Use of Shape Restricted Regression Methods for Fitting Model of Per Capita GDP: A Global Economic Scenario of 2018

The aim of the study is to analyze the pattern of Gross domestic product (GDP) according to Human development index (HDI) for 184 countries of the world. GDP per capita indicates only economic prosperity but not the overall development of the citizens of a country. This research tries to find out the beneath relationship of the financial state and human development of countries using the data of 2018. For demonstrating this analysis several parametric and non-parametric regression methods subject to shape restriction have been used. The study targets to shed light on comparative performance of shape constrained regression with cone projection, polynomial regression, LOESS, Istonic regression with pooled adjacent violators algorithm, Kernel regression, smoothing spline and generalized additive model in convex situation.

Inference in nonparametric/semiparametric moment equality models with shape restrictions

This paper studies the inference problem of an infinite‐dimensional parameter with a shape restriction. This parameter is identified by arbitrarily many unconditional moment equalities. The shape restriction leads to a convex restriction set. I propose a test of the shape restriction, which controls size uniformly and applies to both point‐identified and partially identified models. The test can be inverted to construct confidence sets after imposing the shape restriction. Monte Carlo experiments show the finite‐sample properties of this method. In an empirical illustration, I apply the method to ascending auctions held by the US Forest Service and show that imposing shape restrictions can significantly improve inference.

Nonparametric Instrumental-Variable Estimation

In this article, we introduce the commands npiv and npivcv, which implement nonparametric instrumental-variable (NPIV) estimation methods without and with a cross-validated choice of tuning parameters, respectively. Both commands can impose the constraint that the resulting estimated function is monotone. Using such a shape restriction may significantly improve the performance of the NPIV estimator (Chetverikov and Wilhelm, 2017, Econometrica 85: 1303–1320) because the ill-posedness of the NPIV estimation problem leads to unconstrained estimators that suffer from particularly poor statistical properties such as high variance. However, the constrained estimator that imposes the monotonicity significantly reduces variance by removing nonmonotone oscillations of the estimator. We provide a small Monte Carlo experiment to study the estimators’ finite-sample properties and an application to the estimation of gasoline demand functions.

Lung Segmentation in Feature Images with Gray and Shape Information

Accurate lung segmentation in chest radiography is an important and difficult task in the development of computer-aided diagnosis. Therefore, we proposed a lung segmentation method in feature images with gray and shape information. Firstly, we extracted six feature images, and built an initial shape model. Then, we calculated the gray cost in the feature images. Finally, the lung profile was determined by use of shape restriction. With the feature images and method of shape restriction, the mean overlap rate was improved to 75.60%. Therefore, the method proposed in our study can improve the performance of lung segmentation.

Length and surface area estimation under smoothness restrictions

The problem of estimating the Minkowski content L 0(G) of a body G ⊂ ℝ d is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L 0(G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.

Length and surface area estimation under smoothness restrictions

The problem of estimating the Minkowski content L0(G) of a body G ⊂ ℝd is considered. For d = 2, the Minkowski content represents the boundary length of G. It is assumed that a ball of radius r can roll inside and outside the boundary of G. We use this shape restriction to propose a new estimator for L0(G). This estimator is based on the information provided by a random sample, taken on a square containing G, in which we know whether a sample point is in G or not. We obtain the almost sure convergence rate for the proposed estimator.