R. R. Bahadur, Some approximations to the binomial distribution function, The Annals of Mathematical Statistics, 31:43--54 (1960).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This completes the proof. Remarks. 1. From (2) and the well known continued fraction representation of the incomplete gamma function (cf. 16, p. 136] we have 1 1 1 . Useful estimates for #m (#) may be derived from this representation as in [46, pp. 53 56] and [3] for binomial distribution. 2. The polynomials # j (m) are related to Laguerre polynomials L (#) n (x) z ] 1 z) # 1 exp ( xz (1 z) by # j (m) L (m j) m) and thus satisfy the recurrences (j 1)# j 1 (m) j# j (m) m# j 1 (m) j 1) # 0 (m) 1, # 1 (m) 0. 13) ....

R. R. Bahadur, Some approximations to the binomial distribution function, The Annals of Mathematical Statistics, 31:43--54 (1960).

....This completes the proof. Remarks. 1. From (2) and the well known continued fraction representation of the incomplete gamma function (cf. 16, p. 136] we have 1 1 1 . Useful estimates for Pi m ( may be derived from this representation as in [46, pp. 53 56] and [3] for binomial distribution. 2. The polynomials j (m) are related to Laguerre polynomials L (ff) n (x) z ] 1 Gamma z) Gammaff Gamma1 exp ( Gammaxz= 1 Gamma z) by j (m) L (m Gammaj) j (m) and thus satisfy the recurrences (j 1) j 1 (m) Gammaj j (m) Gamma m j ....

R. R. Bahadur, Some approximations to the binomial distribution function, The Annals of Mathematical Statistics, 31:43--54 (1960).

....Indeed, for small deviations, approximations by the Gaussian distribution (or possibly the Poisson distribution, depending on np) give sharp results, and error estimates have been investigated (c.f. Pr 53] Mo 70] For large deviations, asymptotic expansions have been studied extensively (c.f. [Ba 60], Ne 83] Furthermore, the errors resulting from general Chernoff Hoeffding estimates have also been a matter of study, and asymptotic expansions have been attained. Among the more general results is the fact that when a Chernoff Hoeffding bound for the sum of n independent identically ....

R. R. Bahadur. Some approximations to the binomial distribution function. Ann. Math. Stat., 31(1960), 43--54.

....deepest gate fed by both x i and x j . We say two variables are siblings if they share a common parent (perceptron) and not siblings otherwise. Probabilistic Concepts A probabilistic concept (or p concept) as defined by Kearns and Schapire [10] and Yamanishi [14] is a mapping c : I n # [0, 1]. For each x # I n , c(x)is interpreted as the probability that x is a positive example of the p concept c. Thus, in the p concept model, a labeled example is generated as follows: first, an instance x is chosen according to the target distribution on I n ; then, with probability c(x) the ....

....an OR gate. Then, if P (g =1) P(g=0) #and Q s P (f (s) a (s) #, Inf(x i ) ## n 2 if w i =1 ## n 2 if w i = 1 Proof idea: First note that: Inf(x i ) Y s P(f (s) a (s) P (g(x) 1 x i =1) P(g(x) 1 x i =0) The lemma then follows from Bahadur s expansion [1].# Note that we can assume w.l.o.g. that #, # # 2n, for otherwise we can neglect perceptron g without introducing much error. Under this assumption, the gap is wide enough to be estimated e#ciently using a sample of polynomial size. Once we determine the weight values, we reduce f to a monotone ....

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Bahadur R.,"Some Approximations to the Binomial Distribution Function", Annals Math. Stat., Vol.31, (1960), 43--54.

....lemmas. Lemma 1 Let d and n be two integers. Then, if n 2 # d # n, n X i=d n i # 1 2 n d 1 1 z where z = 1 2 n 1 d 1 . And if 0 # d # n 2 , d X i=0 n i # 1 2 n d 1 1 z # where z # = 1 2 n 1 n d 1 . Proof: follows directly from Bahadur s expansion (Bahadur, 1960).# This lemma is used throughout this paper to approximate binomial distributions. Lemma 2 Let S be the number of successes in m Bernoulli trials each with probability of success p (and probability of failure q =1 p) For any # # 0, Pr S m p # # # 2 exp # 2 m 4pq ....

Bahadur R. (1960). Some Approximations to the Binomial Distribution Function. Annals Math. Stat., Vol. 31, p. 43.

....1, page 304] on the rate of convergence of the central limit theorem. This gives a lower bound on the left hand side of (1) and (2) that is worse by approximately a factor of 2 than the bound 1=19 proven in Fact 3.2. More specialized results on the tails of the Binomial distribution were proven by [1], 2] and [12] We derive (1) and (2) by direct manipulation of the bound in [1] which is in a form suitable for our purposes. Fact 3.3 For every 0 fi ff 1, for every random variable S 2 [0; N ] with ES = ffN , it holds that PrfS fiNg (ff Gamma fi) 1 Gamma fi) Proof. It follows by ....

....a lower bound on the left hand side of (1) and (2) that is worse by approximately a factor of 2 than the bound 1=19 proven in Fact 3.2. More specialized results on the tails of the Binomial distribution were proven by [1] 2] and [12] We derive (1) and (2) by direct manipulation of the bound in [1], which is in a form suitable for our purposes. Fact 3.3 For every 0 fi ff 1, for every random variable S 2 [0; N ] with ES = ffN , it holds that PrfS fiNg (ff Gamma fi) 1 Gamma fi) Proof. It follows by setting z = PrfS fiNg and solving the following for z: ffN = ES = E[S j S ....

R.R. Bahadur. Some approximations to the binomial distribution function. Annals of Mathematical Statistics, 31:43--54, 1960.

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R.R. Bahadur. Some approximations to the binomial distribution function. Annals of Mathematical Statistics, 31:43--54, 1960.

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Bahadur R. (1960) "Some Approximations to the Binomial Distribution Function ", Annals Math. Stat., Vol.31, 43--54.

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