the central limit theorem and monte carlo confidence intervals

Ngày tải lên: 05/08/2014, 09:46

... const.(l) In the following theorems, Theorem 3.4.3 is from [11], Theorem 3.4.4 is from [12], Theorem 3.4.5, Corollary 3.4.6 and Corollary 3.4.7 are from [3] Theorem 3.4.3 Suppose W is a random variable ... ), and the theorem is proved , 30 Chapter Main Results In this chapter, we use the Stein’s method to bound the convergence rate in the central limit theorem for the total magnetization in the ...   and the other entries (2.21) β2 q2 Now we give a limit theorem, which gives the law of large numbers and the breakdown for the empirical vector Ln It was also established in Ellis and Wang...

Ngày tải lên: 03/10/2015, 21:57

... CHAPTER VI.ToC Monte Carlo In the Concert-room The Gambling-saloons The Tables The moth and the candle The true story of Monte Carlo An International grievance and disgrace We reached Monte Carlo in ... form the staple of the weather about the festive time of Christmas and the opening of the new year, for the bright clear skies and sunny days of the south of France and Italy, is so pleasant, and ... shaft in the floor I returned, quietly warned my wife and others of the danger, and soon the master of the hotel and all the servants were on the spot In their excitement to subdue it, before the...

Ngày tải lên: 15/03/2014, 12:20

... approach and method of exchangeable pairs, and considered the case where the elements of A are independent random variables We denote the mean and variance of YA by µA and σA , and use the following ... which we have used above The proof will be presented in Appendix Lemma 2.2 Let x > −1 and < α < and the inductive hypothesis in the proof of Theorem 1.1 holds If n > 25000, and (1 + x)βA /n < 1/84, ... by 1(S) and the cardinal of S is denoted by |S| The following theorem will be proved in this note Theorem 1.1 We have sup |P (WA ≤ x) − Φ(x)| ≤ x∈R 90βA n (1.6) Let aij = aj for ≤ i ≤ m and aij...

Ngày tải lên: 16/10/2015, 09:13

... where > > max x The standard deviation of a random variable X that has the p.d.f or p.m.f f (the subscript is often dropped) The p.d.f for the standard normal The c.d.f for the standard normal ... that there are colons at the end of the first two statements These suppress the respective outputs and only the result for the mean will be seen Also, note that in the execution group below there ... amounts x = Z1 t and y = Z2 t along the x and y axes where Z1 and Z2 are independent standard normal random variables and and are specified positive constants The aim is to investigate the distribution...

Ngày tải lên: 07/03/2014, 15:20

... 2000, which will be used in this example How did Monte Carlo simulation get its name? Monte Carlo simulation was named for Monte Carlo, Monaco, where the primary attractions are casinos containing ... randomly selects a value from each of the distributions and enters it into the proper spreadsheet cell Excel then recalculates the spreadsheet, which displays a new scenario Depending upon the ... produces Crystal Ball® Monte Carlo risk analysis software During his tenure at the company, he has held positions as a Program Manager and as a trainer in Monte Carlo modeling and simulation Mr Goldman...

Ngày tải lên: 08/04/2014, 12:08

... Berkes and E Cs´ ki, “A universal result in almost sure central limit theory,” Stochastic Processes and a Their Applications, vol 94, no 1, pp 105–134, 2001 M Peligrad and S Utev, Central limit theorem ... and W Philipp, “A note on the almost sure central limit theorem, ” Statistics & Probability Letters, vol 9, no 3, pp 201–205, 1990 M Peligrad and Q M Shao, “A note on the almost sure central limit ... limit theorem for weakly dependent random variables,” Statistics & Probability Letters, vol 22, no 2, pp 131–136, 1995 G Khurelbaatar and G Rempala, “A note on the almost sure central limit theorem...

Ngày tải lên: 21/06/2014, 05:20

... everywhere central limit theorem, ” Mathematical Proceedings of the Cambridge Philosophical Society, vol 104, no 3, pp 561–574, 1988 P Schatte, “On strong versions of the central limit theorem, ” Mathematische ... Inequalities and Applications The following lemma is Theorem 2.1 and Corollary 2.1 in Li and Shao Lemma 3.2 (1) Let {ξn } and {ηn } be sequences of standard Gaussian variables with covariance 0 1 rij and ... obtain the desired result , i 1, 3.16 Journal of Inequalities and Applications Lemma 3.4 Let {Xn } be a sequence of standard Gaussian variables satisfying the conditions of Theorem 2.1, then for...

Ngày tải lên: 21/06/2014, 07:20

... satisfies the hypotheses of the Lefschetz-Hopf theorem Thus, the homotopy and additivity properties of the fixed-point index imply that the normalization property follows from the Lefschetz-Hopf theorem ... the characterization of the reduced Lefschetz number in Theorem 1.1 to prove the Lefschetz-Hopf theorem in a quite natural manner by showing that the fixed-point index satisfies the axioms of Theorem ... prove the following theorem Theorem 1.3 (normalization property) If f : X → X is any map of a finite polyhedron, then L( f ) = i(X, f ,X), the fixed-point index of f on all of X The Lefschetz-Hopf theorem...

Ngày tải lên: 23/06/2014, 00:20

... (22) by the same argument as in the renewal theory Applying Theorem 2.4 for the imbedded chain {(Zn , ρn ), n ≥ 1} we obtain the following theorem Theorem 3.2 (Central limit theorem for the imbedded ... Now we state and prove the main theorem as follows Central Limit Theorem for Functional of Jump Markov Processes 459 Theorem 3.3 Assume that the condition (C1 ) of Theorem 3.2 and the following ... section is to establish the conditions for n−1/2 n L ϕ(Xk ) −→ N (μ, σ ) k=1 We need a central limit theorem for martingale differences as follows Theorem 2.3 (Central limit theorem for martingale...

Ngày tải lên: 06/08/2014, 05:20

... of Theorems 1.1, 1.2, and 1.3 In this section, we give combinatorial proofs of the q-binomial theorem and Theorems 1.2 and 1.3 Our combinatorial proof of the q-binomial theorem is based on Theorem ... The following theorem is crucial to prove Theorems 1.1–1.3 combinatorially Theorem 2.1 Given m ≥ 1, the number of partitions of n into at most m parts with no repeated odd parts is equal to the ... 6, 5, 3) Clearly, the number of rows in the diagram of λ is (λ), while the number of columns is λ1 /2 Thus, σ has the required properties and Theorem 2.1 is proved Note that the above involution...

Ngày tải lên: 07/08/2014, 07:21

... identities The proofs are “hybrid” in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) ... n=0 the electronic journal of combinatorics 18 (2011), #P60 (4.3) and (4.2) will then follow from the Identity Theorem, by an argument similar to that used in the proof of the q-Binomial Theorem ... s+r+k ; q k )∞ , (q r+k ; q k )∞ and the result now follows We now give a proof of the q-Binomial Theorem Theorem Let a, z and q be complex numbers with |z|, |q| < Then ∞ n=0 (az; q)∞ (a; q)n z...

Ngày tải lên: 08/08/2014, 14:22

... to the motor The wheel contained holes at various distances from the axis of rotation, which thereby determined the magnitude of the range of the sinusoid motion of the phantom, which is the ... (Case 3), the central point dose was equivalent for both the phantom measurement and 4D dose calculation due to the fact that the central point was well covered by the treatment beams, given the relatively ... the DVH curve for the average dose does not differ much from the static DVH curves (Figure 3) In the Monte Carlo simulations, the statistical uncertainties in the high dose regions, such as the...

Ngày tải lên: 09/08/2014, 08:23

... where > > max x The standard deviation of a random variable X that has the p.d.f or p.m.f f (the subscript is often dropped) The p.d.f for the standard normal The c.d.f for the standard normal ... that there are colons at the end of the first two statements These suppress the respective outputs and only the result for the mean will be seen Also, note that in the execution group below there ... amounts x = Z1 t and y = Z2 t along the x and y axes where Z1 and Z2 are independent standard normal random variables and and are specified positive constants The aim is to investigate the distribution...

Ngày tải lên: 09/08/2014, 16:21

... random number The seed is set using the command ‘randomize(integer)’ before invoking ‘rand()’ Maple also provides another U generator This is ‘stats[random,uniform] (1)’ This is based upon ‘rand’ ... completes the proof This result allows the investigation to be confined to the theoretical properties of type A and B generators only Theoretical tests use the values a c, and m to assess the quality ... the lengths of the smaller and longer sides respectively The larger r2 = l2 /l1 is, the poorer the uniformity of pairs and the poorer the generator This idea can be extended to find the degree of...

Ngày tải lên: 09/08/2014, 16:21

... volatility, and 76 Generation of variates from standard distributions (a) Use the standard results for the mean and variance of a lognormally distributed random variable to show that the mean and standard ... the right-hand side of the inequality in (4.1) by W Suppose that 1−R2 > W Then − ln R2 > W and the prospective variate is accepted Suppose − ln R2 ≯ W but 1/R2 − ≤ W Then − ln R2 ≤ W and the ... variates, and conditional Monte Carlo Several books contain a chapter or more on variance reduction and the reader may find the following useful: Gentle (2003), Hammersley and Handscomb (1964), Law and...

Ngày tải lên: 09/08/2014, 16:21

... i=1 The right-hand side of Equation (5.24) contains the variance removed due to use of the proportional ni rather than the naive estimator, the variance removed due to use of the optimal ni rather ... combined estimator from the primary and antithetic realizations, estimate the expectation of the waiting time of the five customers and its estimated standard error Estimate the variance reduction ... covering the cube If h is again the subinterval length along any of the d axes, then mhd The resulting error is O h2 = O 1/m2/d However, using Monte Carlo, the √ error is still O 1/ m Therefore,...

Ngày tải lên: 09/08/2014, 16:21

... exists if and only if −1 = 1− The < < and then L = / − and the expected cycle length, results are summarized in Table 7.1, where the superscripts p and a refer to the primary and antithetic realizations ... /6 By taking the limit as n → , show that the price for the continuously sampled geometric average is the same as the price of a European √ call where the volatility is / and where the asset earns ... = and is the empty set Let N C and N E denote the number of randomly occuring points in C and E respectively Suppose there exists a positive constant , such that, for all such C and E N C and...

Ngày tải lên: 09/08/2014, 16:21

... of the hyperparameters, , and , is based upon the belief that the prior marginal expectation and standard deviation of any 170 Markov chain Monte Carlo prior and posterior densities prior and ... Section 8.4.2 and Appendix 8.4 so that the prior density of p is beta Run the modified procedure to estimate the posterior expectation of N and p when = and = and the other data and parameters ... separable in s and t and therefore these two random variables are not independent Accordingly, a simulation should use either the cosine form or the sine form but not both (Recall that in the standard...

Ngày tải lên: 09/08/2014, 16:21

... along the x and y axes where Z1 and Z2 are independent standard normal random variables and 1 and 2 are specified positive constants The aim is to investigate the distribution of time until the ... necessary to store (i) the time at which it next deposits a patient at the hospital and (ii) the time at which the patient incurred the emergency Suppose these are called TD j and TA j respectively ... low, being of the order of 2.5% This is because the posterior is so different from the prior which provides a very poor envelope On i the other hand, inspection of the plots for the sequence say,...

Ngày tải lên: 09/08/2014, 16:21