Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bezier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limacon of Pascal. The extension to 3D rational curves is also tackled

Conditional stochastic dominance testing

This article proposes bootstrap-based stochastic dominance tests for nonparametric conditional distributions and their moments. We exploit the fact that a conditional distribution dominates the other if and only if the difference between the marginal joint distributions is monotonic in the explanatory variable for each value of the dependent variable. The proposed test statistic compares restricted and unrestricted estimators of the difference between the joint distributions, and can be implemented under minimal smoothness requirements on the underlying nonparametric curves and without resorting to smooth estimation. The finite sample properties of the proposed tests are examined by means of a Monte Carlo study. We report an application to studying the impact on post-intervention earnings of the National Supported Work Demonstration, a randomized labor training program carried out in the 1970s.Nonparametric testing, Conditional stochastic dominance, Conditional inequality restrictions, Least concave majorant, Treatment effects

Exploring the returns to scale in food preparation (baking penny buns at home)

We show that as household size increases, households substitute away from prepared foods and towards ingredients. They also devote more time to food preparation. These observations (1) are consistent with a simple model with home production, returns to scale in the time input to food preparation, and varieties of food that differ in the required time input; (2) support the idea that returns to scale in home production are an important source of returns to scale in consumption; and (3), mean that across household sizes, household market expenditures on food are not proportional to food consumption quantities. The latter may provide a partial explanation for a puzzle raised by Deaton and Paxson.Household returns to scale, home production, food preparation

Exploring the Returns-to-Scale in Food Preparation

We show that as household size increases, households substitute away from prepared foods and towards ingredients. They also devote more time to food preparation. These observations (1) are consistent with a simple model with home production, returns to scale in the time input to food preparation, and varieties of food that differ in the required time input; (2) support the idea that returns to scale in home production are an important source of returns to scale in consumption; and (3), mean that across household sizes, household market expenditures on food are not proportional to food consumption quantities. The latter may provide a partial explanation for a puzzle raised by Deaton and Paxson.

Exploring the Returns to Scale in Food Preparation (Baking Penny Buns at Home)

We show that as household size increases, households substitute away from prepared foods and towards ingredients. They also devote more time to food preparation. These observations (1) are consistent with a simple model with home production, returns to scale in the time input to food preparation, and varieties of food that differ in the required time input; (2) support the idea that returns to scale in home production are an important source of returns to scale in consumption; and (3), mean that across household sizes, household market expenditures on food are not proportional to food consumption quantities. The latter may provide a partial explanation for a puzzle raised by Deaton and Paxson.household returns to scale; home production; food preparation

Methods of Inference for Nonparametric Curves and Surfaces

Nonparametric regression models offer attractive extensions to the familiar approaches of parametric regression. They adapt to departures from standard parametric forms and therefore have the potential to capture features which may otherwise go unnoticed. This property accounts for the large volume of work in the area of estimation of nonparametric models which has emerged over the last two decades. Inferential techniques using nonparametric model fits, however, have not been so quick to develop. This thesis contributes to this area of research by examining the task of assessing covariate effects via comparisons of nonparametric model fits. In particular, the asymptotic and finite sample bias properties of estimates obtained via local linear smoothers are a major consideration and methods of inference which take into account these properties are developed. Chapter 1 introduces and presents an overview of existing methods of estimation and inference amongst nonparametric regression. Chapter 2 focuses on the task of inference by considering the estimation of the error variance in the nonparametric model context. Special attention is given to the development and assessment of difference based estimators in the presence of two covariates. It is shown that difference based estimators are a viable alternative, in terms of accuracy, to standard residual based estimators. Chapters 3 and 4 employ the estimators of Chapter 2 in the development of test procedures which make comparisons amongst a class of bivariate nonparametric regression models. Chapter 3 develops the theoretical properties of several forms of the test statistic, with particular attention given to statistics based on direct comparisons of fitted values. The theory also highlights the role of centred smoothers and equivalent degrees of smoothing when nonparametric model fits are compared. The simulation studies reported in Chapter 4 compare the novel approaches developed in Chapter 3 with standard approaches based on differences in residual sums of squares, i.e. approximate F-tests. The results show that direct comparisons of fitted values offer an improvement in some settings and never perform less favourably in others. The choice of the error variance estimator is shown to be crucial, with different design spaces requiring different estimators. Specific attention is also given to the effect of correlation amongst the covariates on the tests' performances. Chapter 4 closes with an application of the methods to a real data set describing the spatial distribution of sea bed fauna in the Great Barrier Reef. Chapter 5 extends these methods beyond models with two covariates to models with an unlimited number of additive linear terms and a nonparametric component involving at most two covariates. Recent results which derive the asymptotic properties of models of this form show that the favourable properties of local linear regression in the bivariate setting extend to this multidimensional setting. Results of a simulation study are reported and show that there is much to be gained by making a direct comparison of fitted values in conjunction with a careful choice of the estimator of error variance. Chapters 6 and 7 describe applied projects in environmental and medical contexts respectively. Both of the sets of data contain relationships amongst covariates which are best described using nonparametric models. Chapter 6 considers 14 years of water quality monitoring data from the Firth of Clyde, Scotland. Interest lies in describing relationships between pollutants and environmental factors, including long term trends and seasonal patterns. Chapter 7 investigates the relationship between short term dosage of an immunosuppressive drug and the long term outcome of kidney transplantation patients. Chapter 8 concludes with a summary of the main findings of the thesis and a discussion of potential future work in this area. Although progress has been made in the settings considered in the thesis, further extensions are required before nonparametric modelling will achieve its full potential

Conditional stochastic dominance testing

This article proposes bootstrap-based stochastic dominance tests for nonparametric conditional distributions and their moments. We exploit the fact that a conditional distribution dominates the other if and only if the difference between the marginal joint distributions is monotonic in the explanatory variable for each value of the dependent variable. The proposed test statistic compares restricted and unrestricted estimators of the difference between the joint distributions, and can be implemented under minimal smoothness requirements on the underlying nonparametric curves and without resorting to smooth estimation. The finite sample properties of the proposed tests are examined by means of a Monte Carlo study. We report an application to studying the impact on post-intervention earnings of the National Supported Work Demonstration, a randomized labor training program carried out in the 1970s

Averaged Singular Integral Estimation as a Bias Reduction Technique

This paper proposes an averaged version of singular integral estimators, whose bias achieves higher rates of convergence under smoothing assumptions. We derive exact bias bounds, without imposing smoothing assumptions, which are a basis for deriving the rates of convergence under differentiability assumptions.Publicad

A note on the use of quantile regression in beta convergence analysis

We discuss how to interpret conflicting results obtained by the use of quantile regression methods in growth regression tests of β-convergence hypothesis and the results obtained by nonparametric methods. We show that the assumption of linearity may cause the non-rejection of the β-convergence hypothesis by quantile regression. We also show that using a nonparametric form of quantile regression, we can reject the hypothesis of β-convergence and confirm the results of divergence and formation of convergence clubs. We illustrate the discussion by using the conflicting results on convergence found in the dataset of per-capita income of Brazilian municipalities between 1970 and 1996.