National natural rates of interest and the single monetary policy in the euro area

We estimate time‐varying national natural real rates of interest (r^?) for the four largest economies of the euro area over 1999–2016. We further derive the associated national real interest rate gaps, which gauge the perceived monetary policy stance in each country. We find that the average r^? have been lower after 2008. Furthermore, national r^? were significantly negative in southern countries during the sovereign crisis. As their effective real rates soared, national rate gaps across the euro area diverged. However, a common policy stance has been restored since 2014 as the European Central Bank's unconventional programs gathered pace.     

On Estimators of the Nearest Neighbour Distance Distribution Function for Stationary Point Processes

There are three approaches for the estimation of the distribution function D(r) of distance to the nearest neighbour of a stationary point process: the border method, the Hanisch method and the Kaplan-Meier approach. The corresponding estimators and some modifications are compared with respect to bias and mean squared error (mse). Simulations for Poisson, cluster and hard-core processes show that the classical border estimator has good properties; still better is the Hanisch estimator. Typically, mse depends on r, having small values for small and large r and a maximum in between. The mse is not reduced if the exact intensity λ (if known) or intensity estimators from larger windows are built in the estimators of D(r); in contrast, the intensity estimator should have the same precision as that of λ D(r). In the case of replicated estimation from more than one window the best way of pooling the subwindow estimates is averaging by weights which are proportional to squared point numbers.     

One-Sample Bayesian Predictive Interval of Future Ordered Observations for the Pareto Distribution

Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y _(j) (r < j≤ n ) based on the right type II censored samples Y ₍₁₎ < Y ₍₂₎ < ... < Y _(r) from the Pareto distribution. If some of Y ₍₁₎ < ... < Y _(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y _(j) (r < j ≤ n ) given Y ₍₁₎ <Y ₍₂₎ < ... < Y _(r) is equivalent to the conditional p.d.f. of Y _(j) (r < j ≤ n ) given Y _(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the only ordered observation Y _(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results.     

Sequential point estimation of the powers of a normal scale parameter

We consider the sequential point estimation problem of the powers of a normal scale parameter σ^r with r≠ 0 when the loss function is squared error plus linear cost. It is shown that the regret due to using our fully sequential procedure in ignorance of σ is asymptotically minimized for estimating σ⁻². We also propose a bias-corrected procedure to reduce the risk and show that the larger the distance between r and −2 is, the more effective our bias-corrected procedure is. Received August 2000     

On the exponential law

A characterization of the exponential law is given using its property thatE[(X–a)^r|X>a] is constant for alla0,r being a positive integer. Then this result is applied to the order statistics and some further characterizations are found.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

On approximating the distributions ofFriedman's 75-175-175-1and related statistics

Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x _r ² statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N ^–1/2); an improved bound of ordero (N ^–r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.Supported in part by ONR Contract N00014-72A-0136-003 from the U.S. Office of Naval Research.     

Estimation in the pareto distribution

The unique minimum variance unbiased (UMVU) estimate of the probability distribution function of the Pareto distribution is derived. It is shown that the distribution function and ther ^th moment associated with the UMVU estimate are also UMVU estimators. The p.d.f. and its estimator are compared graphically. An estimate of the 100p ^th percentile is given. It is seen that a function of this estimator has a chi-square distribution.     

Estimation of the location parameter under LINEX loss function: multivariate case

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?^r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

An identity concerning a Wishart matrix

Summary LetS be ap×p Wishart matrix with parametersn and . For a rational number =r/s withr ands integers ands positive, letS denote a positive definite matrix such that (S ) ^s =S ^r . Using a decision theoretic argument, we prove thatE[(trS)² trS ]=(np+2+2)E[trS trS ] when =I andnp+2+2 is positive.     

Locally minimax tests for a multinormal data problem

LetX be ap-normal random vector with unknown mean and unknown covariance matrix and letX be partitioned asX=(X ₍₁₎ ,X ₍₂₎ , ...,X _(r) ) whereX _(j) is a subvector of dimensionp _j such that _j=1 ^r p _j =p. We show that the tests, obtained by Dahel (1988), are locally minimax. These tests have been derived to confront H_o: =0 versusH ₁: 0 on the basis of sample of sizeN, X ₁, ..., X_N, drawn fromX andr additional samples of sizeN _j, U _i ^(j) , i=1, ..., N_j, drawn fromX ₍₁₎, ...X _(r) respectively. We assume that the (r+1) samples are independent and thatN _j>p _j forj=0, 1, ..., r (N _oN andp _op). Whenr=2 andp=2, a Monte Carlo study is performed to compare these tests with the likelihood ratio test (LRT) given by Srivastava (1985). We also show that no locally most powerful invariant test exists for this problem.     

Learning curves—an axiomatic approach

A systematic mathematical analysis of learning curves is presented. It is shown that while a learning curve with learning factor k does necessarily satisfy the functional equation f(2x) = kf(x), this equation admits numerous other analytic, convex solutions as well so that it cannot be used to uniquely characterize learning curves. Rather, it is shown that a pair of functional equations f(rx) = kf(x) and f(sx) = jf(x) where rⁿ ≠ s^m for any positive integers n and m suffice to uniquely determine the learning curve; compatibility of the two equations requires that log_rk = log_sj or there will be no learning curve satisfying the pair of equations. Two classes of almost learning curves are generated and studied by means of suitable perturbation terms in the differential equation y = by/x of the true or standard learning curve and these curves are applied to describe data not exhibiting exact learning-curve behavior. Finally the concept of average marginal hours or cost is introduced and its behavior is found to also exhibit the learning-curve phenomenon except for an initial deviation.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

r-te Momente von Punktprozessen

Zusammenfassung {X (t): tR ⁺} sei ein Punktprozeß,H (x) eine konvexe nicht-negative Funktion. Mit Hilfe der bedingten Wahrscheinlichkeitenp _n (t) für genaun Ereignisse (Punkte) im Zeitpunkt punktt unter der Bedingung, daß im Zeitpunktt mindestens ein Ereignis eintritt, wird eine Beziehung formuliert, die für die Existenz des ErwartungswertesE (H (X (t ₀))) notwendig ist. Hat der Punktprozeß unabhängige Zuwächse, und erfüllt die FunktionH (x) einige weitere Bedingungen, so ist die angegebene Beziehung auch hinreichend für die Existenz dieses Erwartungswertes. Für Punktprozesse mit unabhängigen Zuwächsen ergibt sich als unmittelbare Anwendung dieser Aussagen eine notwendige und hinreichende Bedingung für die Existenz vonE X (t ₀) ^r für reellesr1.

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Summary Let {X (t): tR ⁺} be a point process andH (x) a convex non-negative function. Using the conditional probabilitiesp _n (t) thatn events (points) occur at timet given that at least one event occurs att a condition is formulated which is necessary for the existence ofE (H (X (t ₀))). This condition is sufficient, too, if the point process has independent increments and the functionH (x) fulfils some further conditions. Using these statements one gets a necessary and sufficient condition for the existence ofE X (t ₀) ^r for realr1.

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Herrn ProfessorWeissinger zum 65. Geburtstag am 12. Mai 1978 gewidmet     

A further note on the asymptotic efficiency of friedman’s x r 2 -test

Summary For independently distributed error components, the asymptotic relative efficiency (A.R.E.) ofFriedman’sx _r ² -tests with respect to the classical analysis of variance test has been studied byElteren andNoether andSen [1967]. The present note extends these results to the case of correlated errors arising in some random-effects or mixed-effects models. Work supported by the U.S. Army Research Office, Durham, Grant DA-ARO-D-31-124-G 746.     

Price asymptotics

This paper examines the pricing decisions of a seller facing an unknown demand function. It is assumed that partial information, in the form of an independent random sample of values, is available. The optimal price for the inferred demand satisfies a consistency property—as the size of the sample increases, the maximum profit and price approach the values for the case where demand is known. The main results deduced here are asymptotics for prices. Prices converge at a rate of O _p (n ^−1/3) with a limit that can be expressed as a functional of a Gaussian process. Implications for the comparison of mechanisms are discussed.      

Goodness-of-fit tests for discrete models based on the integrated distribution function

This paper presents a new widely applicable omnibus test for discrete distributions which is based on the difference between the integrated distribution function Ψ(t)=∫_t ^∞ (1−F(x))dx and its empirical counterpart. A bootstrap version of the test for common lattice models has accurate error rates even for small samples and exhibits high power with respect to competitive procedures over a large range of alternatives. Received: July 1998     

Variation diminishing property of densities of uniform generalized order statistics

Let f _*,r , r ≥ 1, denote the density function of rth uniform generalized order statistics as defined by Kamps (1995) or Cramer and Kamps (2003). We prove the following variation diminishing property: the number of zeros in (0,1) of any linear combination does not exceed the number of sign changes in the sequence (a ₁, . . . ,a _r ). This result is applied to study monotonicity and convexity properties of f _*,r .     

Statistische Modelle und Bayessche Verfahren zur Bestimmung einer unbekannten Klassifikation normalverteilter zufälliger Vektoren

Summary Suppose that forN independent normally distributed vectors inR ^p there exists an unknown classification intom disjoint classes such that in each class all vectors have the same (unknown) expectation. For several a-priori-distributions and loss functions Bayesian procedures are developped for the problem of identifying the unknown classification. — The methods can be extended to the case of an unknown numberm of groups and to the case of many-way classifications.

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Zusammenfassung FürN beobachtbare, unabh?ngige, normalverteilte Vektoren desR ^p existiere eine unbekannte Einteilung inm disjunkte Klassen, innerhalb deren die Erwartungswerte gleich, aber unbekannt seien. Es werden Bayessche Verfahren zur Bestimmung dieser unbekannten Klassifikation entwickelt, wobei verschiedene Verlustfunktionen und a-priori-Verteilungen zugrunde gelegt werden. — Die überlegungen k?nnen auf den Fall unbekannter Klassenanzahlm und auf den Fall der Klassifikation nach mehreren Gesichtspunkten erweitert werden.

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Gekürzte Fassung einer von der Mathematisch-Naturwissenschaftlichen Fakult?t der Universit?t Freiburg angenommenen Dissertation (1968).