Optimal Statistical Inference In Financial Engineering 1st Edition Masanobu Taniguchi, Junichi Hirukawa, Kenichiro Tamaki - Solutions

Discover comprehensive solutions and expert insights for "Optimal Statistical Inference in Financial Engineering" by Masanobu Taniguchi and colleagues. Our online platform offers answers key and detailed solutions PDF for each chapter, providing step-by-step answers to solve complex problems. Access the solution manual and instructor manual for in-depth understanding, along with a test bank to test your knowledge. Whether you need answers to textbook questions or chapter solutions, our resources are designed to guide you through solved problems with ease. Enjoy the convenience of free download for quick and efficient learning.

=+contain exactly j observations from the X sample, j ¼ 0; 1; ... ; m.ðaÞ Verify that S0 þ S1 þ þ Sm ¼ n þ 1 and S1 þ 2S2 þ þ mSm ¼ m.ðbÞ If FX ¼ FY, show that the joint distribution of S0; S1; ... ;Sm is given byðn þ 1Þ!s0!s1! sm!m þ n n1
=+available from a continuous cdf FY. Let Sj be the random variable representing the number of Y blocks I1; I2; ... ; Inþ1 (defined in the section on two-sample coverages) that
=+2.27. Suppose that a random sample of size m, X1; X2; ... ; Xm, is available from a continuous cdf FX and a second independent random sample of size n, Y1;Y2; ... ;Yn, is
=+Find the mean and variance of U.
=+2.26. Let the random variable U denote the proportion of the population lying between the two extreme values of a sample of n from some unspecified continuous population.
=+2 ; V3 3 ; ... ; Vn n are independent and identically distributed with the uniform distribution over (0,1).
=+2.22. Let Xð1Þ < Xð2Þ < < XðnÞ denote the order statistics of a sample from a continuous unspecified distribution FX . Define the n random variables Vi ¼ FX ðXðiÞÞFX ðXðiþ1ÞÞ for 14i4n 1 and Vn ¼ FX ðXðnÞÞðaÞ Find the marginal distribution of Vr; 14r4n.ðbÞ Find the
=+ORDER STATISTICS, QUANTILES, AND COVERAGES 71ðcÞ Show that EðXðiÞÞ ¼ Pi j¼1 1=ðn þ 1 jÞ.ðdÞ Interpret the significance of these results if the sample arose from a life test on n light bulbs with exponential lifetimes.
=+2.21. Let Xð1Þ < Xð2Þ < < XðnÞ be order statistics for a random sample from the exponential distribution FX ðxÞ ¼ expðxÞ for x 50.ðaÞ Show that XðrÞ and XðsÞ XðrÞ are independent for any s > r.ðbÞ Find the distribution of Xðrþ1Þ XðrÞ.
=+ðbÞ Verify the probability density function of XðrÞ given in (6.4) by differentiation of the result in (a).ðcÞ By considering PðXðrÞ > t=nÞ in the form of (a), find the asymptotic distribution of XðrÞ for r fixed and n ! 1 if FX ðxÞ is the uniform distribution over(0,1).
=+2.20. Let XðrÞ be the rth-order statistic of a random sample of size n from a continuous distribution FX .ðaÞ Show that PðXðrÞ 4tÞ ¼ Xn k¼r nk½FX ðtÞk½1 FX ðtÞnk.
=+2.19. Let XðnÞ be the largest value in a sample of size n from the population fx.ðaÞ Show that limn!1 Pðn1XðnÞ 4xÞ ¼ expða=pxÞ if fX ðxÞ ¼ a=½pða2 þ x2Þ(Cauchy).ðbÞ Show that limn!1 Pðn2XðnÞ 4xÞ ¼ expða ffiffiffiffiffiffiffiffiffiffi 2=px p Þ if fX ðxÞ ¼ða=
=+(b) The fifth-order statistic of a random sample of size 19 from the exponential distribution fX ðxÞ ¼ expðxÞ for x50.
=+2.18. Use (9.5) and (9.6) to approximate the mean and variance of:(a) The median of a sample of size 2m þ 1 from a normal distribution with mean m and variance a2.
=+2.17. By making the transformation U ¼ nFX ðXð1ÞÞ; V ¼ n½1 FX ðXðnÞÞ in (6.8) with r ¼ 1, s ¼ n, for any continuous FX , show that U and V are independent random variables in the limiting case as n ! 1, so that the two extreme values of a random sample are asymptotically
=+2.16. Give an expression similar to (7.3) for the probability density function of the midrange for any continuous distribution and use it to find the density function in the case of a uniform population over (0,1).
=+2.15. Find the distribution of the range of a random sample of size n from the exponential distribution fX ðxÞ ¼ 4expð4xÞ for x 50.
=+2.14. Find the probability that the range of a random sample of size n from the population fX ðxÞ ¼ 2e2x for x 50 does not exceed 4:
=+2.13. Find the mean and variance of the median of a random sample of n from the uniform distribution over (0,1):(a) When n is odd(b) When n is even and U is defined as in Section 2.7.
=+2.12. Verify the result given in (7.1) for the distribution of the median of a sample of size 2m from the uniform distribution over (0,1) when m ¼ 2. Show that this distribution is symmetric about 0.5 by writing (7.1) in the form fU ðuÞ ¼ 8ð0:5 ju 0:5jÞ2ð1 þ 4ju 0:5jÞ 0 < u < 1
=+2.11. Find the expected value of the largest order statistic in a random sample of size 3 from:(a) The exponential distribution fX ðxÞ ¼ expðxÞfor x50(b) The standard normal distribution
=+(b) Find the number c such that the probability is 0.5 that the point nearest zero will exceed c.70 CHAPTER 2
=+2.10. Ten points are chosen randomly and independently on the interval (0,1).(a) Find the probability that the point nearest 1 exceeds 0.90.
=+2.9. For the n-order statistics of a sample from the uniform distribution over ð0; y), show that the interval ðXðnÞ; XðnÞ=a1=nÞ is a 100 ð1 aÞ percent confidence-interval estimate of the parameter y.
=+functions of Xð1Þ. If the upper limit is Xð1Þ, find the corresponding lower limit gðXð1ÞÞ such that the confidence coefficient is 0.95.
=+2.8. A random sample of size 3 is drawn from the population fX ðxÞ ¼ exp½ðx yÞ for x > 0. We wish to find a 95 percent confidence-interval estimate for the parameter y. Since the maximum-likelihood estimate for y is Xð1Þ, the smallest order statistic, a logical choice for the limits of
=+Suppose a random sample of size 5 is taken with replacement from the discrete distribution fX ðxÞ ¼ 1=6 for x ¼ 1; 2; ... ; 6. Find the probability mass function of Xð1Þ, the smallest order statistic.
=+2.7. The order statistics for a random sample of size n from a discrete distribution are defined as in the continuous case except that now we have Xð1Þ 4Xð2Þ 4 4XðnÞ.
=+2.6. If X is a continuous random variable with probability density function fX ðxÞ ¼ 2ð1 xÞ; 0 < x < 1, find the transformation Y ¼ gðXÞ such that the random variable Y has the uniform distribution over (0,2).
=+2.5. Prove the probability-integral transformation (Theorem 5.1) by finding the moment-generating function of the random variable Y ¼ FX ðXÞ, where X has the continuous cumulative distribution FX and a moment-generating function that exists.
=+(e) The Cauchy distribution defined by the probability density function fðxÞ ¼ b p½b2 þ ðx aÞ2 1 < x < 1; 1 < a < 1; 0 < b < 1
=+(d) The double exponential distribution defined by the probability density function fðxÞ ¼ 1 2b eðjxajÞ=b 1 < x < 1; 1 < a < 1; 0 < b < 1
=+(c) The logistic distribution defined by the probability density function fðxÞ ¼ eðxaÞ=b b½1 þ eðxaÞ=b 2 1 < x < 1; 1 < a < 1; 0 < b < 1
=+(b) Beta distribution with a ¼ 2 and b ¼ 1. The probability density function is given by fðxÞ ¼ 2x 0 < x < 1
=+2.4. Find the transformation to obtain, from an observation U following a uniformð0; 1Þ distribution, an observation from each of the following continuous probability distributions:(a) Exponential distribution with mean 1.
=+for any 0 4p41. The integral on the right is called an incomplete beta integral and written as Ipða; n a þ 1Þ. Thus, if X is a binomial random variable with parameters n and p, the probability that X is less than or equal to a ða ¼ 0; 1; ... ; nÞ is 1 Ipða þ 1; n aÞ ¼ I1pðn a;
=+2.3. Show that Xn x¼a nx pxð1 pÞnx ¼ 1 Bða; n a þ 1ÞZ p 0ya1ð1 yÞna dy
=+2.2. Let X be a discrete random variable taking on only positive integer values. Show that EðXÞ ¼ X1 i¼1 PðX 5iÞ
=+2.1. Let X be a nonnegative continuous random variable with cdf FX . Show that EðXÞ ¼ Z 1 0½1 FX ðxÞ dx
9.2 Show that (9.40) diverges, as ϵ → 0.
9.1 Show that (9.39) converges to zero, as ϵ → 0.
8.5 If ϕK(u) = exp©1 2u′uªin (8.55), then give a no-arbitrage condition for Yn,N given by (8.53) in terms of α( , ), β( , ) and ξ (Kariya and Liu(2003)).
8.4 Prove the equation (8.53).
8.3 Check the formulas (8.42) and (8.44).
8.2 Under (8.32) and (8.33), derive the recursive formula of PN−k,N , k =1, 2, · · · ,N − 1, for the model (8.28).
8.1 Let X be an integrable random variable on (Ω,A, P), and let {An} be an increasing sequence of sub σ-fields of A, and Xn = E(X|An). Then show that {Xn,An} is a martingale.
7.6 Verify the relation (7.77).
7.5 (i) Show that the spectral density (7.66) does not satisfy (7.65), if p1 < p2.(ii) Show that the spectral density (7.67) does not satisfy (7.65).
7.4 Verify that portfolio weights (7.54), (7.55), (7.57) and (7.58) are the solutions of the corresponding criteria of utility.
7.3 From (7.29), derive the Black-Scholes formula (7.30).
7.2 Let X be a random variable on (Ω,A, P), and let A1 and A2 be subσ-fields of A satisfying A1 ⊂ A2. Then, show that E{E(X|A2|A1} = E{X|A1} a.e.
7.1 Let {Xt,At} be a martingale. Then, show that E{Xm|Ak} = Xk a.e., for any m > k.
6.15 Let Xn = (X1, . . . ,Xn)′ be generated from the AR(1) process Xt = θXt−1 + ut, (|θ| < 1), (6.538)where {ut} ∼ i.i.d. N(0, 1). Then, the spectral density function of {Xt}is fθ(λ) = (1/2π)|1 − θeiλ|−2. Here we consider the discriminant problem described by the following
6.14 Confirm assertions (6.236) and (6.237).
6.13 Let the regressor function of Theorem 6.14 have trend (i) or (ii) in Example 6.20. Then, show that ˆ βLS is asymptotically efficient.
6.12 Verify conclusions of (i) and (ii) in Example 6.20.
6.11 Let {Xt} be a stationary process with mean zero and spectral density function g(λ) = (1/2π)|1 − 0.3 eiλ|2. Compute the prediction error PE(g, f)((6.199)) of predictor (6.198) constructed on the basis of incorrectly conjectured spectral density f(λ) =1 2π¯¯1−(0.3 +θ)eiλ +0.3
6.10 Let {Xt} be a Gaussian stationary process satisfying the assumption in Theorem 6.10. Then, show that and E[log {In(A)}]E {log f(A)}, (n ) E [ log {In (\)} d\+> [_*_ 1 (In(\)} log f(A)d, (n ). - -
6.9 Calculate the values of q, κq and R 1−1 w(x)2dx for the Akaike window of Example 6.18.
6.8 Show that the spectral density estimator ˆ fn(λ) given in (6.161) can be written as in the form (6.165).
6.7 Show that the equation (6.151) holds.
6.6 Show that AIC for AR(p) model (6.121) is given by AIC(p) = n log ˆσ2 QGML(p) + 2p. (6.534)
6.5 Verify (6.54).
6.4 Verify the assertion (ii) of Theorem 6.3.
6.3 Show that the matrix Γp defined in (6.44) is positive definite.
6.2. Show that the spectral density function matrix of {Xt} is given by(6.13).6.2 For an autoregressive model (6.33) with p = 1, give the log-likelihood ln(θ)of (6.37) in an explicit form and simplify the likelihood equationas simple as possible. l,(0) = 0, al(0) 2 =0
6.1 Let the VARMA(p, q) process {Xt} defined in (6.12) satisfy Assumption
5.7. Let {Xn,t,An,t} be a martingale difference sequence. Suppose the Lindeberg condition:Xn t=1 E{X2 n,tχ(|Xn,t| > ε)} → 0, (n → ∞), for every ε > 0.Then, show the following (i) and (ii):(i) max 1≤t≤n|Xn,t| p→0,(ii) there exists M > 0 satisfying Xnt M. E max Xntl 1
5.6. Verify Theorem 5.6.
5.5. Show that the following relation holds:uniform mixing ⇒ strong mixing ⇒ mixing.
5.4. Verify that the m-vector general linear process (5.41) has the spectral density matrix (5.42).
5.3. Let L2 ≡ {Y : E(|Y |2) < ∞}. For Yn,Wn ∈ L2, define < Yn,Wn >≡E(YnWn), and write Y = l.i.m.n→∞Yn and W = l.i.m.n→∞Wn. Then, show that< Y,W >= lim n→∞ < Yn,Wn >, (continuity of inner product).
5.2. Verify (5.6) and (5.7).
5.1. Let {Xt : t ∈ Z} be a strictly stationary process. Then, show that the joint distribution function of Xt1 , . . . ,Xtn (t1, . . . , tn ∈ Z) depends only on t2 − t1, . . . , tn − tn−1.
4.9 Verify (4.108).
4.8 Show that S P→Σ, as n1, n2 → ∞ in (4.92).
4.7 Prove that (4.83) holds.
4.6 Verify that the optimal R = {R1,R2} in Example 4.13 is given by (4.77)and (4.78).
4.5 Show that the density function of F-distribution with (m, n) degrees of freedom is given by f(x)= = (mn) mm/2/2 ()() 2m/2-1 (mx + n)(m+n)/2' (x > 0).
4.4 When we tossed a coin twenty times, the following result was obtained, where 0 and 1 represent tails and heads, respectively.1 0 1 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 Then using the test in Example 4.6, test whether the coin is fair or unfair with a significance level of 0.05.
4.3 The following data 0.53, 1.51, 1.55, 2.57, 1.16, 0.66, 1.61, 0.74, 0.59, 2.01 are from a sample of size 10, which are supposed to be independent and identically distributed as N(μ, σ2). Give a confidence interval of μ with confidence coefficient 0.95.
4.2 In (4.8) (i) Derive the joint probability density function of (Y,Z). (ii) Consider the transformation (Y,Z) → (T,Z) and derive the joint probability density function of (T,Z) (using the change of variable formula (Theorem A.6)). (iii) Show that the probability density function of T (the
4.1 Let Z ∼ χ2(n). Show that Z has the probability density function f(2) = 2*r (2)e, z>0.
Prove the inequality (3.73).
Let f(x) and g(x) be probability density functions. Then show that the following inequality holds ZR f(x) log½f(x)g(x)¾dx ≥ 0, and that the equality holds if and only if f(x) = g(x) a.e.
If {Xj} ∼ i.i.d. U(0, θ) (uniform distribution), then find the MLE of θ.
If {Xj} ∼ i.i.d. Po(λ), then find the MLE of λ.
If {Xj} ∼ i.i.d. Exp(1/θ, 0) (exponential distribution), then find the UMVU estimator of θ.
If {Xj} ∼ i.i.d. Po(λ) (Poisson distribution), then find the UMVU estimator of λ.
Let {Xj} ∼ i.i.d. N(μ, σ). Show that³Pn j=1 Xj , Pn j=1 X2 j´is a sufficient and complete statistic for θ = (μ, σ).
Verify (3.50).
Find an orthogonal matrix T whose first row is t1 =¡n−1/2, . . . , n−1/2¢.
(i) Verify (3.44) and (3.46).(ii) Show (3.57).
Verify the equivalent relation (3.37)-(3.39).
Minimize Eθ {T(X) − a(θ)(∂/∂θ) log fθ(X) − b(θ)}2 with respect to a(θ)and b(θ), where a(θ) and b(θ) are independent of X.
Verify (3.23).
If {Xj} ∼ i.i.d. N(θ, 1), then show that the distribution of n−1Pn j=1 Xj is N(θ, n−1).
Prove the statement (3.10).
Prove Theorem 3.1 in the case that X is continuous.
Let {Xn, n = 0, 1, 2 . . . } be a sequence of random variables. If Xn d−→ c for some constantc, then show that Xn p−→ c.

Showing 4500 - 4600 of 5397

First
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
Last

Optimal Statistical Inference In Financial Engineering 1st Edition Masanobu Taniguchi, Junichi Hirukawa, Kenichiro Tamaki - Solutions

Step by Step Answers