Stable Asymptotics for M‐estimators

We review some first‐order and higher‐order asymptotic techniques for M‐estimators, and we study their stability in the presence of data contaminations. We show that the estimating function (ψ) and its derivative with respect to the parameter play a central role. We discuss in detail the first‐order Gaussian density approximation, saddlepoint density approximation, saddlepoint test, tail area approximation via the Lugannani–Rice formula and empirical saddlepoint density approximation (a technique related to the empirical likelihood method). For all these asymptotics, we show that a bounded ψ (in the Euclidean norm) and a bounded (e.g. in the Frobenius norm) yield stable inference in the presence of data contamination. We motivate and illustrate our findings by theoretical and numerical examples about the benchmark case of one‐dimensional location model.     

Moment estimation in discrete shifting level model applied to fast array‐CGH segmentation

We develop a mathematical theory needed for moment estimation of the parameters in a general shifting level process (SLP) treating, in particular, the finite state space case geometric finite normal (GFN) SLP. For the SLP, we give expressions for the moment estimators together with asymptotic (co)variances, following, completing, and correcting Cline (Journal of Applied Probability 20, 1983, 322–337); formulae are then made more explicit for the GFN‐SLP. To illustrate the potential uses, we then apply the moment estimation method to a GFN‐SLP model of array comparative genomic hybridization data. We obtain encouraging results in the sense that a segmentation based on the estimated parameters turns out to be faster than with other currently available methods, while being comparable in terms of sensitivity and specificity.     

Modelling Group Heterogeneity for Small Area Estimation Using M‐Quantiles

Small area estimation typically requires model‐based methods that depend on isolating the contribution to overall population heterogeneity associated with group (i.e. small area) membership. One way of doing this is via random effects models with latent group effects. Alternatively, one can use an M‐quantile ensemble model that assigns indices to sampled individuals characterising their contribution to overall sample heterogeneity. These indices are then aggregated to form group effects. The aim of this article is to contrast these two approaches to characterising group effects and to illustrate them in the context of small area estimation. In doing so, we consider a range of different data types, including continuous data, count data and binary response data.     

ELECTRONIC CURRENCY IN AFRICA: M‐PESA AS PRIVATE INSIDE MONEY

Electronic monies such as Bitcoin and mobile money have become popular in recent times. However, unlike with bitcoin, attempts to classify M‐Pesa theoretically have been rare. This article classifies M‐Pesa as a privately issued inside money, which evolved as an entrepreneurial response to the lack of traditional monies. It describes the institutional framework within which M‐Pesa and other monies evolved while highlighting how competitive forces led to mutual acceptance of competing inside monies, lower prices as well as greater choice for consumers. The case of M‐Pesa and other electronic monies illustrates the potential even today for entrepreneurial and market processes in money given the right institutional setting, especially in developing countries.     

The rate of convergence in distribution of the maxima

Abstract  Assume the normalized maxima, from an independent and identically distributed sequence of random variables, converge in distribution to a non-degenerate random variable with extreme value distribution G ( x ). We derive a pointwise rate for this convergence which holds for all n and x with G(x ) > 0. A closer examination of this result suggests how a uniform rate can be obtained.     

L‐moments skewness and kurtosis as measures of regional convergence and cohesion

The paper deals with the statistical modeling of convergence and cohesion over time with the use of kurtosis, skewness and L‐moments. Changes in the shape of the distribution related to the spatial allocation of socio‐economic phenomena are considered as an evidence of global shift, divergence or convergence. Cross‐sectional time‐series statistical modeling of variables of interest is to overpass the minors of econometric theoretical models of convergence and cohesion determinants. L‐moments perform much more stable and interpretable than classical measures. Empirical evidence of panel data proves that one pure pattern (global shift, polarization or cohesion) rarely exists and joint analysis is required.     

Gamma-admissibility of estimators in the one-parameter exponential family

Summary Admissibility of estimators under vague prior information on the distribution of the unknown parameter is studied which leads to the notion of gamma-admissibility. A sufficient condition for an estimator of the formδ(x)=(ax+b)/(cx+d) to be gamma-admissible in the one-parameter exponential family under squared error loss is established. As an application of this result two equalizer rules are shown to be unique gamma-minimax estimators by proving their gamma-admissibility.     

A central limit theorem for the Hellinger loss of Grenander‐type estimators

We consider Grenander‐type estimators for a monotone function , obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.     

Residual‐based IV estimation of dynamic panel data models with fixed effects

In dynamic panel regression, when the variance ratio of individual effects to disturbance is large, the system‐GMM estimator will have large asymptotic variance and poor finite sample performance. To deal with this variance ratio problem, we propose a residual‐based instrumental variables (RIV) estimator, which uses the residual from regressing Δy_i,t?1 on as the instrument for the level equation. The RIV estimator proposed is consistent and asymptotically normal under general assumptions. More importantly, its asymptotic variance is almost unaffected by the variance ratio of individual effects to disturbance. Monte Carlo simulations show that the RIV estimator has better finite sample performance compared to alternative estimators. The RIV estimator generates less finite sample bias than difference‐GMM, system‐GMM, collapsing‐GMM and Level‐IV estimators in most cases. Under RIV estimation, the variance ratio problem is well controlled, and the empirical distribution of its t‐statistic is similar to the standard normal distribution for moderate sample sizes.     

Asymptotic comparison at optimal levels of reduced‐bias extreme value index estimators

In this article we are interested in the asymptotic comparison, at optimal levels, of a set of semi‐parametric reduced‐bias extreme value (EV) index estimators, valid for a wide class of heavy‐tailed models, underlying the available data. Again, as in the classical case, there is not any estimator that can always dominate the alternatives, but interesting clear‐cut patterns are found. Consequently, and in practice, a suitable choice of a set of EV index estimators will jointly enable us to better estimate the EV index γ, the primary parameter of extreme events.     

Minimax estimators of a normal variance

In the estimation problem of unknown variance of a multivariate normal distribution, a new class of minimax estimators is obtained. It is noted that a sequence of estimators in our class converges to the Stein's truncated estimator. Received: March 1998     

Some asymptotic results of a non‐parametric conditional mode estimator for functional time‐series data

We consider the estimation of the conditional mode function when the covariates take values in some abstract function space. The main goal of this paper was to establish the almost complete convergence and the asymptotic normality of the kernel estimator of the conditional mode when the process is assumed to be strongly mixing and under the concentration property over the functional regressors. Some applications are given. This approach can be applied in time‐series analysis to the prediction and confidence band building. We illustrate our methodology by using El Nio data.     

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H₀ that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H₁ that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.     

Model‐Assisted Estimation of a Spatial Population Mean

This paper deals with the estimation of the mean of a spatial population. Under a design‐based approach to inference, an estimator assisted by a penalized spline regression model is proposed and studied. Proof that the estimator is design‐consistent and has a normal limiting distribution is provided. A simulation study is carried out to investigate the performance of the new estimator and its variance estimator, in terms of relative bias, efficiency, and confidence interval coverage rate. The results show that gains in efficiency over standard estimators in classical sampling theory may be impressive.     

Using Complex Surveys to Estimate the L1‐Median of a Functional Variable: Application to Electricity Load Curves

Mean profiles are widely used as indicators of the electricity consumption habits of customers. Currently, in Électricité De France (EDF), class load profiles are estimated using point‐wise mean profiles. Unfortunately, it is well known that the mean is highly sensitive to the presence of outliers, such as one or more consumers with unusually high‐levels of consumption. In this paper, we propose an alternative to the mean profile: the L ₁ ‐ median profile which is more robust. When dealing with large data sets of functional data (load curves for example), survey sampling approaches are useful for estimating the median profile avoiding storing the whole data. We propose here several sampling strategies and estimators to estimate the median trajectory. A comparison between them is illustrated by means of a test population. We develop a stratification based on the linearized variable which substantially improves the accuracy of the estimator compared to simple random sampling without replacement. We suggest also an improved estimator that takes into account auxiliary information. Some potential areas for future research are also highlighted.     

Asymptotic behaviour of regression pre-test estimators with minimal Bayes risk

Pre-test estimators (PTE) are considered which are optimal under a Bayes risk among PTE with general measurable sets as “regions of significance” for the test statistic t associated with the estimate of a given regression coefficient. Asymptotic and some finite sample results are stated and numerical experiments are commented on.     

Country and industry convergence of equity markets: International evidence from club convergence and clustering

This study employs the panel convergence methodology developed by Phillips and Sul (2007) to explore the convergence dynamics of international equity markets. The analysis considers both country and industry effects. While traditional portfolio management strategies usually follow a top-down procedure, assuming that country-level effects drive financial aggregates (e.g., stock returns) our empirical results suggest that the equity markets of 37 of the 42 counties in our sample do form a unified convergence club. The empirical findings, however, also show more numerous stock-price convergence clubs in certain industries. That is, country factors play a more important role in explaining the actual convergence in real stock prices than industry factors. Conversely, the volatility of stock prices exhibits much more evidence of convergence than stock prices. These findings should assist portfolio managers in the design and implementation of appropriate portfolio management strategies. Regulatory authorities also can benefit in the design of financial regulation.     

Design‐based Estimators Calibrated on Estimated Totals from Multiple Surveys

The use of auxiliary variables to improve the efficiency of estimators is a well‐known strategy in survey sampling. Typically, the auxiliary variables used are the totals of appropriate measurement that are exactly known from registers or administrative sources. Increasingly, however, these totals are estimated from surveys and are then used to calibrate estimators and improve their efficiency. We consider different types of survey structures and develop design‐based estimators that are calibrated on known as well as estimated totals of auxiliary variables. The optimality properties of these estimators are studied. These estimators can be viewed as extensions of the Montanari generalised regression estimator adapted to the more complex situations. The paper studies interesting special cases to develop insights and guidelines to properly manage the survey‐estimated auxiliary totals.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.