kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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If it is close to unity (for example, if it is 0.99 or 0.999), we will be inclined to assume,without considerable hesitation, that ≤ ≤ . Consequently, when the conditionalprobabilities (54) are known for any and , it is natural to assume some probability "sufficiently close to unity and to choose values of (x 1 ; x 2 ; …; x n ) and (x 1 ; x 2 ; …; x n ) foreach system (1) such thatP( ≤ ≤ |x 1 ; x 2 ; …; x n ) = "; (55)and, in addition, that, under this condition, the length of the interval [; ] will be the leastpossible.For example, in Problem 1, assuming that formula (17) is correct, the shortest confidenceinterval for a obeying restriction (55) is given by formulas (41a; 41b). Note, however,concerning this example, that formula (17) may only be justified (even as an approximation)under rather restrictive assumptions specified in §4. As to the strict expression (7) for theconditional probability 1 (a|x 1 ; x 2 ; …; x n ) , it includes the prior density 1 (a) which is usuallyunknown.The same situation exists in most of the other problems of estimating parameters. Thestrict expression of the conditional probabilities (54) usually includes an unknowndistribution of the parameters.There exists an opinion, upheld in the Soviet Union by Bernstein (see, for example, [5]),that in cases in which the prior distribution of the parameters is unknown, the theory ofprobability cannot offer the practitioner anything excepting limit theorems similar to thoseindicated in §4. According to this point of view, if the prior distribution of the parameters isunknown, and given a restricted number of observations, an objective scientific approach tothe most sensible choice of confidence limits for the estimated parameters is simplyimpossible.Here, it is certainly true that the conditional distribution of the parameters, given theresults of the observations, depends on the prior distribution of the same parameters, and wecannot disregard it. But the opinion, that the indication of sensible confidence limits for theestimated parameters is inseparably linked with considering conditional probabilities (54), iswrong.In most practical (in particular, artillery) problems the matter concerns the establishmentof general rules for estimating parameters to be recommended for systematic application tosome vast category of cases. In this section devoted to confidence intervals, we are concernedwith rules such as:Under certain general conditions it is recommended to consider, whatever be theobservational results (1), that the value of parameter is situated within the boundaries(x 1 ; x 2 ; …; x n ) and (x 1 ; x 2 ; …; x n ). When recommending such a rule for futuremathematical application without knowing the values (1) in each separate case, there is noreason to consider the conditional probabilities (54). Instead, it is natural to turn to theunconditional probabilityP[(x 1 ; x 2 ; …; x n ) ≤ ≤ (x 1 ; x 2 ; …; x n )] (56)that no error will occur when applying the rule.Given the type of the functions (x 1 ; x 2 ; …; x n ) and (x 1 ; x 2 ; …; x n ), the unconditionalprobability (56) is generally determined by the distribution of the magnitudes (1) whichdepends on the parameters , 1 , 2 , …, s and by the unconditional (prior) distribution ofthese parameters. Denoting the conditional probability of obeying the inequalities ≤ ≤ when the values of the parameters are given by
P( ≤ ≤ |, 1 , 2 , …, s ) (57)and assuming that the prior distribution of the parameters has density (, 1 , 2 , …, s ), wewill obtain the following expression for the unconditional probability (56):P( ≤ ≤ ) = … P( ≤ ≤ |, 1, …, s )(, 1 , …, s )dd 1 …d s . (58)A particular case of such rules, when the conditional probability (57) remains constant atall possible values of , 1 , …, s , is especially important for practice. If this conditionalprobability is constant and equals ", then, on the strength of (59),P( ≤ ≤ ) = … "(, 1, …, s )d d 1 …d s = ".This means that the unconditional probability (56) does not depend on the unconditionaldistribution of the parameters 10 .We have already indicated in §1 that the very hypothesis on the existence of a priordistribution of the parameters is not always sensible. However, if the conditional probability(57) does not depend on the values of the parameters and is invariably equal to one and thesame number " then it is natural to consider that the unconditional probability (56) exists andis equal to " even in those cases in which the hypothesis on the existence of a priordistribution of the parameters is not admitted 11 .If the conditional probability (57) is equal to " for all the possible values of the parameters(so that, consequently, the same is true with respect to the unconditional probability (56) forany form of the prior distribution of the parameters), we shall say, following Fisher [1; 2],that our rule has a certain confidence probability equal to ". It is easy to see that for Problem1 the rule that recommends to assume that a is situated within the boundariesa ≤ a ≤ a (59)wherea = x + c/hn, a = x + c/hn (60)has a certain confidence probabilityc" = (1/) ′′c′exp (– 2 ) d. (61)Indeed, for any a and h,P(a ≤ a ≤ a|a; h) = P( x + c/hn ≤ a ≤ x + c/hn|a; h) =−−c′′cP(a – c/hn ≤ x ≤ a – c/hn|a; h) = (1/) ′c(1/) ′′c′exp (– 2 ) d.exp (– 2 ) d =For example, if c = – 2 and c = 2,
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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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devoid of quadratic divisors and re
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propositions (B i and C j ) can be
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A ~ A 1 and B = B 1 , we will have
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included in a given totality as equ
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For unconnected totalities we would
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proposition given that a second one
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On the other hand, let x be a parti
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totality is perfect, but that the j
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In this case, all the finite or inf
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probabilities p 1 , p 2 , … respe
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where x is determined by the inequa
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totality of the second type (§3.1.
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x = /2 + /(23) + … + /(23… p n
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that the fall of a given die on any
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infinitely many digits only dependi
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10. (§2.1.5). Such two proposition
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F(x + h) - F(x) = Mh, therefore F(x
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“confidence” probability is bas
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x1+ Lp n (x) x1− Lx1+ Lf(t)dt < x
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|(x 1 ; t 0 ; t 1 ) - 1 t0tf(t)dt|
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5. The distribution ofξ , the arit
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P(x 1i < x) = F(x; a i ) = C(a i )
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egards his promises. Markov shows t
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other solely and equally possible i
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notion of probability and of its re
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However, already in the beginning o
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the revolution. My main findings we
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Nevertheless, Slutsky is not suffic
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path that would completely answer h
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on political economy as well as wit
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scientific merit. Borel was indeed
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[3] Already in Kiev Slutsky had bee
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different foundation. The difficult
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5. On the criterion of goodness of
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--- (1999, in Russian), Slutsky: co
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Here also, the author considers the
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second, it is not based on assumpti
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experimentation and connected with
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Russian, and especially of the Sovi
Page 119 and 120: station in England. This book, as h
Page 121 and 122: Uspekhi Matematich. Nauk, vol. 10,
Page 123 and 124: variety and detachment of those lat
Page 125 and 126: 46. On the distribution of the regr
Page 127 and 128: 119. On the Markov method of establ
Page 129 and 130: No lesser difficulties than those e
Page 131 and 132: Separate spheres of work considerab
Page 133 and 134: 10. Anderson, O. Letters to Karl Pe
Page 135 and 136: Hier sind, im Allgemeinen, ganz ana
Page 137 and 138: Jedenfalls, glaube ich erwiesen zu
Page 139 and 140: werde ich das ganze Material in kur
Page 141 and 142: considered as the limiting case of
Page 143 and 144: and, inversely,] = m ...1 2 N[ ch h
Page 145 and 146: µ 2 2 = m 2 2 - 2m 2 m 1 2 + m 1 4
Page 147 and 148: (x k - x k+1 ) … (x k - x +) = E(
Page 149 and 150: the thus obtained relations as pert
Page 151 and 152: [1/S(S - 1)(S - 2)][(Si = 1Sx i ) 3
Page 153 and 154: ( N −1)((S − N )(2NS− 3S− 3
Page 155 and 156: µ 5 + 2µ 2 µ 3 = U [S/S] 5 + 2U
Page 157 and 158: case, the same property is true wit
Page 159 and 160: It follows that the question about
Page 161 and 162: then expressed my doubts). And Gned
Page 163 and 164: For Problem 1, formula (7) shows th
Page 165 and 166: Let us calculate now, by means of f
Page 167 and 168: ϕ′1(x)1E(a|x 1 ; x 2 ; …; x n
Page 169: Theorem 3. If the prior density 3
Page 173 and 174: 6. A Sensible Choice of Confidence
Page 175 and 176: 0 = A 0 n, = B2, = B2, 0 = C 0 n
Page 177 and 178: Note also that (95),(96), (83),(85)
Page 179 and 180: Γ(n / 2)Γ [( n −1) / 2]k = (1/2
Page 181 and 182: f (x 1 , x 2 , …, x n ) = 1 if x
Page 183 and 184: and the probability of achieving no
Page 185 and 186: E = kEµ. (14)In many particular ca
Page 187 and 188: a = np, b = np 2 = a 2 /n, = a/nand
Page 189 and 190: with number (2k - 2), we commit an
Page 191 and 192: (67)which is suitable even without
Page 193 and 194: " = 1/[1 - e - ], = - ln [1 - (1/
Page 195 and 196: Such structures are entirely approp
Page 197 and 198: 11. As a result of its historical d
Page 199 and 200: exaggeration towards a total denial
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kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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