Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

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 .     

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

Eine zweifache Verallgemeinerung der Pólya-Verteilung und ihre verschiedenen Deutungen

Zusammenfassung Die erste Verallgemeinerung der Pólya-Verteilung durch Ziehungen aus einer Urne, die Kugeln ins verschiedenen Farben enthält, wird aufr Ziehungsgruppen verallgemeinert und entspricht damit einer Überschreitungswahrscheinlichkeit beir+1 Serien von Stichproben und einer Wartezeitverteilung (Gruppe vons–1 Warteperioden) bei einer geeignet gefüllten Urne mit Kugeln inr Farben sowie der Transponierungswahrscheinlichkeit bei einem Kugelziehungsproblem und weiteren Deutungen; die interessierenden Varianzen und Kovarianzen werden angegeben und mit denen einer verallgemeinerten hypergeometrischen Verteilung verglichen.

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Summary The first generalization ofPólya's distribution by taking balls from an urn with balls ins colours is generalized tor series of takings and thus corresponds to a generalized distribution of exceendances, a certain waiting-problem, to a law of succession and certain other interpretations. Variances and covariances are calculated and compared with those of a generalized hypergeometric distribution.

     

Über eine Variante der empirischen Verteilungsfunktion

Summary If ₁, ₂,..., _n and ₁, ₂,..., _–1 are two ordered samples from a population with continuous distribution functionF(x), then the points ( _r ,r/n),r=1, 2,..., n–1 provide a better approximation ofF(x) than the points ( _r ,r/n),r=1, 2,..., n, in the following sense:A maximal upper deviation and a maximal lower deviation of more theny have — contrary to the points ( _r ,r/n) — equal probability for anyy0, if we deal with the points ( _r ,r/n). This probability is at least for ally in the interval , 1 less than the probability for a maximal upper deviation of more thany in the case of the points ( _r ,r/n). This is shown by a comparison of the Smirnow-Birnbaum-Tingey — formula with an analogous formula for the maximal one-sided deviations of the points( _r ,r/n).     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

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.

     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

A motivation for $$1/ \sqrt{n}$$-shrinking neighborhoods

In this paper we give a motivation for the shrinking rate let p ₀ and q _n be the outlier probability under the ideal model, and some member of a neighborhood about this ideal model of radius r _n , respectively. Assuming n i.i.d. observations, the critical rate of r _n may be defined such that the minimax test for outlier probability q _n =p ₀ versus q _n >p ₀ has asymptotic error probabilities bounded away from 0 and 1/2. Summarizing the neighborhoods to their upper probability, this leads to r _n of the exact rate . The result makes precise and simplifies ideas in Bickel (1981), Rieder (1994), and Huber (1997). Considering general probabilities of exact Hellinger distance r _n to P, this shrinking rate translates into , but leads to the same optimality theory as in the corresponding setup.     

Asymptotic distributions of some scale estimators in nonlinear models

Often in the robust analysis of regression and time series models there is a need for having a robust scale estimator of a scale parameter of the errors. One often used scale estimator is the median of the absolute residuals s ₁. It is of interest to know its limiting distribution and the consistency rate. Its limiting distribution generally depends on the estimator of the regression and/or autoregressive parameter vector unless the errors are symmetrically distributed around zero. To overcome this difficulty it is then natural to use the median of the absolute differences of pairwise residuals, s ₂, as a scale estimator. This paper derives the asymptotic distributions of these two estimators for a large class of nonlinear regression and autoregressive models when the errors are independent and identically distributed. It is found that the asymptotic distribution of a suitably standardizes s ₂ is free of the initial estimator of the regression/autoregressive parameters. A similar conclusion also holds for s ₁ in linear regression models through the origin and with centered designs, and in linear autoregressive models with zero mean errors.  This paper also investigates the limiting distributions of these estimators in nonlinear regression models with long memory moving average errors. An interesting finding is that if the errors are symmetric around zero, then not only is the limiting distribution of a suitably standardized s ₁ free of the regression estimator, but it is degenerate at zero. On the other hand a similarly standardized s ₂ converges in distribution to a normal distribution, regardless of the errors being symmetric or not. One clear conclusion is that under the symmetry of the long memory moving average errors, the rate of consistency for s ₁ is faster than that of s ₂.     

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.     

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     

Strong uniform convergence of the recursive regression estimator under φ-mixing conditions

Suppose the observations (X_i, Y_i) taking values in R^d×R, are -mixing. Compared with the i.i.d. case, some known strong uniform convergence results for the estimators of the regression function r(x)=E(Y_i|X_i=x) need strong moment conditions under -mixing setting. We consider the following improved kernel estimators of r(x) suggested by Cheng (1983): Qian and Mammitzsch (2000) investigated the strong uniform convergence and convergence rate for to r(x) under weaker moment conditions than those of the others in the literature, and the optimal convergence rate can be attained under almost the same conditions as stated in Theorem 3.3.2 of Györfi et al. (1989). In this paper, under the similar conditions of Qian and Mammitzsch (2000), we study the strong uniform convergence and convergence rates for (j=2,3) to r(x), which have not been discussed by Qian and Mammitzsch (2000). In contrast to , our estimators and are recursive, which is highly desirable for practical computation.     

Comparison of sampling strategies

Summary Horvitz andThompson [1952] considered varying probability sampling method in general and furnished an unbiased estimator of the population total.Rao, Hartley andCochran [1962] proposed a simple procedure of unequal probability sampling with replacement. It leads to an estimator of the population total having smaller variance than is obtained by sampling with replacement. An attempt has been made in the present paper to compare efficiencies ofHorvitz-Thompson's estimator with that due toRao, hartley andCochran. It is demonstrated that the generalized ps sampling strategy consisting of the design with _i , the probability of inclusion of thei-th population unit in the sample proportional to the modified size together withHorvitz-Thompson's estimator is superior toRao, Hartley andCochran's sampling strategy under a general super-population model.     

Asymptotic inference for a one-dimensional simultaneous autoregressive model

A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n ², while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n ^1/2.     

A Generalized Method of Moments Estimator for a Spatial Panel Model with an Endogenous Spatial Lag and Spatial Moving Average Errors

Abstract

This paper proposes a new generalized method of moments (GMM) estimator for spatial panel models with spatial moving average errors combined with a spatially autoregressive dependent variable. Monte Carlo results are given suggesting that the GMM estimator is consistent. The estimator is applied to English real estate price data.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

Forecasting a long memory process subject to structural breaks

We develop an easy-to-implement method for forecasting a stationary autoregressive fractionally integrated moving average (ARFIMA) process subject to structural breaks with unknown break dates. We show that an ARFIMA process subject to a mean shift and a change in the long memory parameter can be well approximated by an autoregressive (AR) model and suggest using an information criterion (AIC or Mallows’ _Cp$C_p$) to choose the order of the approximate AR model. Our method avoids the issue of estimation inaccuracy of the long memory parameter and the issue of spurious breaks in finite sample. Insights from our theoretical analysis are confirmed by Monte Carlo experiments, through which we also find that our method provides a substantial improvement over existing prediction methods. An empirical application to the realized volatility of three exchange rates illustrates the usefulness of our forecasting procedure. The empirical success of the HAR-RV model can be explained, from an econometric perspective, by our theoretical and simulation results.     

Moment convergence of M‐estimators

This study extends the rate of convergence theorem of M‐estimators presented by van der Vaart and Wellner (weak convergence and empirical processes: with applications to statistics, Springer‐Verlag, Newyork, 1996) who gave a result of the form r  to a result of the form sup_nE | r , for any p≥1. This result is useful for deriving the moment convergence of the rescaled residual. An application to maximum likelihood estimators is discussed.     

Minimal sufficient statistics for the group divisible partially balanced incomplete block design (GD-PBIBD) with interaction under an Eisenhart Model II

Summary In this paper, an Eisenmhart Model II with interaction for a GD-PBIB design withp replicates per cell is considered. Specifically the Model Y_ijl=µ+_i+_j+()_ij+e_ijl is assumed, wherei=1, 2, ...,b; j=1, 2, ...,t andl=0, 1, 2, ...p s _ij wheres _ij=1, if treatmentj appears in blocki, 0, otherwise.If _i, _j, ()_ij ande _ijl are normally and independently distributed, then a minimal sufficient (Vector-valued) statistic for the class of densities for this model is found, together with the distribution of each component in the minimal sufficient statistic. It is also shown that the minimal sufficient statistic for this class densities is not complete. Hence the solution of the problem of finding minimum variance unbiased estimators of the variance components is not straightforward.