Statistical learning with ranked set samples has shown promising results in estimating various po... more Statistical learning with ranked set samples has shown promising results in estimating various population parameters. Despite the vast literature on rank-based statistical learning methodologies, very little effort has been devoted to studying regression analysis with such samples. A pressing issue is how to incorporate the rank information of ranked set samples into the analysis. We propose two methodologies based on a weighted least squares approach and multilevel modeling to better incorporate the rank information of such samples into the estimation and prediction processes of regression-type models. Our approaches reveal significant improvements in both estimation and prediction problems over already existing methods in the literature and the corresponding ones with simple random samples. We study the robustness of our methods with respect to the misspecification of the distribution of the error terms. Also, we show that rank-based regression models can effectively predict simple random test data by assigning ranks to them a posteriori using judgment post-stratification. Theoretical results are augmented with simulations and an osteoporosis study based on a real data set from the Bone Mineral Density (BMD) program of Manitoba to estimate the bone mineral density level of patients using easy to obtain covariates. This article is protected by copyright. All rights reserved.
Communications in Statistics - Theory and Methods, 2017
ABSTRACT We consider robust Bayesian prediction of a function of unobserved data based on observe... more ABSTRACT We consider robust Bayesian prediction of a function of unobserved data based on observed data under an asymmetric loss function. Under a general linear-exponential posterior risk function, the posterior regret gamma-minimax (PRGM), conditional gamma-minimax (CGM), and most stable (MS) predictors are obtained when the prior distribution belongs to a general class of prior distributions. We use this general form to find the PRGM, CGM, and MS predictors of a general linear combination of the finite population values under LINEX loss function on the basis of two classes of priors in a normal model. Also, under the general ε-contamination class of prior distributions, the PRGM predictor of a general linear combination of the finite population values is obtained. Finally, we provide a real-life example to predict a finite population mean and compare the estimated risk and risk bias of the obtained predictors under the LINEX loss function by a simulation study.
In clinical research, one of the key problems is to estimate the effect of the best treatment amo... more In clinical research, one of the key problems is to estimate the effect of the best treatment among the given k treatments in two-stage adaptive design. Suppose the effects of two treatments have normal distributions with means θ1 and θ2, respectively, and common known variance σ2. In the first stage, random samples of size n1 with means X1 and X2 are chosen from the two populations. Then the population with the larger or smaller sample mean XM is selected, and a random sample of size n2 with mean YM is chosen from this population in the second stage of design. Our aim is to estimate the mean θM or θJ of the selected population based on XM and YM in two-stage adaptive design under the reflected normal loss function. We obtain minimax estimators of θM and θJ, and then provide some sufficient conditions for the inadmissibility of estimators of θM and θJ. Theoretical results are augmented with a simulation study as well as a real data application.
Judgment post‐stratification is used to supplement observations taken from finite mixture models ... more Judgment post‐stratification is used to supplement observations taken from finite mixture models with additional easy to obtain rank information and incorporate it in the estimation of model parameters. To do this, sampled units are post‐stratified on ranks by randomly selecting comparison sets for each unit from the underlying population and assigning ranks to them using available auxiliary information or judgment ranking. This results in a set of independent order statistics from the underlying model, where the number of units in each rank class is random. We consider cases where one or more rankers with different ranking abilities are used to provide judgment ranks. The judgment ranks are then combined to produce a strength of agreement measure for each observation. This strength measure is implemented in the maximum likelihood estimation of model parameters via a suitable expectation maximization algorithm. Simulation studies are conducted to evaluate the performance of the esti...
The problem of estimation after selection arises when we select a population from the given k pop... more The problem of estimation after selection arises when we select a population from the given k populations by a selection rule, and estimate the parameter of the selected population. In this paper we consider the problem of estimation of the scale parameter of the selected Pareto population $$\theta _{M}$$θM (or $$\theta _{J}$$θJ) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of $$\theta _{M}$$θM and $$\theta _{J}$$θJ are obtained. In the case of $$k=2,$$k=2, we give a sufficient condition for minimaxity of an estimator of $$\theta _{M}$$θM and $$\theta _{J},$$θJ, and show that the UMRU and natural estimators of $$\theta _{J}$$θJ are minimax. Also the class of linear admissible estimators of $$\theta _{M}$$θM and $$\theta _{J}$$θJ are obtained which contain the natural estimator. By using the Brewester–Ziedeck technique we find sufficient condition for inadmissibility of some scale and permutation invariant estimators of $$\theta _{J},$$θJ, and show that the UMRU estimator of $$\theta _{J}$$θJ is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.
Robust Bayesian analysis is concerned with the problem of making decisions about some future obse... more Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.
A subclass of the scale-parameter exponential family is considered and for the rth power of the s... more A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a ...
Journal of Statistical Planning and Inference, 1996
... Inference 52 (1996) 7791 journal of statistical planning and inference Estimation of scale pa... more ... Inference 52 (1996) 7791 journal of statistical planning and inference Estimation of scale parameter under entropy loss function Ahmad Parsian*, Nader ... with = a2; inverse Gaussian with zero drift and q = 1 2. Estimation of the scale parameter, introduced by Pitman, dates back ...
Statistical learning with ranked set samples has shown promising results in estimating various po... more Statistical learning with ranked set samples has shown promising results in estimating various population parameters. Despite the vast literature on rank-based statistical learning methodologies, very little effort has been devoted to studying regression analysis with such samples. A pressing issue is how to incorporate the rank information of ranked set samples into the analysis. We propose two methodologies based on a weighted least squares approach and multilevel modeling to better incorporate the rank information of such samples into the estimation and prediction processes of regression-type models. Our approaches reveal significant improvements in both estimation and prediction problems over already existing methods in the literature and the corresponding ones with simple random samples. We study the robustness of our methods with respect to the misspecification of the distribution of the error terms. Also, we show that rank-based regression models can effectively predict simple random test data by assigning ranks to them a posteriori using judgment post-stratification. Theoretical results are augmented with simulations and an osteoporosis study based on a real data set from the Bone Mineral Density (BMD) program of Manitoba to estimate the bone mineral density level of patients using easy to obtain covariates. This article is protected by copyright. All rights reserved.
Communications in Statistics - Theory and Methods, 2017
ABSTRACT We consider robust Bayesian prediction of a function of unobserved data based on observe... more ABSTRACT We consider robust Bayesian prediction of a function of unobserved data based on observed data under an asymmetric loss function. Under a general linear-exponential posterior risk function, the posterior regret gamma-minimax (PRGM), conditional gamma-minimax (CGM), and most stable (MS) predictors are obtained when the prior distribution belongs to a general class of prior distributions. We use this general form to find the PRGM, CGM, and MS predictors of a general linear combination of the finite population values under LINEX loss function on the basis of two classes of priors in a normal model. Also, under the general ε-contamination class of prior distributions, the PRGM predictor of a general linear combination of the finite population values is obtained. Finally, we provide a real-life example to predict a finite population mean and compare the estimated risk and risk bias of the obtained predictors under the LINEX loss function by a simulation study.
In clinical research, one of the key problems is to estimate the effect of the best treatment amo... more In clinical research, one of the key problems is to estimate the effect of the best treatment among the given k treatments in two-stage adaptive design. Suppose the effects of two treatments have normal distributions with means θ1 and θ2, respectively, and common known variance σ2. In the first stage, random samples of size n1 with means X1 and X2 are chosen from the two populations. Then the population with the larger or smaller sample mean XM is selected, and a random sample of size n2 with mean YM is chosen from this population in the second stage of design. Our aim is to estimate the mean θM or θJ of the selected population based on XM and YM in two-stage adaptive design under the reflected normal loss function. We obtain minimax estimators of θM and θJ, and then provide some sufficient conditions for the inadmissibility of estimators of θM and θJ. Theoretical results are augmented with a simulation study as well as a real data application.
Judgment post‐stratification is used to supplement observations taken from finite mixture models ... more Judgment post‐stratification is used to supplement observations taken from finite mixture models with additional easy to obtain rank information and incorporate it in the estimation of model parameters. To do this, sampled units are post‐stratified on ranks by randomly selecting comparison sets for each unit from the underlying population and assigning ranks to them using available auxiliary information or judgment ranking. This results in a set of independent order statistics from the underlying model, where the number of units in each rank class is random. We consider cases where one or more rankers with different ranking abilities are used to provide judgment ranks. The judgment ranks are then combined to produce a strength of agreement measure for each observation. This strength measure is implemented in the maximum likelihood estimation of model parameters via a suitable expectation maximization algorithm. Simulation studies are conducted to evaluate the performance of the esti...
The problem of estimation after selection arises when we select a population from the given k pop... more The problem of estimation after selection arises when we select a population from the given k populations by a selection rule, and estimate the parameter of the selected population. In this paper we consider the problem of estimation of the scale parameter of the selected Pareto population $$\theta _{M}$$θM (or $$\theta _{J}$$θJ) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of $$\theta _{M}$$θM and $$\theta _{J}$$θJ are obtained. In the case of $$k=2,$$k=2, we give a sufficient condition for minimaxity of an estimator of $$\theta _{M}$$θM and $$\theta _{J},$$θJ, and show that the UMRU and natural estimators of $$\theta _{J}$$θJ are minimax. Also the class of linear admissible estimators of $$\theta _{M}$$θM and $$\theta _{J}$$θJ are obtained which contain the natural estimator. By using the Brewester–Ziedeck technique we find sufficient condition for inadmissibility of some scale and permutation invariant estimators of $$\theta _{J},$$θJ, and show that the UMRU estimator of $$\theta _{J}$$θJ is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.
Robust Bayesian analysis is concerned with the problem of making decisions about some future obse... more Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.
A subclass of the scale-parameter exponential family is considered and for the rth power of the s... more A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a ...
Journal of Statistical Planning and Inference, 1996
... Inference 52 (1996) 7791 journal of statistical planning and inference Estimation of scale pa... more ... Inference 52 (1996) 7791 journal of statistical planning and inference Estimation of scale parameter under entropy loss function Ahmad Parsian*, Nader ... with = a2; inverse Gaussian with zero drift and q = 1 2. Estimation of the scale parameter, introduced by Pitman, dates back ...
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