Biometric technology in rural credit markets

Identity theft is a common crime the world over. In developing countries, the damage caused by identity theft and identity fraud goes far beyond the individual victim, however, and ultimately creates a direct impediment to progress, particularly in credit markets. Recent research reveals that biometric technology can help reduce these problems. A biometric is a measurement of physical or behavioral characteristics used to verify or analyze identity. Common biometrics include a person’s fingerprints; face, iris, or retina patterns; speech; or handwritten signature. These are effective personal identifiers because they are unique and intrinsic to each person, so, unlike conventional identification methods (such as passport numbers or government-issued identification cards), they cannot be forgotten, lost, or stolen. Recent advances in recognition technology coupled with increases in both digital storage capacity and computer processing speeds have made biometric technology (for example, ocular or fingerprint scanners) feasible in many applications, from controlling restricted building access to allowing more effective delivery of targeted government programs with large-scale identification systems, such as those being implemented in India by the Unique Identification Authority of India. Biometric technology can also improve access to credit and insurance markets, especially in countries that do not have a unique identification system, where identity fraud—the use of someone else’s identity or a fictitious one—to gain access to services otherwise unavailable to an individual is rather common. For example, lenders in Malawi describe past borrowers who purposefully defaulted then tried to obtain a fresh loan from the same or another institution under a false identity. And, although less common in developing countries because markets are less developed, the potential for sick individuals without healthcare coverage to use the insurance policy of a friend or relative does exist. The response of lenders and insurance companies has been to restrict the supply of such services to the detriment of the greater population, not just those people committing identity fraud.Biometric technology, Commodities, conditional cash transfers, credit, Insurance, rural areas, Subsidies,

On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

It is shown that the variable bandwidth density estimator proposed by McKay (1993a and b) following earlier findings by Abramson (1982) approximates density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum norm over bounded sets where the preliminary density estimates on which they are based are bounded away from zero. A somewhat more complicated estimator proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb R)$ is also shown to attain minimax rates in sup norm over the same kind of sets. These estimators are strict probability densities.Comment: 29 page

Uniform limit theorems for wavelet density estimators

Let $p_n(y)=\sum_k\hat{\alpha}_k\phi(y-k)+\sum_{l=0}^{j_n-1}\sum_k\hat {\beta}_{lk}2^{l/2}\psi(2^ly-k)$ be the linear wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support), $\hat{\alpha}_k$, $\hat{\beta}_{lk}$ are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density $p_0$ on $\mathbb{R}$, and $j_n\in\mathbb{Z}$, $j_n\nearrow\infty$. Several uniform limit theorems are proved: First, the almost sure rate of convergence of $\sup_{y\in\mathbb{R}}|p_n(y)-Ep_n(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that $\sup_{y\in\mathbb{R}}|p_n(y)-p_0(y)|$ attains the optimal almost sure rate of convergence for estimating $p_0$, if $j_n$ is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of $p_n$, that is, for the stochastic processes $\sqrt{n}(F_n ^W(s)-F(s))=\sqrt{n}\int_{-\infty}^s(p_n-p_0),s\in\mathbb{R}$, are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int(p_n-p_0)f$, $f\in\mathcal{F}$, for other Donsker classes $\mathcal{F}$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508--539].Comment: Published in at http://dx.doi.org/10.1214/08-AOP447 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Concentration inequalities and asymptotic results for ratio type empirical processes

Let $\mathcal{F}$ be a class of measurable functions on a measurable space $(S,\mathcal{S})$ with values in $[0,1]$ and let $$P_n=n^{-1}\sum_{i=1}^n\delta_{X_i}$$ be the empirical measure based on an i.i.d. sample $(X_1,...,X_n)$ from a probability distribution $P$ on $(S,\mathcal{S})$. We study the behavior of suprema of the following type: $$\sup_{r_n<\sigma_Pf\leq \delta_n}\frac{|P_nf-Pf|}{\phi(\sigma_Pf)},$$ where $\sigma_Pf\ge\operatorname {Var}^{1/2}_Pf$ and $\phi$ is a continuous, strictly increasing function with $\phi(0)=0$. Using Talagrand's concentration inequality for empirical processes, we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior, expressing the conditions in terms of expectations of localized suprema of empirical processes. We also prove new bounds for expected values of sup-norms of empirical processes in terms of the largest $\sigma_Pf$ and the $L_2(P)$ norm of the envelope of the function class, which are especially suited for estimating localized suprema. With this technique, we extend to function classes most of the known results on ratio type suprema of empirical processes, including some of Alexander's results for VC classes of sets. We also consider applications of these results to several important problems in nonparametric statistics and in learning theory (including general excess risk bounds in empirical risk minimization and their versions for $L_2$-regression and classification and ratio type bounds for margin distributions in classification).Comment: Published at http://dx.doi.org/10.1214/009117906000000070 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Empirical graph Laplacian approximation of Laplace--Beltrami operators: Large sample results

Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators $$\Delta_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M$$ where ${K(u):={\frac{1}{(4\pi)^{d/2}}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and ${h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of ${M,}$ ${\Delta_Mf}$ (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ${\Delta_{h_n,n}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ for smooth functions ${f}$ both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)}$ and ${\mu}$ is the Riemannian volume measure). In particular, we show that for any class ${{\cal F}}$ of three times differentiable functions on ${M}$ with uniformly bounded derivatives $$\sup_{p\in M}\sup_{f\in F}\Big|\Delta_{h_n,p}f(p)-\frac{1}{|\mu|}\Delta_Mf(p)\Big|= O\Big(\sqrt{\frac{\log(1/h_n)}{nh_n^{d+2}}}\Big) a.s.$$ as soon as $$nh_n^{d+2}/\log h_n^{-1}\to \infty and nh^{d+4}_n/\log h_n^{-1}\to 0,$$ and also prove asymptotic normality of ${\Delta_{h_n,p}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ (functional CLT) for a fixed ${p\in M}$ and uniformly in ${f}.$Comment: Published at http://dx.doi.org/10.1214/074921706000000888 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Integrability of Liénard systems with a strong saddle

We study the local analytic integrability for real Li\'{e}nard systems, $\dot x=y-F(x),$ $\dot y= x$, with $F(0)=0$ but $F'(0)\ne0,$ which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the $[p:-q]$ resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the $[p:-q]$ resonant saddle into a strong saddle.The first author is partially supported by a MINECO/FEDER grant number MTM2014- 53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR-1204. The second author is partially supported by a FEDER-MINECO grant MTM2016-77278-P, a MINEC0 grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568