Lectures i n Mathematical Statistics Parts 1 and 2

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 22 9

Lectures i n Mathematical Statistics

Parts 1 and 2

Yu. N . Lin'ko v Translated b y Ole g Klesov an d Vladimi r Zayat s

^//I^^=\\o America n Mathematica l Societ y / | Providence , Rhod e Islan d

10.1090/mmono/229

E D I T O R I A L C O M M I T T E E

A M S S u b c o m m i t t e e

Robert D . MacPherso n Grigori i A . Marguli s Jame s D . Stashef f (Chair )

A S L S u b c o m m i t t e e Steffe n Lemp p (Chair )

I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair )

K) . H . JIHHBKO B

J I E K I I H H n o M A T E M A T M M E C K O H C T A T M C T M K E

"MCTOKM", HOHEIIK , 200 1

This wor k wa s originall y publishe d i n Russia n b y Istoki , Donets k unde r th e titl e

"JTeKijHH n o MaTeMaTH^ecKO H CTaracTHKe , ^acT H 1,2 " © K) . H . JIUHLKOB , 1999 . Th e

present translatio n wa s create d unde r licens e fo r th e America n Mathematica l Societ y an d

is published b y permission .

Translated fro m th e Russia n b y Ole g Kleso v an d Vladimi r Zayats .

2000 Mathematics Subject Classification. Primar y 62-01 .

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Library o f Congres s Cataloging-in-Publicatio n D a t a

Lin'kov, IU . N. [Lektsii p o matematicheskoi statistike . English ] Lectures i n mathematical statistic s : parts 1 and 2 / Yu . N. Lin'kov ; translated b y Oleg Kleso v

and Vladimi r Zayats . p. cm . - (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 229)

Includes bibliographica l reference s an d index . ISBN 0-8218-3732- X (alk . paper) 1. Mathematica l statistics . I . Titl e II . Serie s

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Contents

Foreword t o th e Englis h Translatio n vi i

Part 1 1

Preface t o Par t 1 3

Chapter 1 . Sample s fro m One-Dimensiona l Distribution s 5 1.1. Empirica l distributio n functio n an d it s asymptoti c behavio r 5 1.2. Sampl e characteristic s an d thei r propertie s 8 1.3. Orde r statistic s an d thei r propertie s 1 3 1.4. Th e distribution s o f som e function s o f Gaussia n rando m vector s 2 0

Chapter 2 . Sample s fro m Multidimensiona l Distribution s 2 5 2.1. Empirica l distributio n function , samplin g moments , an d thei r

properties 2 5 2.2. Samplin g regressio n an d it s propertie s 3 1

Chapter 3 . Estimatio n o f Unknown Parameter s o f Distribution s 3 9 3.1. Statistica l estimator s an d thei r qualit y measure s 3 9 3.2. Estimatio n o f a locatio n paramete r 4 9 3.3. Estimatio n o f a scal e paramete r 5 6 3.4. Th e Cramer-Ra o inequalit y an d efficien t estimator s 6 1 3.5. Th e Cramer-Ra o inequalit y fo r a multidimensiona l paramete r 8 0 3.6. Integra l inequalitie s o f Cramer-Ra o typ e 8 8

Chapter 4 . Sufficien t Statistic s 9 9 4.1. Sufficien t statistic s an d a theorem o n factorizatio n 9 9 4.2. Sufficien t statistic s an d optima l estimator s 11 3

Chapter 5 . Genera l Method s fo r Constructin g Estimator s 13 1 5.1. Metho d o f moment s 13 1 5.2. Th e maximu m likelihoo d metho d 13 3 5.3. Baye s an d minima x method s 14 2 5.4. Confidenc e interval s an d region s 14 7

References t o Par t 1 15 3

vi C O N T E N T S

Part 2 15 5

Preface t o Par t 2 15 7

Chapter 1 . Genera l Theor y o f Hypotheses Testin g 15 9 1.1. Testin g tw o simpl e hypothese s 15 9 1.2. Distinguishin g a finite numbe r o f simple hypothese s 17 3 1.3. Distinguishin g composit e hypothese s 18 2

Chapter 2 . Asymptoti c Distinguishabilit y o f Simpl e Hypothese s 20 3 2.1. Statistica l hypothese s an d test s 20 3 2.2. Type s o f th e asymptoti c distinguishabilit y o f familie s o f hypothe -

ses. Th e characterizatio n o f types 20 5 2.3. Complet e asymptoti c distinguishabilit y unde r th e stron g la w o f

large number s 21 8 2.4. Complet e asymptoti c distinguishabilit y unde r th e wea k conver -

gence 23 8 2.5. Contiguou s familie s o f hypotheses 24 8

Chapter 3 . Goodness-of-Fi t Test s 26 3 3.1. Th e settin g o f the problem . Kolmogoro v tes t 26 3 3.2. Th e Pearso n tes t 26 6 3.3. Smirno v tes t 27 5 3.4. Othe r goodness-of-fi t test s 28 2

Chapter 4 . Sequentia l Test s 29 3 4.1. Baye s sequentia l test s o f hypotheses 29 3 4.2. Wal d sequentia l test s 30 0 4.3. Th e optimalit y o f a sequentia l Wal d tes t 31 0

References t o Par t 2 31 7

Index 319

Foreword t o th e Englis h Translatio n

Parts 1 and 2 of "Lecture s i n Mathematica l Statistics " b y Yu. N . Lin'kov wer e originally publishe d i n Russia n a s two separate books . Fo r the Englis h translation , the tw o part s ar e combine d int o on e book . Eac h par t ha s it s ow n prefac e an d lis t of references , wit h chapters , sections , theorems , etc. , numbere d independentl y i n each part .

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References t o Par t 2

1. R . R . Bahadur , Some limit theorems in statistics, SIAM , Philadelphia , PA , 1971 . 2. J.-R . Barra , Notions fondamentales de statistique mathematique, Dunod , Paris , 1971 ; English

transl., Mathematical basis of statistics, Academi c Press , Ne w York-London , 1981 . 3. P . Billingsley , Convergence of probability measures, Wiley , Ne w York-London-Sydney , 1968 . 4. D . Blackwel l an d M . A . Girshick , Theory of games and statistical decisions, Wile y an d Chap -

man an d Hall , Ne w Yor k an d London , 1954 . 5. M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-based methods in linear statistical

models, "Nauka" , Moscow , 1997 ; Englis h transl. , Amer . Math . Soc , Providence , RI , 1997 . 6. L . N . Bol'shev an d N . V. Smirnov , Tables of mathematical statistics, "Nauka" , Moscow , 1965 .

(Russian) 7. A . A . Borovkov , Mathematical statistics. Estimation of parameters. Testing of hypotheses,

"Nauka", Moscow , 1984 ; Englis h transl. , Mathematical statistics, Gordo n & Breach , Amster -dam, 1998 .

8. , Mathematical statistics. Supplementary chapters, "Nauka" , Moscow , 1984 ; Englis h transl., Gordo n & Breach , Amsterdam , 1998 .

9. , Mathematical statistics, "Nauka" , Sibirsko e otdeleni e RAN , Novosibirsk , 1997 . (Rus -sian)

10. A . A . Borovko v an d A . A . Mogul'skiT , Large deviations and the testing of statistical hypothe-ses, Proceeding s o f th e Institut e o f Mathematics , 19 , "Nauka" , Sibirsko e otdeleni e RAN , Novosibirsk, 1992 . (Russian )

11. N . N . Chencov , Statistical decision rules and optimal inference, "Nauka" , Moscow , 1972 ; English transl. , Amer . Math . Soc , Providence , RI , 1982 .

12. D . M . Chibisov , Certain tests of the chi-square type for continuous distributions, Teor . Vero -yatnost. i Primenen . 1 6 (1971) , no . 1 , 3-20 ; Englis h transl . i n Theor . Probabilit y Appl . 1 6 (1971), no . 1 , 1-22 .

13. Y . S . Chow , H . Robbins , an d D . Siegmund , The theory of optimal stopping, Correcte d reprin t of th e 197 1 original , Dover , Ne w York , 1991 .

14. H . Cramer , Mathematical methods of statistics, reprin t o f th e 194 6 original , Princeto n Univ . Press, Princeton , NJ , 1999 .

15. M . H . DeGroot, Optimal statistical decisions, McGraw-Hill , Ne w York-London-Sydney, 1970 . 16. D . Dugue , Traite statistique theorique et appliquee: analyse aleatoire, algebre aleatoire, Mas -

son e t Cie , Paris , 1958 . (French ) 17. R . S . Ellis, Entropy, large deviations, and statistical mechanics, Springer-Verlag , Berlin , 1985 . 18. W . Feller , An introduction to probability theory and its applications, Thir d edition , vol . 1 ,

Wiley, Ne w York-London-Sydney , 1968 ; vol . 2 , 1971 . 19. I . I . Gikhman , An introduction to the general theory of measure and integral, Donets k Uni -

versity Press , Donetsk , 1971 . (Russian ) 20. B . V . Gnedenk o an d A . N . Kolmogorov , Limit distributions for sums of independent random

variables, Gostekhizdat , Leningrad-Moscow , 1949 ; English transl. , Addison-Wesley , Reading , MA, 1968 .

21. P . E . Greenwoo d an d A . N . Shiryayev , Contiguity and the statistical invariance principle, Gordon & Breach , Ne w York , 1985 .

22. J . Haje k an d Z . Sidak , Theory of rank tests, Academi c Pres s an d Academi a Publishin g Hous e of th e Czechoslova k Academ y o f Sciences , Ne w York-Londo n an d Prague , 1967 .

23. P . R . Halmos , Measure theory, Va n Nostrand , Ne w York , 1950 . 24. P.-L . Hennequi n an d A . Tortrat , Theorie des probabilites et quelques applications, Masso n e t

Cie, Editeurs , Paris , 1965 . (French )

317

318 R E F E R E N C E S T O PAR T 2

25. I . A . Ibragimo v an d R . Z . Khas'minskii , Statistical estimation. Asymptotic theory, "Nauka" , Moscow, 1979 ; Englis h transl. , Springer-Verlag , Ne w York-Berlin , 1981 .

26. G . I . Ivchenk o an d Yu . I . Medvedev , Mathematical statistics, "Vysshay a shkola" , Moscow , 1984. (Russian )

27. , Decomposable statistics and hypotheses testing for grouped data, Teor . Veroyatnost . i Primenen . 2 5 (1980) , no . 3 , 549-560 ; Englis h transl . i n Theor y Probab . Appl . 2 5 (1981) , no. 3 , 540-551 .

28. L . Jaco d an d A . N . Shiryaev , Limit theorems for stochastic processes, 2n d edition , Berlin , Springer-Verlag, 2003 .

29. M . G . Kendal l an d A . Stuart , The advanced theory of statistics. Inference and relationship, vol. 2 , Griffin , London , 1961 .

30. N . Kligen e an d L . Telksnis , Methods of detecting instants of change of random processes properties, Avtomat . i Telemekh . (1983) , no . 10 , 5-56 ; Englis h transl . i n Automat . Remot e Control 4 4 (1984) , no . 10 , par t 1 , 1241-1283 .

31. A . N . Kolmogoro v an d S . V . Fomin , Introductory real analysis, "Nauka" , Moscow , 1968 ; English t rans l , Prentice-Hall , Ne w York , 1970 .

32. A . M . Kolodii , Basics of the general theory of measure and integral, Volgogra d Universit y Press, Volgograd , 1999 . (Russian )

33. S . Kullback , Information theory and statistics, Dover , Ne w York , 1968 . 34. E . L . Lehmann , Theory of point estimation, Wiley , Ne w York , 1983 . 35. F . Lies e an d I . Vajda , Convex statistical distances, Teubner , Leipzig , 1987 . 36. Yu . N . Lin'kov , The asymptotic distinguishability of two simple statistical hypotheses, Pre -

print 86.45 , Institut e o f Mathematic s o f Ukrainia n Academ y o f Sciences , Kiev , 1986 . 37. , Asymptotic statistical methods for stochastic processes, "Naukov a dumka" , Kiev ,

1993; Englis h transl. , Amer . Math . Soc , Providence , RI , 2001 . 38. , Lectures on mathematical statistics, vol . 1 , "Istoki" , Donetsk , 1999 ; Englis h transl. ,

Part 1 of thi s book . 39. , Large deviation theorems for extended random variables and some applications, J .

Math. Sci . 9 3 (1999) , no . 4 , 563-573 . 40. G . V . Martynov , Omega-square tests, "Nauka" , Moscow , 1978 . (Russian ) 41. I . P . Natanson , Theory of functions of real variable, GITTL , Moscow , 1957 ; Englis h transl. ,

vol. 1 , Ungar , Ne w York , 1955 ; vol . 2 , 1961 . 42. J . Neveu , Mathematical foundations of the calculus of probability, Masso n e t Cie , Editeurs ,

Paris, 1964 ; Englis h transl. , Holden-Day , Sa n Francisco-London-Amsterdam , 1965 . 43. C . R . Rao , Statistical inference and its applications, Wiley , Ne w York-London-Sydney, 1965 . 44. R . T . Rockafellar , Convex analysis, Princeto n Univ . Press , Princeton , NJ , 1970 . 45. G . G . Roussas , Contiguity of probability measures: some applications in statistics, Cambridg e

Univ. Press , London-Ne w York , 1972 . 46. A . N . Shiryaev , Optimal stopping rules, "Nauka" , Moscow , 1976 ; Englis h transl. , Springer -

Verlag, Ne w York-Heidelberg , 1978 . 47. , Probability, "Nauka" , Moscow , 1989 ; Englis h transl. , Springer-Verlag , Ne w York ,

1996. 48. A . V . Skorokhod , Random processes with independent increments, "Nauka" , Moscow , 1964 ;

English transl. , Kluwer , Dordrecht , 1991 . 49. J.-L . Soler , Notion de liberte en statistique mathematique, Thes e d e Docteu r d e Troiseim e

Cycle, Universit e d e Grenoble , 1970 . (French ) 50. F . P . Tarasenko , Nonparametric statistics, Toms k Universit y Press , Tomsk , 1976 . (Russian ) 51. A . Wald , Sequential analysis, Wile y an d Chapma n an d Hall , Ne w Yor k an d London , 1947 . 52. A . Wald, Statistical decision functions, Wile y an d Chapma n an d Hall , Ne w York an d London ,

1950. 53. S . S . Wilks , Mathematical statistics, Wiley , Ne w York-London , 1967 . 54. S . Zacks , The theory of statistical inference, Wiley , Ne w York-London-Sydney , 1971 .

Index

cr-algebra sufficient, 102 , 11 6

minimal, 11 6

a prior i probability , 17 6

Bayes approac h complete, 18 4 partial, 18 4

Bayes estimatio n method , 14 3 bias (o f th e estimator) , 4 0

canonical equation , 3 3 conditional expectation , 9 9 conditional probability , 10 0 confidence bounds , 4 0 confidence interval , 39 , 14 7 confidence level , 14 7 confidence limits , 14 7 confidence probability , 40 , 14 7 confidence region , 15 0

asymptotic, 15 0 confidence set , 19 4

uniformly mos t precise , 19 6 unbiased, 19 8

convergence weak, 1 2

correlation coefficient , 2 7 sampling, 2 9

Cramer-Rao bound , 8 8 critical set , 26 3

decision function , 159 , 31 1 distance

in variance , 20 5 Kakutani-Hellinger, 20 6

distribution chi-square, 2 1 Fisher, 2 4 least favorable , 17 9 Snedekor, 2 4 standard normal , 2 0 Student, 2 4

distribution functio n empirical, 5 , 2 5 Kolmogorov, 8 , 26 4

entropy relative, 21 8

error probability , 20 3 of typ e I , 159 , 186 , 26 3 of typ e II , 16 0

estimator absolutely admissible , 4 4 admissible, 4 4 asymptotically Bayes , 9 6 asymptotically efficient , 77 , 8 5

in th e stron g (weak ) sense , 7 7 asymptotically minimax , 9 7 asymptotically R-Bayes , 9 6 asymptotically unbiased , 4 0 Bayes, 4 5

a posteriori , 14 2 generalized, 4 5

consistent, 4 2 efficient, 76 , 8 5 equivariant, 49 , 5 7 likelihood, 13 4 maximum likelihood , 13 4

polynomial, 27 3 minimax, 14 6 optimal, 4 4 Pitman, 50 , 5 7 point, 3 9 statistical, 3 9 strongly consistent , 4 3 superefficient, 9 0 unbiased, 4 0

excess, 1 2

families o f hypothese s completely asymptoticall y

distinguishable, 20 8 completely asymptoticall y

indistinguishable, 21 1 mutually contiguous , 21 5 mutually noncontiguous , 21 5

family o f function s dense, 21 3 uniformly integrable , 21 4

319

320 INDEX

family o f hypothese s contiguous, 21 3

family o f measure s complete, 11 8 dominated b y a measure , 10 2 exponential, 12 2 relatively compact , 24 8 tight, 24 8

Fisher information , 62 , 25 7 matrix, 8 0

Gamma distribution , 1 6

Neyman-Fisher factorizatio n criterion , 10 3

observation, 101 , 15 9 operating characteristi c (o f a test) , 31 0 order (o f th e moment) , 2 9 order statistic , 5

central, 1 6

power function , 18 2

quantile, 16 , 24 0 sampling, 1 6

Hellinger integral , 20 6 hypothesis, 15 9

composite, 15 9 main, 26 3 null, 26 3 one-sided, 18 7 simple, 15 9 two-sided, 18 7

inequality Barankin-Kiefer, 8 8 Bhattacharyya, 8 7 Chapman-Robbins, 75 , 8 8 Cramer-Rao, 6 8

matrix analog , 8 1

Kullback-Leibler divergence , 21 8

least variance , 3 3 lemma

Neyman-Pearson, 16 7 Stein, 22 3

likelihood function , 13 3 logarithmic, 13 3

likelihood ratio , 16 3 location parameter , 4 9

mean squar e approximation , 3 2 measure

absolutely continuous , 16 1 measures

equivalent, 16 1 singular, 16 2

method o f moments , 13 1 minimax, 17 9 mixed moment , 2 6

central, 2 6 sampling, 2 8

sampling, 2 8 moment, 8

central, 8 , 2 9 sampling, 9 , 3 0

central, 9 , 3 0

random variabl e uncorrelated, 2 6

random vecto r Gaussian, 2 0 normal, 2 0

random walk , 27 9 rank statistic , 28 6 reflection method , 27 9 regression, 3 1

linear, 3 2 coefficient of , 3 2 sampling, 3 5 sampling coefficien t of , 3 5

parabolic, 3 4 sampling, 3 7

regularity condition s Cramer-Rao (CR) , 6 1 Cramer-Rao (CR)* , 6 9

relative stability , 21 9 risk

a posteriori , 14 2 of th e estimator , 4 5 of th e test , 19 9

risk function , 4 4

sample, 2 5 sampling space , 3 9 scale parameter , 5 6 sequence

asymptotically normal , 7 Sheppard correction , 27 5 skewness, 1 2 Spearman ran k correlatio n coefficient , 28 9 statistic, 39 , 10 1

complete, 11 8 minimal, 11 6 of th e test , 26 3 subordinated, 11 6 sufficient, 10 1

statistics equivalent, 11 6

stopping rule , 29 3 Bayes, 29 4 truncated, 29 5

INDEX 321

test Bayes, 166 , 175 , 183 , 31 1 chi-square, 27 0 empty blocks , 28 5 empty boxes , 28 3 for independence , 28 8 goodness-of-fit, 26 3

Kolmogorov, 26 4 Pearson, 27 0 Smirnov, 28 1 symmetric, 28 2

Kendall, 29 0 likelihood ratio , 20 4 Mann-Whitney, 28 7 maximum likelihood , 167 , 18 2 minimax, 167 , 18 4 Moran, 29 1 Neyman-Pearson, 170 , 20 4 nonrandomized, 159 , 17 4 of series , 28 6 Pearson, 27 3 g-Bayes, 31 1 quantile, 27 1 randomized, 159 , 17 4 rank, 28 6 sequential, 29 3

Wald, 30 0 sign, 27 1 Spearman, 28 9 statistical, 17 4 unbiased, 19 1 uniformly mor e powerful , 18 3 uniformly mos t powerfu l (UMP) , 18 3 von Mises-Smirnov , 29 1 Wilcoxon, 28 7

theorem Glivenko, 6 Kolmogorov, 8 , 26 4 Le Cam , first, 25 1 Lehmann-SchefTee, 12 1 Pearson, 26 7 Rao-Blackwell-Kolmogorov, 11 3

trajectory, 27 7

Wald identity , 30 4

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Titles i n Thi s Serie s

229 Yu . N . Lin'kov , Lecture s i n mathematica l statistics , 200 5

228 D . Zhelobenko , Principa l structure s an d method s o f representatio n theory , 200 5

227 Takahir o Kawa i an d Yoshitsug u Takei , Algebrai c analysi s o f singula r perturbatio n

theory, 200 5

226 V . M . Manuilo v an d E . V . Troitsky , Hilber t C*-modules , 200 5

225 S . M . Natanzon , Modul i o f Rieman n surfaces , rea l algebrai c curves , an d thei r

superanaloges, 200 4

224 Ichir o Shigekawa , Stochasti c analysis , 200 4

223 Masatosh i Noumi , Painlev e equation s throug h symmetry , 200 4

222 G . G . Magaril-Il'yae v an d V . M . Tikhomirov , Conve x analysis : Theor y an d

applications, 200 3

221 Katsue i K e n m o t s u , Surface s wit h constan t mea n curvature , 200 3

220 I . M . Gelfand , S . G . Gindikin , an d M . I . Graev , Selecte d topic s i n integra l

geometry, 200 3

219 S . V . Kerov , Asymptoti c representatio n theor y o f th e symmetri c grou p an d it s

applications t o analysis , 200 3

218 Kenj i U e n o , Algebrai c geometr y 3 : Furthe r stud y o f schemes , 200 3

217 Masak i Kashiwara , D-module s an d microloca l calculus , 200 3

216 G . V . Badalyan , Quasipowe r serie s an d quasianalyti c classe s o f functions , 200 2

215 Tatsu o Kimura , Introductio n t o prehomogeneou s vecto r spaces , 200 3

214 L . S . Grinblat , Algebra s o f set s an d combinatorics , 200 2

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegativ e matrices , 200 2

212 A . V . Mel'nikov , S . N . Volkov , an d M . L . Nechaev , Mathematic s o f financial

obligations, 200 2

211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2

210 Toshitak e Kohno , Conforma l field theor y an d topology , 200 2

209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2

208 Yuki o Matsumoto , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i Ueno , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l equation s an d

Secondary Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d coding , 200 2

202 V . P . Maslo v an d G . A . Omel'yanov , Geometri c asymptotic s fo r nonlinea r PDE . I ,

2001

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology , 200 1

197 Kenj i U e n o , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 200 1

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c operator s i n differentia l topology, 200 1

TITLES I N THI S SERIE S

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n geometry , 200 0

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces , 200 0

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena , 200 0

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f rando m variable s

and rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d applications , 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r theor y 1 :

Fermat's dream , 200 0

185 Kenj i U e n o , Algebrai c Geometr y 1 : Fro m algebrai c varietie s t o schemes , 199 9

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d conservatio n

laws fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finite groups . Par t 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d optima l control ,

1998

179 V . E . Voskresenski i , Algebrai c group s an d thei r birationa l invariants , 199 8

178 Mi tsu o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e Hamiltonia n

systems wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g Wu , Secon d orde r ellipti c equation s an d ellipti c

systems, 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l propertie s o f

distributions o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud 7, Character s o f finite groups . Par t 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t coefficient , 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s o f fractals , 199 7

166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n

Galois theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko ,

Probability theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r

statistical models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 199 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .