On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

Six impossible things before breakfast

The financial world is in meltdown—and it is the fault of modern financial theory. How did the theorists manage to believe so many impossible things for so long? William Janeway marvels and explains.     

Best ever athlete?

Who was the best athlete the modern world has ever seen? Must we refer to ancient history to decide? Joseph Hilbe looks at the contenders.     

A semidefinite programming study of the Elfving theorem

The theorem of Elfving is one of the most important and earliest results which have led to the theory of optimal design of experiments. This paper presents a fresh study of it from the viewpoint of modern semidefinite programming. There is one-to-one correspondence between solutions of the derived semidefinite programming problem (SDP) and c-optimal designs. We also derive a uniqueness theorem which ensures a unique optimal design without assuming the linear independence property over the largest set of supporting points. The SDP can also be cast as an ?₁‐convex program which has recently been extensively studied and often yields sparse solutions. Our numerical experiments on the trigonometric regression model confirm that the SDP does produce a sparse optimal design.     

Who wins the football?

Portsmouth have dumped their manager—will it change their results? Chelsea are on a winning streak—does it mean anything? Media pundits and sports fans—and punters—devote time and energy to speculation about who will win Saturday's beautiful game. The discussions—in pubs, newspapers and dressing rooms—are usually based on subjective opinion, or evidence that is at best partial or impressionistic. Can statisticians do better? John Goddard offers a large-scale analysis of patterns in English league results from the past 35 football seasons—new managers, winning and losing streaks, home advantages and all.     

The Langevin distribution on high dimensional spheres

The Langevin (or von-Mises) distribution L(μ,k,q) of a unit vector in Rq has been well studied for the cases of large samples (n? 8) and high concentrations (K? 8). Here we study it as q? 8. Interesting results are only obtained when we use L(μ,q^½,q).     

Life in the Trenches of Print and Web Publishing: An Interview with Jenna Freedman, Curator of the Barnard Zine Collection

For this issue of “From Picas to Pixels,” incoming co-editor Jessamyn West talks to fellow rogue librarian Jenna Freedman, curator of the Barnard Zine Collection. Are zines serials or monographs? Should they circulate or not? Jenna and Jessamyn discuss the difficulties of getting a zine collection started, the difficulties of keeping them alive, and much more.     

Defective Galton–Watson processes

The Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k ? 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑_{k ? 0}p_k = 1 ? ? for some ? ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ?. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ? → 0.     

Minimum Cost Linear Trend Free 2n-(n-k) Designs of Resolution IV

This article extends the resolution of time trend free designs for sequential 2^n-p experiments from III into IV and minimizes the number of factor level changes between runs (i.e., cost) by constructing a catalog of (2^k?2 ?1) minimum cost linear trend free resolution IV 2^n?(n?k) designs (2^k?2 ≤ n ≤ 2^k?1?2) from the full 2^k factorial experiment using the interactions-main effects assignment technique. Each systematic 2^n?(n?k) design in the catalog is economic in minimum number of factor level changes and allows for the estimation of all n main effects unbiased by either the linear time trend (which may be present in the 2^n?(n?k) sequentially generated responses) or the non negligible two-factor interactions. This article provides for each 2^n?(n?k) design: (1) the defining relation or the alias structure; (2) the k independent generators for sequencing the 2^n?(n?k) runs by the generalized foldover scheme; and (3) the minimum cost represented by the total number of factor level changes between the 2^n?(n?k) runs. All k main effects of the 2^k experiment are excluded from the selection assignment process due to their nonlinear time trend resistance as well as excluding a total of (2^k?1 –k +1) interactions violating the resolution IV requirement.     

London murders: a predictable pattern?

Four stabbings to death in a single day. Ninety murders in 7 months. Shocking figures—or are they? Knife crime makes the headlines almost daily but are Londoners really at increased risk of being murdered? David Spiegelhalter and Arthur Barnett investigate—and find a predictable pattern of murder.     

Safety in numbers

How do we behave in emergencies? If fire breaks out in a building do we act as thinking individuals, or as chaotic panicking automata? Safety officers have to know—and Steven Gwynne says that they lack the data.     

Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Because of the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of the largest eigenvalue. A simulation‐based method is proposed to compute the 1 ? α quantile of this eigenvalue. With the computation power of modern computers, the simultaneous confidence tube can be computed fast and accurately. A real‐data example is used to illustrate the method, and many potential research problems have been pointed out.     

Procrustes analysis on some special manifolds

We pnwnt some ilicorciical results obtained by applying Procrustes methods to the statistical analysis on the two special manifolds, the Stiefel manifold V_k,m and the Grassmann manifold G_k,m?k or, equivalently, the manifold P_k,m?k of all m × m orthogonal projection matrices idempoteut of rank k. Procrustes representations of V_k,m and P_k,m?k by means of equivalence classes of matrices are considered, and Procrustes statistics and means are defined via the ordinary, weighted and generalized Procrustes methods. We discuss perturbation theory in Procrustes analysis on V_k,m and P_k,m?k. Finally, we give a brief discussion of embeddings of the Stiefel and Grassmann manifolds.     

Nonnegative piecewise linear histograms *

We introduce a modified version ?_nof the piecewiss linear hisiugrimi uf Beirlant et al. (1998) which is a true probability density, i.e., ?_n[d] 0 and [d]?_n=1. We prove that ?_nestimates the underlying densitv ? strongly consistently in the L₁mmn, derive large deviation inequalities for the t\ error \?_n- f\ and prove that £||/"-/|| tends to zero with the rate n ^-1\3, We also show that the derivative lf'n estimates consistently in ine expected Lx error the derivative/ of sufficiently smooth density and evaluate the rate of convergence n-i/5 for Epf'n -f'% The estimator/" thus enables to approximate/in the Besov space with a guaranteed rate of convergence. Optimization of the smoothing parameter is also studied. The theoretical or experimentally approximated values of the expected errors E\\?_n- f\\ and E||2?'_n-?' are compared with tiie errors aCiiieveu u-y t"e histogram of Beirlant et ah, and other nonparametric methods.     

Distributions of certain variables in samples from two-parameter exponential distribution

Cumulative distribution function of the variable Y=(U+c)/(Z/2ν)) is given. Here U and Z are independent random variables, U has the exponential distribution (1.1) with θ=0, σ=1, Z has the distribution χ² (2ν) and c is a real quantity. The variable Y with U and Z given by (2.2) and (2.3) is used for inference about the parametric functions ?=θ?kσ of a two-parameter exponential distribution (1.1) with k or ? known. Special cases of ? or k are: the parameter θ, the Pth quantile Xp, the mean θ+σ and the value of the cumulative distribution function or of the reliability function at given point a. Also one-sided tolerance limits for a two-parameter exponential distribution can be derived from the distribution of the variable Y. The results are also applied to the Pareto distribution.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Statistics and the Bayesian mind

Did German flying bombs really target the poorer areas of London—or did the poor people underneath them just think that they did? How many millions of dollars will the current re-make of King Kong really gross? Oh, and how many taxis are there in London? The human mind can make very good guesses, it seems—and it uses Bayesian analysis to do so. Tom Griffiths and Joshua Tenenbaum explain.     

Change-Point Detection With Non-Parametric Regression

We consider the regression model y_i = ?(x_i ) + ε in which the function ? or its pth derivative ?^(p) may have a discontinuity at some unknown point τ. By fitting local polynomials from the left and right, we test the null that ?^(p) is continuous against the alternative that ?^(p)(τ?) ≠ ?^(p)(τ+). We obtain Darling-Erdös type limit theorems for the test statistics under the null hypothesis of no change, as well as their limits in probability under the alternative. Consistency of the related change-point estimators is also established.     

A test of linearity against functional coefficient autoregressive models

Our object in this paper is to propose a powerful test for detecting a broad class of nonlinearity of time series as one application of the method by Matsuda (1998). Consider autoregresive models X_t=Φ₁X_t?1+…+Φ_ρX_t?ρ+ε_t and we propose a statistic for testing whether or not Φ_i depends on delayed values X_t?d for some i. We compare the power of our test with that of tests proposed by Luukkonen, Saikkonen and Teräsvirta (1988a) and Hjellvik and TjΦstheim (1995) by simulation studies and our test is shown to be satisfactory.     

Fehlende und fehlerhafte Daten in der amtlichen Statistik. Neue Herausforderungen und L?sungsans?tze

Zusammenfassung Die Datenerhebungsprozesse in der amtlichen Statistik unterliegen einem tief greifenden Wandel. Zu diesem Wandel haben verschiedene Faktoren beigetragen: So verlangen die Auskunftgebenden oftmals eine spürbare Verringerung der Belastung. Gleichzeitig wünschen viele Nutzer tiefer gegliederte Informationen in einer wachsenden Zahl von Bereichen. Die statistischen ?mter sehen sich dabei mit immer neuen Budgetkürzungen konfrontiert, die ihre Reaktionsm?glichkeiten nicht selten einschr?nken. Schlie?lich ist im Europ?ischen Statistischen System eine weitere Harmonisierung der Datenerhebungsprozesse erforderlich. Der Wandel der Datenerhebungsprozesse hat nahezu alle Bereiche der amtlichen Statistik erfasst. Vielfach ersetzen heute Verwaltungsregister (wie im Fall des Unternehmensregisters oder des geplanten registergestützten Zensus) traditionelle Totalerhebungen. Bei neuen Statistiken tritt nicht selten die Freiwilligkeit der Auskunftserteilung an die Stelle der Auskunftspflicht. In der Unternehmensstatistik werden die Abl?ufe bei Befragung und Aufbereitung darüber hinaus grundlegend ver?ndert, wenn etwa die Unternehmen ihre Angaben automatisch aus dem betrieblichen Rechnungswesen an das statistische Amt senden. In anderen Bereichen erfordert eine gesunkene Teilnahmebereitschaft an Erhebungen den Einsatz zunehmend komplexer Stichprobenverfahren, wie etwa im Fall der Dauerstichprobe befragungsbereiter Haushalte. All diese Ver?nderungen haben erhebliche Auswirkungen auf die Messung, insbesondere auf die Behandlung fehlender und fehlerhafter Daten. Die Konzepte zur Messung von Fehlern in statistischen Erhebungen sind h?ufig noch immer ausgerichtet an der traditionellen Prim?rstatistik. Um eine Bewertung der ver?nderten Ans?tze in der Datenerhebung vornehmen zu k?nnen, bedürfen die Fehlertypen daher einer konzeptionellen Neuausrichtung. So unterscheiden sich z. B. registergestützte und traditionelle Erhebungen hinsichtlich der Konstruktion der Grundgesamtheit grundlegend. Mit den genannten Ver?nderungen des Erhebungsprozesses sind zugleich neue Ans?tze zur Erkennung und Korrektur von Antwortausf?llen erforderlich, z. B. im Fall fehlender oder fehlerhafter Werte in Verwaltungsregistern. Unterschiede zwischen den Merkmalsdefinitionen in Verwaltungsregistern und der amtlichen Statistik k?nnen schlie?lich zu neuen Typen von Messfehlern führen, die bislang nur selten Beachtung finden. Dieser Beitrag entwirft ein “Fehlerportfolio”, das bei der ?nderung von Erhebungsprozessen die Auswirkungen auf den Fehler der Ergebnisse bewerten hilft.

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Data collection processes in official statistics are currently facing many pressures. Respondents ask for a reduced response burden. Users ask for more detailed information in a growing number of subject matter areas. Governments restrict the budget, thus at the same time restricting the possible use of data collection methods. And the European Statistical System requires a further harmonisation of the data collection methods used within the member states. These pressures are also reflected by fundamental changes in the way official statistics are collecting data. Administrative registers replaced traditional censuses in many areas (like the population census or in the whole field of business statistics). Voluntary surveys more and more replace surveys with mandatory response. In some fields, the logic of the response process is being radically reshaped: For example, in German business surveys, enterprises can now report their data directly from their Enterprise Resource Planning (ERP) systems. And in many areas increasingly complex sampling designs are used in order to enhance the efficiency of the fieldwork and to counteract a growing reluctance to participate in surveys. The paper focuses on the implications of these changes on survey errors. We argue that the concepts used to measure and assess survey errors still reflect the perspective of the ‘traditional’ survey with primary data collection. With the current changes in the data collection processes these concepts are no longer fully appropriate. For example, an assessment of the coverage errors has to take into account the differences in the construction of the target population in registers and traditional censuses. The changes in the response process necessitate changes in the way we measure and correct for nonresponse errors (e. g. detection of erroneous or missing values in registers). Measurement errors have to be conceived differently in order to cover, e. g., errors due to lacks in compliance of statistical concepts with the concepts used in public administration. We propose an ‘error portfolio’ enabling an assessment the impacts of changes in data collection processes on the survey error.

JEL classification C80