lectures in mathematical statistics
TRANSCRIPT
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 .
For additiona l informatio n an d update s o n thi s book , visi t
www.ams.org/bookpages /mmono-229
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
QA276.16.L5513 200 5 519.5-dc22 200505266 1
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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 / .