# Asymptotics, Nonparametrics, and Time Series

DOI link for Asymptotics, Nonparametrics, and Time Series

Asymptotics, Nonparametrics, and Time Series book

# Asymptotics, Nonparametrics, and Time Series

DOI link for Asymptotics, Nonparametrics, and Time Series

Asymptotics, Nonparametrics, and Time Series book

Edited BySubir Ghosh

Edition 1st Edition

First Published 1999

eBook Published 18 February 1999

Pub. location Boca Raton

Imprint CRC Press

Pages 854 pages

eBook ISBN 9780429175343

SubjectsMathematics & Statistics

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#### Get Citation

Ghosh, S. (Ed.). (1999). Asymptotics, Nonparametrics, and Time Series. Boca Raton: CRC Press, https://doi.org/10.1201/9781482269772

"Contains over 2500 equations and exhaustively covers not only nonparametrics but also parametric, semiparametric, frequentist, Bayesian, bootstrap, adaptive, univariate, and multivariate statistical methods, as well as practical uses of Markov chain models."

## TABLE OF CONTENTS

chapter 1|3 pages

#### Some Examples of Empirical Fourier Analysis in Scientific Problems

WithUniversity of California, Berkeley, California

chapter |6 pages

#### + u,y +

t) Y(x,y, t)} =

Withof the source of interest in the sky and/( a, of (a, (3). In other words, the Fourier and Wolf(\964). abound in nature. They have the property

chapter |1 pages

#### = ±I, ±2,...

With= 8(>.- < A,J.l:::; 1r,f(·) being the power spectrum of and 8(·) the Dirac delta of taking one and thereby making some operations on

chapter |11 pages

#### < is approximately (complex) normal for each Under station-

Y,, the variance of Eq. (4.2) is

With27r7f(>-) of Eq. (4.1) at frequency Surprisingly, the of ST(>-) at distinct frequencies of the form of Eq. (3.13) may be shown to be asymptotically and variance of Eq. (4.4). Further

chapter 0|7 pages

#### E. and D. R. Cox, Asymptotic Techniques for Use in

WithChapman and Hall, London, 1989. and S. Moss de Olivierra, Experimental demon- of the mathematical properties of Fourier transforms using

chapter 2|2 pages

#### Modeling and Inference for Periodically Correlated Time Series

Robert B. Lund and Jslnrar V. Ba.1·mra

WithLUND and ISHWAR V. BASAWA The University of 1. INTRODUCTION

chapter |19 pages

#### = Corr[XnT+=

Withor equal to x. Then a PC univariate series { X of PC series through

chapter |5 pages

#### AN ESTIMATING EQUATION APPROACH

WithPARMA parameter PARMA parameters and 9 of a parameter vector p; hence, we allow for the pos- of parametric constraints such as those in Eq. (4.27).

chapter 3|4 pages

#### Modeling Time Series of Count Data Richard A. Dal'is, William T. M. Dunsmuir. and Ying Wang

WithFort DUNSMUIR University of New South Wales,

chapter 1|12 pages

#### 3 Generalized State-Space Models

Withof linear systems of this subject see Davis and Vinter, 1985, and Hannan and For the parameter-driven model, S+given (S,S(t-ll, y(ll) is assumed to be of (S(t-l l, y(tl) with conditional density function, := p(s+ds 1,2,.... (1.2) = P(Ynlsn)p(snlscn-1 (1.3)

chapter |4 pages

#### = using the GLM fitted values,

Withx,x.; exp[(xi + x.;)p]'yJs- Use of the correct standard errors for the trend term would lead to the conclusion that the trend is not significant whereas use of the standard errors produced by the GLM analysis would lead to declaring the trend to be significant.

chapter 2|9 pages

#### 4 Estimating the Variance of the Latent Process

Withof no latent process. Zeger proposed the estimate """ ,J of which is approximately unbiased for and would be exactly unbiased , )2 , """ '4 on its approximate expected value of (Recall

chapter |1 pages

#### = flf(Y;I6;).f(o)

r. As noted above, the likelihood of the observed

With[rr" exp{-exp(8;+xTJl)}exp{(6;+xTJl)r;}] v-'1'12 of o. Our objective is to estimate the model parameters 9 data y does not have a simple closed form and maximum likelihood estima- of the complete of the complete data (y' o) is then given by

chapter |15 pages

#### OBSERVATION DRIVEN MODELS

cis defined implicitly. The log-likelihood for w is

With3.1 Review of Existing Models and b,,_ < I. Using this formulation the stochastic mechanism for and the forecast function E(Yr+dy(T))is shown to follow an ex- of past Y's. Similar models, based on TnliT" [max(Yr-i,c)]''' (3.1) < c < 1 is a constant which prevents Y = 0 becoming an absorb- (xip) = I, c is interpreted as an immigration rate adding to

chapter 4|13 pages

#### Seasonal and Cyclical Long Memory

Josu Arteche and Peter M. Robinson

Withof the Basque Country, Bilbao, Spain of Economics and Political Science,

chapter |7 pages

#### ... dand

... d ... -d". KLl'h-l cos(jwk) /j have slow decay with oscillations depending on the

Withof Eq. (3.8) impedes calculation of explicit If there is more than one and Leipus (1995) showed that the autocovariances of Eq. (3.8) satisfy as}-> oo. of and 'Yr

chapter 5|3 pages

#### Nonparametric Specification Procedures for Time Series

WithJ0STHEIM University of Bergen, Bergen, Norway of time series data requires formal or informal speci- of the model building. For example,

chapter |1 pages

#### = is carried out using a Lagrange multiplier test. The

exp(iwt)f(w) dw. K(s, t) K(s, t) exp( t=-x

Withof optimality and b), Saikkonen and Luukkonen if it is used against a specific alternative (specific out that this particular alternative is in fact true. This seldom (never) of course, and one could resort to a battery of tests for of alternatives it must also be taken into = E(X,XX = -I

chapter |6 pages

#### Lxt+IK,(XI-

K"(-) /z- K"(X,-x) _ {M. ( )}2 (X, - z lying in a neighborhood of x. Here Mii) (x) denotes the ith derivative .... 'Mr)(x)V r, (n- (X;+k-

Withat a point x we approximate Mk(x) by a poly- of order T, so that of Mk (x). Consider the following least squares problem: let i;, = I, ... , i {Mu(x), Kxr' xr = (Xk+l, ... , X I) matrix with ... (X;+k- = 0 cor-

chapter |1 pages

#### = -f'"(k)-

f'"(k)=[ChT(u,v)](u,v= l, ChT(-h + I,k)J', a"(O) chT(u,v) N-l LXI-u+lxt-v+l·

With1\(k), + 1), the subscript D stands for For h > I, however, the direct and plug-in estimators do not coincide. As

chapter |9 pages

#### = = ... = =

,(,j) = O,j Vp(l,k)

Withif k < m, finite or infinite, then for h I, of the linear least-squares pre- of incorrect models, see also of this point.

chapter |4 pages

#### MODEL SELECTION

K) (4.1)

WithIt is now standard practice to use an order selection criterion for selecting of an autoregressive model. As is well-known, Bhansali (l993b), a of currently available model selection criteria may be written as of a general criterion of the following form: + ... > 0 is either a fixed constant or a function of T, and the order is of Eq. (4.1), with = 2 and so is a Bayesian criterion of Schwarz ( 1978), with a = In T. If m in Eq. (4.1) is finite, the order selected by minimizing the AIC"

chapter |1 pages

#### CONCLUDING REMARKS

Withof the present paper is to provide a review of the current state- of lead-time dependent model and/or parameter estimation for multistep prediction. This method

chapter |6 pages

#### of Time Series. Wiley, New York, tor-!, 61-73 (1974).

With1971, p. 467. Ansley, C. F. and P. Newbold, On the bias in the estimates of forecast mean square error. J. A mer. Statist. Assoc .. 76: 569-578 (1981 ). Berk, K. N., Consistent autoregressive spectral estimate. Ann. Statist., 489-502 (1974). Bhansali, R. J., Effects of not knowing the order of an autoregressive pro- cess on the mean squared error of prediction-I, J. A mer. Statist. Assoc .. 588-597 (1981 ). Bhansali, R. J., The inverse partial correlation function of a time series and its applications, J. Mult. Analysis, 13:310-327 (1983).

chapter 7|11 pages

#### Nonlinear Estimation for Time Series Observed on Arrays

Robert H. Shumway, Sung-Etm Kim, and Robert R. Blandford

WithROBERT H. SHUMWAY and SUNG-EUN KIM University of California, Davis, California ROBERT R. BLANDFORD Center for Monitoring Research, Arlington, Virginia

chapter |4 pages

#### It follows that the analysis of variance components

Within Table I have the same approximate expected sums of squares, except that now given by Eq. (43). Also,

chapter 8|5 pages

#### Some Contributions to Multivariate Nonlinear Time Series and to Bilinear Models

T. Subba Rao and W. K. Wong

WithUniversity of Manchester Institute of Science and Technology, Manchester, England

chapter |3 pages

#### = (I= l, ... L)

Proof See Wong (1993, p35).

Withof arbitrary finite dimensions. Then ... y(L)} .. · cum{X(m,) E v("')}]Q To illustrate the result of Eq. (2.12) consider the evaluation of )0X(l,x(Jl}, X(l),l= of dimensions d x d. The original partition is { 12!3} and the indecom-

chapter |9 pages

#### ... , dZ,(wk)}

H(w2) ... ,wk) f*(wl ,w2, ... g(w2) ... g(wk)} - ... ,wk-1)

With=cum{ H(w of Eq. (3.8) is equal to -".ck and and (3.7), we arrive at the following useful result. THEOREM 3.1 Let the time series {Xsatisfy Eq. (3.1). Then /(wl, w2, ... 0 ck (3.9)

chapter 9|7 pages

#### Optimal Testing for Semi-Parametric AR Models-From Gaussian Lagrange Multipliers to Autoregression Rank Scores and Adaptive Tests

Marc Hallin and Bas J. M. Werker

WithMultipliers to Autoregression Rank Scores and Adaptive Tests Institute of Statistics and Operations Research, European Center for Advanced Research in Economics, and Universite Libre de Bruxelles, Brussels, Belgium

chapter |14 pages

#### < l,

With(2.5) It is easily seen that the g, 's are equicontinuous functions of 6 of 9, con- 9E vlt(O), and define N(N'N)-N' stands for the orthogonal (Euclidean) projection of the positive

chapter 4|11 pages

#### --8 (R-= --8

WithR-=--8:::; R-=--8 (4.20) with score function = tan This score function is not square-integrable, and the cosine density does not satisfy assumption (A3) (the Fisher information is infinite). The results of Proposition 4.3 thus do not hold under this cosine density, and the corre- sponding figures in Table I are not, strictly speaking, asymptotic relative efficiencies. However, the cosine density can be obtained as the limit of a

chapter |21 pages

#### ,f i

(f;g).f

Withof Ranked and Double Exponential Densities, Respectively. (f) Normal g Logistic g

chapter 10|3 pages

#### Statistical Analysis Based on Functionals of Nonparametric Spectral Density Estimators

Masanobti Taniguchi

WithJapan It seems that the origin of semiparametric estimation stems from Hannan

chapter |1 pages

#### = ... ,X = ...

G(j) = {Gah(j); a= ... ,m, b ... ,r}'s trG(j)QG(j)' < oo. Then the process

Withand e(t) ,e,(t))' such that = 0 and E{e(t)e(s)'} = 8(t,s)Q with n ={nab} a nonsingular {X( t); t E Z} is a second-order stationary process with spectral density

chapter |7 pages

#### ... r and

EIE{ea(tl)e,(t2)e,(t3)ed(t4)IF(t,- E{ea(ti)e,(t2)ec(t3)ed(t4)}l

With> 0, = O(r- > 0.

chapter |4 pages

#### ap R# = ap

LJ=o 0 for lzl S I, choose . (x) cos(j- RE(j- 1rb(j, £),

With= [r#(C)J. ln the. case fe(>..) = -t = rrr./,(>..)- In the case jlj( = exp[L_j=8cosj>.J, 8= I, (cf. Bloomfield = (log.xi, then the non-iterative estimator is [-1r,1r], choose Klj!(x) = [w(x)- mator is given by Iff =/

chapter |4 pages

#### eff(T) = lim {Eg,.(T")}

Withof "parametric alternatives'', eff(T For another test eff(T eff(Tt). If we take the parametric alternative P};H.fii (i.e., g = this asymp- Hannan Kondo (1993).

chapter |2 pages

#### = 0.05

K{g(..\)}d..\,

Withof OW of the test can be also applied to many problems in time series. a non-

chapter |1 pages

#### > 0 we choose category II

(],,f). Because e f and f

With> 0 implies en(./ (f,g) repre- of distance between of B": THEOREM 5.3 Zhang and Taniguchi (1995). Under the assumptions /ii), 8 C and h = (h and all the derivatives are bounded on [-n, n]. THEOREM 5.4 Zhang and Taniguchi (1995). Under Eq. (5.20) and the (0.4) and (0.5),

chapter |2 pages

#### + > 0

/(.f :g) is sensitive. Thus B" is better than

Withan interval in [-1r, for sufficiently small > I. Suppose thatf(>.) ¢ g(>.) on a set of positive Lebesgue measure. (0.4), IBn(/,f,g)- (f,g)l = 0, for a E (0, I), 11(/,f:

chapter |1 pages

#### J(f: g) and JB(f: g) work well, especially

With(f:g) does quite well. 6. HIGHER ORDER ASYMPTOTIC THEORY FOR SEMIPARAMETRIC ESTIMATION on an R"+ K: R" of partial derivatives K'(u) = (8/8u)K(u). For > 0, consider the statistic £( YIX),

chapter |5 pages

#### f + and( +

g is ( M qf, q'l q.f' f, gf, and ,ud)', is differentiable, /ii X ( li) by ( ·). h-d-+ j/i 1/

With+ I (M + I) of e are bounded, for M I. x) is dif- on the boundaries of their convex (possibly K(u), u llui!L)jK(u)j + IIK'(u)JI} du + I!K'(u)ll < oo, I, if e, + ... · · · K(u) du + ... + < (6.2) + ... + (, For E R" we denote the distribution function of THEOREM 6.1 Robinson (1995). Assume that (E.I )-(E.9) hold. Then IF-'(.:)- O(n-t/+ n-

chapter 11|2 pages

#### Efficient Estimation in a Semiparametric Additive Regression Model with ARMA Errors

Anton Schick

WithBinghamton University, Binghamton, New York This chapter characterizes and constructs efficient estimates of the finite

chapter |5 pages

#### = 1,2,

(3TV;- o"(n- v.), f normal.

Withwhere is an unknown vectbr in "f is an unknown Lipschitz-continuous function from [0, I] (UV), (VV), •.. are x [0, 1]- valued random vectors with common distribution G and are independent of the errors XX.•• which have zero means and finite variances. This model has received considerable attention in the case when -I Uj( with l' 2, As pointed out by Chen (1988) such estimates are efficient (in the sense of being least dispersed regular estimates) if the error density

chapter |7 pages

#### 1, l) into

f, c G, (r,, fh, G,.) a fh(x) d>.(x) is continuous at 0 and there exists a

WithI) into such that fo map from ( -1, into such that G= G and has a density g, with < 00 and = o(a map b 0 < .f < oo and

chapter |11 pages

#### ( }Pe) .N(o,

With= [I with I the 3 x 3 identity matrix. The desired result now of Le Cam's Third Lemma. Le N(O,J;

chapter 12|3 pages

#### Efficient Estimation in Markov Chain Models: An Introduction

WithUniversity of Siegen, Siegen, Germany outline the theory of efficient estimation for semiparametric Markov chain models, and illustrate in a number of simple cases how the theory

chapter |1 pages

#### = and assume that 8 is smooth in the following sense.

K, the tangent space, and a linear map D: K H, rdecreases to 0 pointwise and is 1r-integrable for large This version d:;k

With(2.5) of the form

chapter |12 pages

#### = and

WithhE H, the perturbed transition distribution Qthrough Eq. (2.1). product in Eq. (2.6) is = Q(hh'), of Qr at = Q€' the Fisher information. The inner product is not the nat-

chapter |1 pages

#### With£'= p' jp,

Q= is Qnk(x, dy) = Pnb(Y- Q(x, dy)( 1 -axe' o:x)) ). (Dk)(x,y) = -ax£'(y- o:x). Eb(c) +op(i).

Withof Qnk is (k,k') = aa'EXE£'(c) + Eb(c)b'(c), (7.1) -1/2" E£(c) - For a proof see Huang (1986) or Kreiss (1987b). For known innovation and Akritas and Johnson ( 1982). of estimating the autoregression parameter

chapter |5 pages

#### pis symmetric see Kreiss (l987a).

WithThe asymptotic variance of the efficient estimator is (EXEe'(c) Hence the relative efficiency of the least squares estimator is (EcEe'(c) This equals the relative efficiency of the empirical estimator in the i.i.d. location model generated by the density Consider now the problem of estimating the distribution of the in- novations c;. To be specific, we estimate the expectation E.f(c) of some

chapter |10 pages

#### ....

Withof Xi is the same as for the diffusion and Sorensen (1995) and Bibby and Sorensen (1996). In Markov chain Monte Carlo procedures is the = QQk, the one with random = (I jk) If we denote the simulations from the corre-

chapter |16 pages

#### .f. Suppose.{ is

dth derivative_/") L(R). Assume that there exists D2 0 and 2 0, 0 d such

Withas.f in the following discussion and the problem of estima- of _/ l(x)l for lxl > 4d and its derivatives ¢Ul up to l, there exists a

chapter |4 pages

#### of Experiments (Ed. R. R. Bahadur), Wiley Eastern, New

WithSer. A 52: 1-15. Pub/. Inst. Stat. Univ. Paris 35:51-

chapter 14|10 pages

#### Minimum Distance and Nonparametric Dispersion Functions

Omer Oztti'rk, Thomas P. Hettmansperger, and Jz'irg Hasler

WithOMER OZTURK Ohio State University, Marion, Ohio THOMAS P. HETTMANSPERGER Pennsylvania State University, JURG HUSLER University of Bern, Bern, Switzerland

chapter |3 pages

#### ={X;-

JL(F)}ja-(F) and r is the expected value of e{F(e)- K0(e). ../ii( (F)_ foEFofo(t)t2 Q(F) = 4a-(F)

WithI/2}- It is obvious that Q(F a test statistic. In order to find the asymp- totic distribution of Q(F we need to find an asymptotically equivalent expression for fo{a-(F From the linearization of the scale estimat- ing equation, we can write Ko(e;)} (12) combining Eqns (ll) and (12), for large n have l/2}-(e·)-

chapter |1 pages

#### er =

2Ko(Y;- 2F(t) and x(t) 2K(t)

WithCis needed for correct centering of the scale estimating

chapter 15|5 pages

#### Estimators of Changes

J. Antoch and M. Hu.fkova

WithJ. ANTOCH and M. HUSKOV Charles University, Prague, Czech Republic This chapter concerns point estimators of the change point in various mod- els. The least squares type estimators, M-estimators and R(rank)-estimators

chapter |4 pages

#### = I, 2, not depending on n such that, as

(j); j = 0, g,n(.) and W (M) are inde-

WithTHEOREM 5.1 (Fixed change). Let assumptions (M.l)-(M.6) be satisfied and let m argmax{6bu W(j, Z ±I, ±2, Z;(M), i 0, ±L ±2, cdf as hY fori 0 and as hY

chapter |6 pages

#### 'IE (0, 1).

(j, Z1(R)) - (j,Z(R))'s are defined by Eqns (2.20) and (2.16), ZJ R), i = 0, ± ±2, ... , of ZR) are inde- ti1R(G)- 8satisfying Eq. (2.24) and let m = [n1],

Withcdf as k cdf ,3, of Zn(R) are independent ±I, ±2, ... i::; cdf (Yand with Zi.3,i=0,±1,±2, ... ,i>O, cdf THEOREM 6.2 (Local change). Let assumptions (R.l)-(R.4) be satisfied 'IE (0, 1). Then the assertion

chapter |1 pages

#### = a > 0. Then, as

Jli<T(t(l- Proof The proof is, e.g., in Csorgo and Horvath (1993). P(j21og!!_ Proof Assertion (i) is the Erdos-Darling theorem, for the proof see, e.g., sLsi

With< oo with some [B(t)f } 'IE 1/2], where {B(t), t E is a Brownian bridge. THEOREM 9.2 Let assumptions of Theorem 9.1 be satisfied. 2loglogn max + 2log + exp{-2exp{-.r}}. If moreover (2.28) is satisfied, then, as n oo, for any y E R + Lsi} + 2log 1 1 47r) -2exp{ -y}}. =I, ... (9.4)

chapter |5 pages

#### of Theorem 2.4. The proof follows the lines of the proof of

k-G+ Gjn-+ 0. of Theorem 2.6. The proof is omitted since the assertions can be

Withand (9.5) taking into account the assumption 6oo. of the proof are in Antoch and Huskova (1994) and Huskova k) ISk.Lsl} = Op(l) > E > Laf!J'art of th; theorem from the of, the ... , standard arguments.

chapter 16|5 pages

#### On Inverse Estimation

Amoud C. M. van RooU and Frits H. Ruymgaart

WithUniversity of Nijmegen, Nijmegen, The Netherlands Texas Tech University, Lubbock, Texas 1. INTRODUCTION, MODEL, AND EXAMPLES Inverse estimation concerns the recovery of an unknown input signal from

chapter |3 pages

#### r R,

Withh(Z) r(•,Zk), and VIz-consistent estimator of q. of models A and B arise when of functions = {B,, t E Y},.Yc

chapter |20 pages

#### > 0}

L(/Lf[}) L l 0, for all > 0, we obtain a regularized inverse by

With/II 0, as a 1 0, (2.2) (2.2) we might, e.g., write For suitable families of functions 'Pn : (0, oo) [0, oo ), > 0, that in particular have the prop-

chapter |8 pages

#### {r [-A, for some 0 <A:= A(a) <

?A x, X e , t E nv 2n

Withon [0, oo). of the estimator reduces to (7.5) of the identity with kernel =::!sine of e. To remedy this, one could of a.s. convergence of d(iJ of r. An open problem is to determine the of this convergence. of operators: the Wiener-Hopf equation

chapter 17|5 pages

#### Approaches for Semiparametric Bayesian Regression

Alan E. Ge((and

WithUniversity of Connecticut, Storrs, Connecticut Developing regression relationships is a primary inferential activity. We consider such relationships in the context of hierarchical models incorpor-

chapter 0|1 pages

#### , the support of 9. That

. Then, partition Binto Band Bot. partition B Band E , etc. In this way a tree-like structure {Bo,Bl,Boo,Bol,B ,Bool•· .. } by II. In order to complete the Be(o:o,o

Withand B 0 and we denote the associated collection of sets II it is necessary to P(9 E BIOol9 E BJo) = ... = (I - · vw( I -

chapter |1 pages

#### ... such that = = =

With0 w.p. v I w.p. I - Then given say 0 w.p. vo, 1 w.p. 1 - etc. Let Then 9 is the desired realization. In application if we stop the tree at level r, then realizations are obtained with accuracy of measurement T'", prior to inversion to the 9 scale. apparent that, by adjusting the a's, we need not confine ourselves to

chapter |2 pages

#### p "'.f(p) G G

'ell such that = 'T};- g(t) increases in t and g(t) arises as a r's connect to g, the q's connect to the underlying

Withp"' G. Assuming¢;= ¢/H'; as usual, we can develop a Gibbs sampler to obtain of p, ¢, l) (equivalently and G. In particular, the G is obtained by GAMMA PROCESSES of modeling the monotonic function g, where of g is assumed to be R+. In fact, to suggest a time scale we write of g(t) as a stochastic process which is nondecreasing, = Pr(T 2 aIT a_J), -logPr(T 2 a;J add randomness to the structure such that the r; are of independent increments, i.e., g is a Levy process.

chapter |11 pages

#### of the event for the ith individual in the interval

g(t) g(t) arises by integrating ((J(s))- g(t) defined by Eq. (9) is called an extended gamma g is again an

Withor I, the likelihood associated with t; E [ae-l ,ctis of h. "'f(h

chapter 18|4 pages

#### Consistency Issues in Bayesian Nonparametrics

S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi

WithGHOSAL Vrije Universiteit, Amsterdam, The Netherlands J. K. GHOSH Indian Statistical Institute, Calcutta, India RAMAMOORTHI Michigan State University, East Lansing,

chapter |1 pages

#### Jf. The expected value under II, Err(P) is the prob-

P(B) P(B)II(dP). We will refer to the expec- Pin any of the above senses, then the Bayes estimate converges to Po in P(i) = 1}. Let Pbe in (j) } Jl(x) ensures two things: P in the sense, for any c

Withof P under II(·IXX or as the It is not hard to see that if the posterior is consistent 3. GENERAL CONSISTENCY THEOREMS of II. For any c > 0, let < c . and compactness of (')I > 0, I P(X;) c O

chapter xl|6 pages

#### , x2' ... , be iid U(O, e) where eE8 = (0, In this example the

Withof every U(O, from U(O, I) is ov. Thus the if II is a prior with support all of [0, 1], then it is Ghosh and Ramamoorthi (1997), that there is a prior with I) in its weak support such that the posterior fails to be consistent I). In this example too, Schwartz's condition fails to hold. If U is a strong neighborhood, then Le Cam (1973) and Barron ( 1989) III - < 6}. THEOREM 5. Let II be a prior on with II(.F) = I. Suppose foE L(tt) and II(KJ/ > 0 for all > 0. If for each > 0 there is a

chapter |17 pages

#### P"' PT(T, and given P if XX.•• Xare iid P, then the poster-

a:,+ 2:ak" Jfo logj

With< oo, then the of all distributions absolutely of the location model and strong consistency we eta!. (1998) shows and further a:,, = If < then the densityj < oo, we have, for > 0, PT(T,a.)(K<(fo)) > 0. a:,) provides an example of priors = then the

chapter 19|14 pages

#### Breakdown Theory for Estimators Based on Bootstrap and Other Resampling Schemes

Gutti Jogesh Babu

With1. INTRODUCTION

chapter 20|3 pages

#### On Second-Order Properties of the Stationary Bootstrap Method for Studentized Statistics

S. N. Lahiri

WithLAHIRI Iowa State University, Ames, Iowa

chapter |2 pages

#### y'ii;(H(Xi,)-H(X,J)/iw

L;;:,,(at'V'L- at'Vi,a1,'Vh, of d x I vector of jax;' a:r)'. For a =D"H(X )/o!, a E (71..+)".

Withof the studentized statistic of the of an individual resampled block, and where ai, is the of H(x) at x = Xi,. of the and Ti,, which is apparently different from the +a", a:.- = llxll = + ... + of A and for a square matrix A, let of A defined by IIAII Axil llxll 1}. Write

chapter 6|6 pages

#### > that k = 6-... < k <

(it) d,E*S(Ij,L}t LJ)')

Withand for all 1 E with < for some of assumptions (A.2)-(A.5) in terms of the a-fields fiJ/s, of the random vectors X/s themselves provides more flexibility in of these conditions. Choosing '!}i = a(Xi) of (A.2) and (A.4) rather trivial but makes of the conditional Cramer condition (A.5) quite difficult. In of '!}i (from a(Xi)) is often more suit- For example, if Xi = :Lr=o c" Yi-k for some of iid random vectors { Y )/+") + 3i(n+})- L:Jn+.J(\c,(cov*(S(J ,L)",S(/,L)') cov*(S(/L cov*(S(/Lt)'', S(/1, LJ)') · cov*(S(/1, LI)'S(/1, L1 )'))}]

chapter |2 pages

#### > 0 such that for all

(np)- P(K (np) 1ogn. Then, by the definition of K and the formula for the f(t) r)/n] r/ + of Theorem 3.1. To prove the theorem, it is enough to show that

With1ogn):::; exp( (logn) + (np) 1ogn):::; exp( (logn)). We only prove the first inequality. The other part can be proved by using similar arguments. Let be the largest integer not exceeding = log{e- qe')- )"},0 < t < -logq. Then, it is easy to see thatf(t) attains its minimum at log[(n- -logq (0, -logq). Next, using the fact that r(np)- o(l) as n---+ oo, and using Taylor's expansion, after some algebra, one can show that expCf(to)) + )), where 1ogn. This completes the proof of Lemma 4.5. 5. PROOFS We continue to use the notation and the conventions adopted in earlier sections. In particular, set = 0 unless otherwise stated.

chapter |3 pages

#### b>O be such that (D ... ,DdH(x))'-j.O for all

H(x), **an li x/1)" (

Withltll 28. Hence, using Taylor's expansion and some simple algebra, [D"+' H(XnlJ(x;;- 'hl*-

chapter |1 pages

#### Bin we get

v" B - E*S (/3) and by E**St(/3) and r,;, respect-

With"*-1/2 + b"): + (np)-1 I",' ((DB)2c)] of Bhattacharya and Ghosh IP**(n":::; EIIL,J

chapter |2 pages

#### o,(n-

IP**(Ti, ?**(Tin:::; x)l P*(As;;) C(d, £11Zxll

WithTo see this, note that > C(d, EIIZ,J )(npf (log + 2 sup IP**(Tj ){(np)- + P*(A5;;)}. To complete the proof of the theorem, it is now enough to show that = o(i) as oo (5.20) and that on the set ( 5.21)

chapter 21|5 pages

#### Convergence to Equilibrium of Random Dynamical Systems Generated by liD Monotone Maps, with Applications to Economics

Rabi Bhattacharya and Mukul Majumdar

WithBHATTACHARYA* Indiana University, Bloomington, Indiana MUKUL MAJUMDAR Cornell University, Ithaca, New York

chapter |1 pages

#### oo, x] = tt(0) = 0 = (v o y-

(1-b)d(ft,v) d(T*"tt, T*"v) d(T*N (T*(Il-N) ft), T*N (T*(Il-N)v)) Ej2.

With-oo, x] V1 E fx· (2.14) that -oo, -oo, x])]P(d1)1 = d(tt,v). (2.15) That is, T*N is a strict contraction and T* is a contraction. As a conse- > N, one has (l _ fJ)i"/Nid(T*("-1"/NJN) fl, T*(11-[11/NjN)v) (2.16) 9(S) is a complete metric that T*N has a It follows that G(yc)

chapter x|9 pages

#### , S such that G(x,J <E. Then v(S):::: G(J'o:)->

WithREMARK 2.2. Suppose that a if the {x}) of is continuous), then o 'Y_, is nonatomic E of zero P-probability). It follows that if Xhas a continuous d.f., then so and so on. Since, by Theorem of continuous d. f.'s (of X of the latter is continuous. Thus is nonatomic if a

chapter V|1 pages

#### _, by

ct+d h satisfies the following property: for each

Withand the optimal transition function (6.7) Vis concave and nondecreasing on !Pl+. of identifying conditions under which the function his mono- Majumdar and Mitra ( 1994a), and Mitra and Ray ( 1984). Instead of present- -11·dx, c)/'(x) ll'dx, c) n°. (6.9) on c (as in the "standard" Ram- = 0, so that Eq. (6.9) is satisfied if < 0.

chapter 6|9 pages

#### 2 Complex Dynamics

x E [0, 0.5) x 2: 0.5, px(i- X,p E I b{az-

Withof economies indexed by a parameter (where I )-(F3)) and the same discount factor 8 E (0, I). of their felicity or one-period return functions: w : x I ( w depending on the + [h(z.rt)]

chapter |5 pages

#### J a=

a.•• b ... For simpli- =a, "f(b) = x =a, "f(b) =b. On the space S

With:= {'y E Extend"( E f1 \(f, n f2) b} ) by setting = =b. r, n f2 by setting "f(a) proof given above shows that is the unique

chapter 22|14 pages

#### Chi-Squared Tests of Goodness-of-Fit For Dependent Observations

Kamal C. Chanda

Withof goodness-of-fit of a sequence of observations to a given distribution of distributions usually requires that the sequence of obser-

chapter 23|11 pages

#### 3 Applications

George G. Roussas

Withof California, Davis, California of this chapter is to provide a selective, as opposed to an of some modes of positive and negative dependence,

chapter |2 pages

#### p.d.f.f of a real-valued r.v. X is said to be a Polya frequency function of

(PFif, for all x PFif log ,xk(k>l) ..• Xbe a random sample without replacement

Withof Joag-Dev and Proschan, 1983). If X NA (a.s.). DEFINITION 3.7. The distribution of the vector ... ,xk) of Joag-Dev and Proschan, 1983). A permutation For :S N, let X

chapter |2 pages

#### liMIT THEOREM UNDER POSITIVE OR

X"= 0, n E

WithNEGATIVE ASSOCIATION PA r.v. 's CIT for a random field of PA r.v.'s were obtained by C: R In For u and v u:::; v, means, of course, that vi, i = I, ... , d, and < For any two such points, define the of the discussion here makes our results of atmospheric applications, environmental science, geostatistical radar and sonars. Such applications are discussed in a recent publication,

chapter |5 pages

#### > c > n

'n-'. F(x)]ja(x)}, n I, converges in distribu- f(x) of X J,,(x) f(x) a.s. on I, and also I, and x E J]----+ 0 a.s., where J is any compact subset of K(lx-

With> c]; x > c]; x E J}----+ 0 a.s. over of and, for every x for which 0 < F(x) < I, the of r.v.'s { P(x)]. Now, letf,(x) be the usual kernel esti- of the p.d.f. -f(x)l; x E 0 a.s. Similar results also hold for the usual of the hazard rate r(x); namely, r > 0}. Similar results have been obtained in Roussas (1991) = Jx (5.2)

chapter 24|19 pages

#### Second-Order Information Loss Due to Nuisance Parameters: A Simple Measure

Bruce G. Lindsay and Richard Waterman

Withof Pennsylvania, Philadelphia,

chapter |3 pages

#### f That

_g-f -f')-

With£[/(x)/f(x)] of this as a nuisance parameter problem, with the of interest being e, and the nuisance parameter being the infinite of functions[(-). The concept of efficient score has been

chapter 1966|1 pages

#### of multivariate multi-sample rank-order tests. Sankhya

of dispersion matrices. Sankhya, Series A, 30, 1-22. Co-

WithSeries A, 28. 353-376. Co-author: K. Sen. 1967 Mathematical Statistics, 38, 523-549. Co-author: K. L. Mehra. author: K. Sen. Annals of the Institute of Statistical Mathematics, 22, 99-106.

chapter |2 pages

#### of rank order tests for a genera/linear hypothesis. The Annals of

of location parameters in the multivariate one-

Withof Mathematical Statistics, 41, 87- K. Sen.

chapter |2 pages

#### A* MAG= A* M and

of order statistics for mix-

Withof Indian Statistical Association, 17, 103- 1980 of Multivariate Analysis, 10, 405-425.

chapter |4 pages

#### of convergence in the central limit theorem for signed rank

of convergence to asymptotic normalityfor generalized linear

Withof the Institute of Statistical Mathematics, 37, 95-108. Co-author: of the Institute of Statistical Mathematics, 37,

chapter |3 pages

#### of convergence in normal approximation and large deviation

Withof Probability and its Applications, Translation

chapter |2 pages

#### of the iterated logarithm for perturbed empirical distribution func-

WithJournal of Theoretical Probability, 7, 831-855. Co-author: Michel HareI . 200. Berry-Esseen rate in asymptotic normality for perturbed sample quan- Metrika, 41, 83-98. Co-author: Munsup Seoh.