Further results on asymptotic normality I

Summary Starting fromLe Cam [1956], it was shown inMichel andPfanzagl [1970] that — under certain regularity conditions — a dominated family of probability measures withEuclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components. It is the purpose of this paper to estimate the accuracy of such a normal approximation.     

On the measurability and consistency of minimum contrast estimates

Summary The concept of minimum contrast (m.c.) estimates used in this paper covers maximum likelihood (m.l.) estimates as a special case. Section 1 contains sufficient conditions for the existence of measurable m.c. estimates and for their consistency. The application of these results to m.l. estimates (section 2) yields the existence of m.l. estimates for families ofp-measures (probability measures) which are compact metric or locally compact with countable base, admitting upper semicontinuous densities, whereas the classical results refer to continuous densities. This generalization is insofar of interest as upper semicontinuous versions of the densities exist whenever the densities areμ-upper semicontinuous (whereasμ-continuity does not, in general, entail the existence of continuous versions). Under appropriate regularity conditions, consistency of asymptotic maximum likelihood estimates is proven for compact (and also locally compact) separable metric families ofp-measures with upper semicontinuous densities and for arbitrary families having uniformly continuous densities with respect to the uniformity of vague convergence. The conditions sufficient for consistency are shown “indispensable” by counterexamples. Section 3 contains auxiliary results. Besides their relevance for sections 1 and 2, some of them may also be of interest in themselves, e.g. Theorem (3.4) on the selection of semicontinuous functions from semicontinuous equivalence classes.     

Zur Schätzung eines Dichtefunktionals

Summary Fór the estimation of the functional (f)=f ²(x)dx Bhattacharyya andRoussas [1969] proposed an estimator based on the kernel-technique for density estimation. This paper describes a method, which rests on density estimations by orthogonal expansions. In the main we show the considered estimator to be consistent in the quadratic mean.     

Über die Existenz überall trennscharfer Tests im nicht-dominierten Fall

Einleitung Es sei (,A) ein Meßraum undP eine Familie von Wahrscheinlichkeitsmaßen auf . IstP durch ein -endliches Maß dominiert, so ist nachPfanzagl [1960] für die Existenz eines überall trennscharfen Tests zu jedem Niveau für die HypotheseP=P _o (P, P _o P) gegen die AlternativePP ₀ notwendig und hinreichend, daßP/{P ₀} isotonen Likelihood-Quotienten bzgl.P ₀ besitzt. Für den Fall, daßP total geordnet und dominiert ist, gilt nach [Pfanzagl, 1963] eine entsprechende Aussage: Genau dann existiert zu jedem Niveau ein überall trennscharfer Test, wennP isotonen Likelihood-Quotienten besitzt.Die vorliegende Arbeit zeigt, daß auf die Annahme der Dominiertheit verzichtet werden kann, und liefert darüber hinaus einen einheitlichen Beweis für die beiden oben zitierten Sätze vonPfanzagl.

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Summary Two theorems ofPfanzagl [1960, 1963] about necessary and sufficient conditions for the existence of uniformly most powerful tests are generalized to the undominated case. Moreover a unified proof for the two theorems ofPfanzagl is given.

     

On the inequality Σp i f(p i )≥ p i f(q i )

Summary We shall deal with the inequality (2) and we prove among others the following results. Every solution of the inequality (2) is monotone increasing. Every solution of the inequality (2) is differentiable ifn≥3. A functionf satisfies the inequality (3) if and only if its saltus part and absolutely continuous part satisfy also the inequality (3). We give the general solution of (3) in the field of saltus function. Finally by the help of [Muszély] we give the general solution of (3). This results are generalisations of one ofJ. Aczél andJ. Pfanzagl.

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Zusammenfassung Wir betrachten die Ungleichung (2) und beweisen die folgenden Ergebnisse. Die allgemeine L?sung der Ungleichung (2) ist monoton wachsend. Die allgemeine L?sung der Ungleichung (2) ist differenzierbar im Falle wennn≥3 ist. Wir geben also die allgemeine L?sung der Ungleichung (3) mit Hilfe von [Muszély]. Diese Ergebnisse sind Verallgemeinerungen des Resultats vonJ. Aczél undJ. Pfanzagl.

     

On a linear stochastic differential equation

Summary Stochastic differential (s. d.) equations had been considered in [Nasr, 1960] and [Nasr]. We consider here, the s. d. equationf(D)x(t)=m(t)+v(t)z(t) wherem(t),v(t) are real functions oft,f(D) is a polynomial inD withD=d/dt, andz(t) is a random function. In particular,z(t) is assumed here, to be of the stationary type, while other types namely whenz(t) is of theGaussian or of thePoisson type, are considered in [Nasr]. A particular integral of the stated equation, and an associated covariance function of this integral are given in the form of generalized (g-)functions; [Nasr, 1965]. The equationdx/dt=v(t)z(t) wherez(t) is stationary in the wide sense is considered as a special case.     

Median unbiased estimates for M.L.R.-families

Summary Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family. The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general case.     

Eine Momentengleichung bei Exponentialfamilien

Zusammenfassung Eine beiKendall undStuart [S. 122 bzw. S. 126] für Binomial- bzw. Poissonverteilungen und beiNoack [S. 128] für Potenzreihenverteilungen aufgestellte Momentengleichung wird auf einparametrige Exponentialfamilien verallgemeinert, und es wird gezeigt, da? eine weitere Verallgemeinerung nicht m?glich ist. Als Anwendung ergibt sich eine Charakterisierung der Normalverteilung.

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Summary A recurrence relation for the moments about the mean, which has been given byKendall andStuart [p. 122 and p. 126] for Binomial and Poisson distributions and byNoack [p. 128] for power series distributions, is generalized to exponential families, and it is shown, that a further generalization is impossible. As an application a characterization of the normal distribution is given.

     

On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

On unordering Horvitz and Thompson'sT 1 class of linear estimators

Summary When elements of a finite population are sampled with varying probability selection at each draw,Horvitz andThompson [1952] have formulated certain classes of linear estimators to bear on the problem of providing a smaple appraisal of the population total.Horvitz andThompson's T ₁ class is an ordered one, which was examined by the present author [1967 b]. For some sampling procedures a best estimator exists for theT ₁ class. Subsequently the present author [1967 c] appliedMurthy's technique [Murthy 1967] of unordering an ordered estimator and derived a more efficient estimator. The present paper is concerned with applyingMurthy's technique to theT ₁ class itself, and examining the unorderedT ₁ class. Curiously enough, it is noted that the condition of unbiasedness is sufficient to completely specify the unorderedT ₁ class for the sampling procedure considered here.Research sponsored by Marathwada University, Aurangabad, India; under Grant No. Research-12-68-69/3314-16.     

Una classe di funzioni monotone generalizzate

Si considera la classe delle funzioni realiF(x,y) definite inS×S, conSR ^N , che soddisfano per ognix,yS la condizione di monotoniaF(x,y)+F(y,x)0. Indebolendo la precedente disuguaglianza si introducono classi di funzioni monotone generalizzate e, supponendo soddisfatta una opportuna condizione di omogeneità, si caratterizzano tali funzioni in base alla struttura del segno delle funzioni _{x, v} (t, s) = F(x + tv, x + sv), x S, v R ^N \{0}. Infine dopo aver definite le funzioni F-differenziabili, si introducono classi di funzioni conversse generalizzate, rispetto ad F, e si studiano i collegamenti tra queste classi e la monotonia generalizzata diF.

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Summary We consider the class of real valued functionF(x,y) defined inS×S, withSR ^N , satisfying x,yS the monotone conditionF(x,y)+F(y,x)0. Weakening the previous inequality we introduce the class of quasi-monotone, pseudo-monotone and strictly pseudo-monotone functions. Under a suitable assumption of homogeneity we characterize the generalized monotone functions studying the sign structure of the functions _{x, v} (t, s) = F(x + tv, x + sv), x S, v R ^N \{0}.Finally by means of the notion ofF-differentiability we introduce new classes of generalized convex functions (with respect toF) and we study the relationship between these classes and the generalized monotonicity ofF.

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Questa ricerca è stata parzialmente finanziata dal Ministero per l'Università e la Ricerca Scientifica.     

Zur Theorie der Rangtests

Zusammenfassung Bekanntlich lassen sich eine Reihe wichtiger einseitiger Rangtests für verschiedene Problemklassen als lokal beste invariante Tests herleiten [vgl. etwaLehmann], so u. a. für den Vergleich zweier Stichproben, das Symmetrietestproblem und für die Prüfung auf Unabh?ngigkeit. In 2.1 bis 2.5 werden die Voraussetzungen angegeben, die für die Herleitung einer für den Vergleich vonk-Stichproben zuerst vonHoeffding angegebenen Formel ben?tigt werden und die u. a. bei den obigen drei Testproblemen erfüllt sind (vgl. hierzu 3.1 bis 3.3). Die in 4.5 angegebene, vom speziellen Problem unabh?ngigeHoeffding-Formel erm?glicht, wie in 4.2 kurz angedeutet werden soll, nach der Reduktion durch Invarianz eine einheitliche (optimale) Herleitung von Rangtests für die verschiedenen Problemklassen. Eine entsprechende Systematisierung, die überdies die bei der praktischen Durchführung von Rang-und Permutationstests bestehenden Analogien widerspiegelt, ist bei Permutationstests m?glich; jedoch sind hierzu gewisse Zusatzüberlegungen erforderlich [vgl.Witting 1969]. Diese Arbeit ist aus Diskussionen mitG. N?lle † bei der Abfassung vonWitting u.N?lle entstanden. Ihm sei hierfür auch an dieser Stelle nochmals gedankt.

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Summary Assumptions are formulated which are necessary for deriving a formula originally given byHoeffding for the comparison ofk samples. An analogous formula, which is independent of the special problem, makes it possible to derive in a consistent manner locally most powerful invariant tests for several classes of one-sided nonparametric test problems.

     

Remarks on the strong law of large numbers for a triangular array of associated random variables

A strong law of large numbers for a triangular array of strictly stationary associated random variables is proved. It is used to derive the pointwise strong consistency of kernel type density estimator of the one-dimensional marginal density function of a strictly stationary sequence of associated random variables, and to obtain an improved version of a result by Van Ryzin (1969) on the strong consistency of density estimator for a sequence of independent and identically distributed random variables.     

Minimum variance unbiased estimation of the parameters of the Pareto distribution

Summary This paper presents a simpler proof of some results concerning estimation of the Pareto distribution due toLike [1969].     

The limit distribution of the estimates in cointegrated regression models with multiple structural changes

We study estimation and inference in cointegrated regression models with multiple structural changes allowing both stationary and integrated regressors. Both pure and partial structural change models are analyzed. We derive the consistency, rate of convergence and the limit distribution of the estimated break fractions. Our technical conditions are considerably less restrictive than those in Bai et al. [Bai, J., Lumsdaine, R.L., Stock, J.H., 1998. Testing for and dating breaks in multivariate time series. Review of Economic Studies 65, 395–432] who considered the single break case in a multi-equations system, and permit a wide class of practically relevant models. Our analysis is, however, restricted to a single equation framework. We show that if the coefficients of the integrated regressors are allowed to change, the estimated break fractions are asymptotically dependent so that confidence intervals need to be constructed jointly. If, however, only the intercept and/or the coefficients of the stationary regressors are allowed to change, the estimates of the break dates are asymptotically independent as in the stationary case analyzed by Bai and Perron [Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78]. We also show that our results remain valid, under very weak conditions, when the potential endogeneity of the non-stationary regressors is accounted for via an increasing sequence of leads and lags of their first-differences as additional regressors. Simulation evidence is presented to assess the adequacy of the asymptotic approximations in finite samples.     

z-Transform and identification of linear econometric models with autocorrelated errors

The paper consists of two main parts. In the first part we derive the solution of systems of linear stochastic difference equations by means of thez-transform. In the second part thisz-transform is used to treat the problem of identification of linear econometric systems (the term econometric is used to stress the special aspects of the identification problem dealt with in econometrics). It is shown, that under suitable restrictions observationally equivalent structures are related by unimodular matrices. Using this result, we state (rank-) conditions which ensure, that the unimodular matrices are constant, such that the classical econometric identification theorems can be applied. These conditions are given for stationary errors in the general case as well as in the MA, AR and ARMA case.     

On the variance of the ratio estimator for midzuno-sen sampling scheme

Summary For the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. An expression for the same is derived in this note.     

Formtest räumlicher Punkthaufen bei beliebiger Population

In an eralier work [Landhäusser, 1973] the densities for the form variables ₂₁ ² , ₃₁ ² , ₃₂ ² were derived by means of the assumption of a population sphere. The variables cited represent the ratio of each two eigenvalues of the 3×3 matrixX, whose elements satisfy theWishart-density.In this wrok the densities for the form variables are invesigated while assuming an arbitrary populaion ellipsoid (Concentration ellipsoid) and for the resulting series an estimate fo the remainder is given. With the help of this estimate it is possible to establish an upper limit for the summation index. An evaluatio nis only possible when very fast machines are used.For the rotation ellipsoides one gains relatively simple results. The densities are give for ₂₁ ² . Possiblities for application of these densities can be found in the fields of geodesy and ballistics.A given target (e.g. an object in flight) is enveloped by a rotation ellipsoid whose half-axial rations are introduced into the densities as populations parameters. After determination of an error-probability the form of cluster of points can be tested for significance. Additional tests with regard to volume and center of a random-sample are possible.     

Inference robustness of ARIMA models under non-normality —Special application to stock price data

Wold's decomposition theorem [Wold] states that every weakly stationary stochastic process can be written as a linear combination of orthogonal shocks. For practical reasons, however, it is desirable to employ models which use parameters parsimoniously.Box andJenkins [1970] show how parsimony can be achieved by representing the linear process in terms of a small number of autoregressive and moving average terms (ARIMA-models). The Gaussian hypothesis assumes that the shocks follow a normal distribution with fixed mean and variance. In this case the process is characterized by first and second order moments. The normality assumption seems reasonable for many kinds of series. However, it was pointed out byKendall [1953],Mandelbrot [1963, 1967],Fama [1965],Mandelbrot andTaylor [1967] that particularly for stock price data the distribution of the shocks appears leptokurtic: In this paper we investigate the sensitivity of ARIMA models to non-normality of the distribution of the shocks. We suppose that the distribution function of the shocks is a member of the symmetric exponential power family, which includes the normal as well as leptokurtic and platikurtic distributions. A Bayesian approach is adopted and the inference robustness of ARIMA models with respect to

1. the estimation of parameters
2. the forecasts of future observations is discussed.

     

On the variance of the ratio estimator

Summary In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.