As a result, the issues of mathematical statistics began to interest me, <strong>and</strong> it seemed to methat, once I return to this field <strong>and</strong> focus all my power there, I would to a larger extent benefitmy mother country <strong>and</strong> the cause of the socialist transformation of social relations. Afteraccomplishing a few works which resulted from my groping for my own sphere of research, Iconcentrated on generalizing the stochastic methods to the statistical treatment of observationsnot being mutually independent in the sense of the theory of probability.It seemed to me, that, along with theoretical investigations, I ought to study some concreteproblems so as to check my methods <strong>and</strong> to find problems for theoretical work in a number ofresearch institutes. For me, the methodical approach to problems <strong>and</strong> the attempts to preventdeviations from the formulated goal always were in the forefront. In applications, I consider asmost fruitful my contributions, although not numerous, in the field of geophysics.I have written this in December 1938, when compiling my biography on the occasion of myfirst entering the Steklov Mathematical Institute at the Academy of Sciences of the SovietUnion. I described in sufficient detail the story of my life <strong>and</strong> internal development up to thebeginning of my work at Moscow State University <strong>and</strong> later events are sufficiently welloutlined in my completed form. I shall only add, that, while working at the University, mymain activity had been not teaching but work at the Mathematical Research Institute there.When the Government resolved that that institution should concentrate on pedagogic work({monitoring} postgraduate studies) with research being mainly focussed at the SteklovInstitute, my transfer to the latter became a natural consequence of that reorganization.Notes1. {It had been extremely dangerous to maintain ties with foreigners, <strong>and</strong> even with relativesliving abroad, hence this lengthy explanation. A related point is that Slutsky passed over insilence his work at the Conjuncture Institute, an institution totally compromised by the savagepersecution of its staff.}2. {Vannovsky as well as Bogolepov mentioned in the same connection by Chetverikov inhis essay on Slutsky (also translated here) are entered in the third edition of , vols 4 <strong>and</strong> 3 respectively, whose English edition is called GreatSoviet Encyclopedia. It is not easy, nor is it important, to specify which of them was actuallyresponsible for expelling the students.}3. {This unpublished composition is kept at the Vernadsky Library, Ukrainian Academy ofSciences.}6b. E.E. Slutsky. [Later] AutobiographyI was born on 7(19) April 1880 in the village Novoe of the former Mologsky District,Yaroslavl Province, to a family of an instructor of a teacher’s seminary. After graduating in1899 from a classical gymnasium in Zhitomir with a gold medal, I entered the MathematicalDepartment of the Physical <strong>and</strong> Mathematical Faculty at Kiev University. I was several timesexpelled for participating in the student movement <strong>and</strong> therefore only graduated in 1911, fromthe Law Faculty. Was awarded a gold medal for my composition on political economy, but,owing to my reputation of a Red Student, I was not left at the University for preparing myselffor professorship. I passed my examinations in 1917 at Moscow University <strong>and</strong> became Masterof Political Economy <strong>and</strong> <strong>Statistics</strong>.I wrote my student composition for which I was awarded a gold medal from the viewpointof a mathematician studying political economy <strong>and</strong> I continued working in this direction formany years. However, my intended {summary?} work remained unfinished since I lost interestin its essence (mathematical justification of economics) after the very subject of study (aneconomic system based on private property <strong>and</strong> competition) disappeared in our country with
the revolution. My main findings were published in three contributions ([6; 21; 24] in theappended list {not available}). The first of these was only noticed 20 years later <strong>and</strong> itgenerated a series of Anglo-American works adjoining <strong>and</strong> furthering its results.I became interested in mathematical statistics, <strong>and</strong>, more precisely, in its then new directionheaded by Karl Pearson, in 1911, at the same time as in economics. The result of my studieswas my book (Theory of Correlation), 1912, the first systematicexplication of the new theories in our country. It was greatly honored: Chuprov published acommendable review of it <strong>and</strong> academician Markov entered it in a very short bibliography tohis (Calculus of <strong>Probability</strong>). The period during which I had beenmostly engaged in political economy had lasted to ca. 1921 – 1922 <strong>and</strong> only after that Idefinitively passed on to mathematical statistics <strong>and</strong> theory of probability.The first work [8] of this new period in which I was able to say something new was devotedto stochastic limits <strong>and</strong> asymptotes (1925). Issuing from it, I arrived at the notion of a r<strong>and</strong>omprocess which was later destined to play a large role. I obtained new results, which, as Ithought, could have been applied for studying many phenomena in nature. Other contributions[22; 31; 32; 37], apart from those published in the C.r. Acad. Sci. Paris (for example, on thelaw of the sine limit), covering the years 1926 – 1934 also belong to this cycle. One of these[22] 1 includes a certain concept of a physical process generating r<strong>and</strong>om processes <strong>and</strong>recently served as a point of departure for the Sc<strong>and</strong>inavian {Norwegian} mathematicianFrisch <strong>and</strong> for Kolmogorov. Another one [37], in which I developed a vast mathematicalapparatus for statistically studying empirical r<strong>and</strong>om processes, is waiting to be continued.Indeed, great mathematical difficulties are connected with such investigations. They dem<strong>and</strong>calculations on a large scale which can only be accomplished by means of mechanical aids thetime for whose creation is apparently not yet ripe.However, an attempt should have been made, <strong>and</strong> it had embraced the next period of mywork approximately covering the years 1930 – 1935 <strong>and</strong> thus partly overlapping the previousperiod. At that time, I had been working in various research institutions connected withmeteorology <strong>and</strong>, in general, with geophysics, although I had already begun such work whenbeing employed at the Central Statistical Directorate.I consider this period as a definitive loss in the following sense. I aimed at developing <strong>and</strong>checking methods of studying r<strong>and</strong>om empirical processes among geophysical phenomena.This problem dem<strong>and</strong>ed several years of work during which the tools for the investigation, soto say, could have been created <strong>and</strong> examined by issuing from concrete studies. It is naturalthat many of the necessary months-long preparatory attempts could not have been practicallyuseful by themselves. Underst<strong>and</strong>ably, in research institutes oriented towards practice thegeneral conditions for such work became unfavorable. The projects were often suppressed aftermuch work had been done but long before their conclusion. Only a small part of theaccomplishment during those years ripened for publication. I have no heart for grumbling sincethe great goal of industrializing our country should have affected scientific work by dem<strong>and</strong>ingconcrete findings necessary at once. However, I was apparently unable to show that myexpected results would be sufficiently important in a rather near future. The aim that Iformulated was thus postponed until some later years.The next period of my work coincides {began} with my entering the research collective ofthe Mathematical Institute at Moscow State University <strong>and</strong> then {<strong>and</strong> was continued}, whenmathematical research was reorganized, with my transfer to the Steklov Mathematical Instituteunder the Academy of Sciences of the Soviet Union. In the new surroundings, my plans, thatconsumed the previous years <strong>and</strong> were sketchily reported above, could have certainly met withfull underst<strong>and</strong>ing. However, their realization dem<strong>and</strong>ed means exceeding any practicalpossibilities. I had therefore moved to purely mathematical investigations of r<strong>and</strong>om processes[43; 44]; very soon, however, an absolutely new for me problem of compiling tables of
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of All Countries and to the Entire
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(Coll. Works), vol. 4. N.p., 1964,
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individuals of the third class, the
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From the theoretical point of view
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Second case: Each crossing can repr
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On the other hand, for four classes
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f i = i S + i , i = 1, 2, 3, 4, (
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f 1 = C 1 P(f 1 ; …; f n+1 ), C 1
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ut in this case f = 2 , f 1 = 2 ,
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I also note the essential differenc
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A 1 23n1 + 1 A 1 A 1 … A 11A 2 A
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coefficient of 2 in the right side
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h(A r h - c h A r 0 ) = - A r0we tr
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Notes1. Our formulas obviously pres
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Bernstein’s standpoint regarding
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Corollary 1.8. A true proposition c
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It is important to indicate that al
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ut for the simultaneous realization
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and, inversely,] = m ...1 2 N[ ch h
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µ 2 2 = m 2 2 - 2m 2 m 1 2 + m 1 4
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(x k - x k+1 ) … (x k - x +) = E(
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the thus obtained relations as pert
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[1/S(S - 1)(S - 2)][(Si = 1Sx i ) 3
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( N −1)((S − N )(2NS− 3S− 3
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µ 5 + 2µ 2 µ 3 = U [S/S] 5 + 2U
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case, the same property is true wit
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It follows that the question about
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then expressed my doubts). And Gned
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For Problem 1, formula (7) shows th
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Let us calculate now, by means of f
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ϕ′1(x)1E(a|x 1 ; x 2 ; …; x n
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Theorem 3. If the prior density 3
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P( ≤ ≤ |, 1 , 2 , …, s )
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6. A Sensible Choice of Confidence
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0 = A 0 n, = B2, = B2, 0 = C 0 n
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Note also that (95),(96), (83),(85)
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Γ(n / 2)Γ [( n −1) / 2]k = (1/2
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f (x 1 , x 2 , …, x n ) = 1 if x
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and the probability of achieving no
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E = kEµ. (14)In many particular ca
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a = np, b = np 2 = a 2 /n, = a/nand
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with number (2k - 2), we commit an
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(67)which is suitable even without
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" = 1/[1 - e - ], = - ln [1 - (1/
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Such structures are entirely approp
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11. As a result of its historical d
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exaggeration towards a total denial