution with unknown mean and unknown scale-like parameter , respectively. We study the problem of classification of an observation when prior information suggests some orderings on parameters. When the means are equal but unknown, we derive plug-in Bayes classification rules based on the maximum likelihood estimator (MLE), Graybill-Deal type estimator and shrinkage estimator of the common mean. When all parameters are unknown and unequal, we also derive likelihood ratio-based classification rules. For more than two populations, we suggest ordered rules when s follow an ordering. When the means are unequal, we also derive rules assuming ordering among either s or s. Extensive simulations are carried out to compare the proposed rules with respect to expected probabilities of correct classification. Applications of these classification rules are described using real data sets.
International journal of applied mathematics and statistics, 2008
... Subscription or Fee Access. Admissibility of the Pitman Estimator of Ordered Location Paramet... more ... Subscription or Fee Access. Admissibility of the Pitman Estimator of Ordered Location Parameters. Somesh Kumar, Ajay Kumar, Tanuja Srivastava. Abstract. Abstract available in paper due to mathematical expressions. Full Text: PDF.
Estimation of the scale parameter of the scale mixture of a location–scale family under the scale... more Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.
Communications in Statistics - Simulation and Computation, 2015
ABSTRACT A Langevin distribution with two parameters (mean direction and concentration parameter)... more ABSTRACT A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this article, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Jeffreys’ priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behavior of Bayes estimators is carried out with the help of simulations.
A Langevin distribution with two parameters (mean direction and concentration parameter) has been... more A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this paper, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Je ffreys priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behaviour of Bayes estimators is carried out with the help of simulations.
The main purpose of this work is to identify the determinants for Attainment in Schooling in a ru... more The main purpose of this work is to identify the determinants for Attainment in Schooling in a rural setting. The Ordinary Least Square Method (O.L.S.) of the regression analysis has been the main plank of the present paper. The estimates of multiple regression analysis have been tested for multicollinearity, heteroscedasticity and serial correlation.
In this paper, the problem of estimating the mean of a two parameter lognormal distribution is co... more In this paper, the problem of estimating the mean of a two parameter lognormal distribution is considered under the assumption that one of its parameter is restricted a priori. Several classical and Bayes estimators are proposed. Some of them are shown to be inadmissible using Rao-Blackwellization. Finally, risk performance of all estimators is compared numerically.
In many real life situations, prior information about the parameters is available, such as the or... more In many real life situations, prior information about the parameters is available, such as the ordering of the parameters. Incorporating this prior information about the order restrictions on parameters leads to more efficient estimators. In the present communication, we investigate estimation of the ordered scale parameters of two shifted exponential distributions with unknown location parameters under a class of bowl-shaped loss functions. We have proved that the best affine equivariant estimator (BAEE) is inadmissible. Various non smooth and smooth estimators has been obtained which improve upon the BAEE. In particular we have derived the improved estimators for some well known loss functions. Finally numerical comparison is carried out to compare the risk performance of the proposed estimators.
Abstract In the present article, we have studied the estimation of the reciprocal of scale parame... more Abstract In the present article, we have studied the estimation of the reciprocal of scale parameter , that is, hazard rate of a two parameter exponential distribution based on a doubly censored sample. This estimation problem has been investigated under a general class of bowl-shaped scale invariant loss functions. It is established that the best affine equivariant estimator (BAEE) is inadmissible by deriving an improved estimator. This estimator is non-smooth. Further, we have obtained a smooth improved estimator. A class of scale equivariant estimator is considered and sufficient conditions are derived under which these estimators improve upon the BAEE. In particular, using these results we have obtained the improved estimators for three special loss functions. A simulation study is conducted to compare the risk performance of the proposed estimators. Finally, we analyze a real data set.
There are various situations where prior information is available on parameters in the form of or... more There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: <inline-formula> <tex-math notation="LaTeX">$(i)$ </tex-math></inline-formula> i.i.d. sampling, <inline-formula> <tex-math notation="LaTeX">$(ii)$ </tex-math></inline-formula> record values, <inline-formula> <tex-math notation="LaTeX">$(iii)$ </tex-math></inline-formula> Type-II censoring and <inline-formula> <tex-math notation="LaTeX">$(iv)$ </tex-math></inline-formula> progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.
Uploads
Papers by SOMESH KUMAR