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principles of managerial statistics

Questions and Answers of Principles Of Managerial Statistics

10. Suppose Y = (Y1,...,Yd) has a meta-Gaussian distribution with continuous marginal distributions and copula CGauss(·|Ω). Show that ifρτ (Yi, Yj ) = 0 then Yi and Yj are independent.
9. A convex combination of k joint CDFs is itself a joint CDF (finite mixture), but is a convex combination of k copula functions a copula function itself?
8. Let ϕ(u|θ) = (1 − u)θ, for some θ ≥ 1, and show that for the twodimensional case this generates the copula C(u1, u2|θ) = max[0, 1 − {(1 − u1)θ + (1 − u2)θ}1/θ].Further, show that
7. Suppose that ϕ1,...,ϕk are k strict generator functions and define a new generator ϕ as a convex combination of these k generators, that isϕ(u) = a1ϕ1(u) + ··· + akϕk(u), in which
6. Show that as θ → ∞, CFr(u1, u2|θ) → min(u1, u2), the co-monotonicity copula C+.
5. Show that the generator of a Frank copulaϕFr(u|θ) = − loge−θu − 1 e−θ − 1, θ ∈ {(−∞, 0) ∪ (0,∞)}, satisfies assumptions 1–3 of a strict generator.
4. The co-monotonicity copula C+ is not an Archimedean copula; however, in the two-dimensional case, the counter-monotonicity copula C−(u1, u2) =max(u1 + u2 − 1, 0) is. What is its generator
3. Show that an Archimedean copula with generator function ϕ(u) = − log(u)is equal to the independence copula C0. Does the same hold when the natural logarithm is replaced by the common logarithm,
2. Suppose that X is Uniform(0,1) and Y = X2. Then the Spearman rank correlation and the Kendall’s tau between X and Y will both equal 1, but the Pearson correlation between X and Y will be less
1. Kendall’s tau rank correlation between X and Y is 0.55. Both X and Y are positive. What is Kendall’s tau between X and 1/Y ? What is Kendall’s tau between 1/X and 1/Y ?
7. Suppose Y = (Y1, Y2, Y3) has covariance matrix COV (Y) =⎛⎝1.0 0.9 a 0.9 1.0 0.9 a 0.9 1.0⎞⎠for some unknown valuea. Use Eq. (7.7) and the fact that the variance of a random variable is
6. Verify the following results that were stated in Sect. 7.3:
5. Show that if X is uniformly distributed on [−a, a] for any a > 0 and if Y = X2, then X and Y are uncorrelated but they are not independent.
4. (a) Show that E{X − E(X)} = 0 for any random variable X.(b) Use the result in part (a) and Eq. (A.31) to show that if two random variables are independent then they are uncorrelated.
3. Verify formulas (A.24)–(A.27).
2. Let X1, X2, Y1, and Y2 be random variables.(a) Show that Cov(X1 + X2, Y1 + Y2) = Cov(X1, Y1) + Cov(X1, Y2) +Cov(X2, Y1) + Cov(X2, Y2).(b) Generalize part (a) to an arbitrary number of Xis and Yis.
1. Suppose that E(X) = 1, E(Y )=1.5, Var(X) = 2, Var(Y )=2.7, and Cov(X, Y )=0.8.(a) What are E(0.2X + 0.8Y ) and Var(0.2X + 0.8Y )?(b) For what value of w is Var{wX + (1−w)Y } minimized? Suppose
3. The following R code was used to bootstrap the sample standard deviation.( code to read the variable x )sampleSD = sd(x)n = length(x)nboot = 15000 resampleSD = rep(0, nboot)
2. In the following R program, resampling was used to estimate the bias and variance of the sample correlation between the variables in the vectors x and y.samplecor = cor(x, y)n = length(x)nboot =
1. To estimate the risk of a stock, a sample of 50 log returns was taken and s was 0.31. To get a confidence interval for σ, 10,000 resamples were taken. Let sb,boot be the sample standard deviation
13. In this problem, you will fit a t-distribution to daily log returns of Siemens.You will estimate the degrees-of-freedom parameter graphically and then by maximum likelihood. Run the following
12. In this problem you will fit a t-distribution by maximum likelihood to the daily log returns for BMW. The data are in the data set bmw that is part of the evir package. Run the following
11. The number of small businesses in a certain region defaulting on loans was observed for each month over a 4-year period. In the R program below,
10. Assume that you have a sample from a t-distribution and the sample kurtosis is 9. Based on this information alone, what would you use as an estimate of ν, the tail-index parameter?
9. Suppose that X1,...,Xn iid∼ Normal(μ, σ2), with 0 < σ2 < ∞, and defineμˆ = 1 nn i=1 Xi. What is Bias(ˆμ)? What is MSE(ˆμ)? What if the distribution of the Xi is not Normal, but
8. For any univariate parameter θ and estimator θ, we define the bias to be Bias(θ) = E(θ) − θ and the MSE (mean square error) to be MSE(θ) =E(θ− θ)2. Show that MSE(θ) = {Bias(θ)}2 +
7. Suppose that X1,...,Xn are i.i.d. exponential(θ). Show that the MLE ofθ is X.
6. Fit the F-S skewed t-distribution to the gas flow data. The data set is in the file GasFlowData.csv, which can be found on the book’s website.
5. (a) What is the kurtosis of a normal mixture distribution that is 95 %N(0, 1) and 5 % N(0, 10)?(b) Find a formula for the kurtosis of a normal mixture distribution that is 100p% N(0, 1) and 100(1
4. Let X be a random variable with mean μ and standard deviation σ.(a) Show that the kurtosis of X is equal to 1 plus the variance of {(X −μ)/σ}2.(b) Show that the kurtosis of any random
3. Show that f ∗(y|ξ) given by Eq. (5.15) integrates to (ξ + ξ−1)/2.
2. Suppose that Y1,...,Yn are i.i.d. N(μ, σ2), where μ is known. Show that the MLE of σ2 is n−1 n i=1(Yi − μ)2.
1. Load the CRSPday data set in the Ecdat package and get the variable names with the commands library(Ecdat)data(CRSPday)dimnames(CRSPday)[[2]]Plot the IBM returns with the commands r = CRSPday[
7. Suppose that Y1,...,Yn are i.i.d. with a uniform distribution on the interval (0,1), with density function f and distribution function F defined as f(x) = 1 if x ∈ (0, 1), 0 otherwise, and
6. Use the following fact about the standard normal cumulative distribution function Φ(·):Φ−1(0.025) = −1.96.(a) What value is Φ−1(0.975)? Why?(b) What is the 0.975-quantile of the normal
5. Let diffbp be the changes (that is, differences) in the variable bp, the U.S. dollar to British pound exchange rate, which is in the Garch data set of R’s Ecdat package.(a) Create a 3 × 2
4. Suppose in a normal plot that the sample quantiles are plotted on the vertical axis, rather than on the horizontal axis as in this book.(a) What is the interpretation of a convex pattern?(b) What
3. This problems uses the Garch data set in R’s Ecdat package.(a) Using a solid curve, plot a kernel density estimate of the first differences of the variable dy, which is the U.S. dollar/Japanese
2. Column seven of the data set RecentFord.csv on the book’s web site contains Ford daily closing prices, adjusted for splits and dividends, for the years 2009–2013. Repeat Problem 1 using these
1. This problem uses the data set ford.csv on the book’s web site. The data were taken from the ford.s data set in R’s fEcofin package. This package is no longer on CRAN. This data set contains
22. A coupon bond matures in 4 years. Its par is $1,000 and it makes eight coupon payments of $21, one every one-half year. The continuously compounded forward rate is r(t)=0.022 + 0.005 t − 0.004
21. A par $1,000 bond matures in 4 years and pays semiannual coupon payments of $25. The price of the bond is $1,015. What is the semiannual yield to maturity of this bond?
20. Par $1,000 zero-coupon bonds of maturities of 0.5-, 1-, 1.5-, and 2-years are selling at $980.39, $957.41, $923.18, and $888.489, respectively.(a) Find the 0.5-, 1-, 1.5-, and 2-year semiannual
19. The 1/2-, 1-, 1.5-, and 2-year spot rates are 0.025, 0.029, 0.031, and 0.035, respectively. A par $1,000 coupon bond matures in 2 years and has semiannual coupon payments of $35. What is the
18. Suppose that the forward rate is r(t)=0.03 + 0.001t + 0.0002t 2(a) What is the 5-year spot rate?(b) What is the price of a zero-coupon bond that matures in 5 years?
17. A coupon bond has a coupon rate of 3 % and a current yield of 2.8 %.(a) Is the bond selling above or below par? Why or why not?(b) Is the yield to maturity above or below 2.8 %? Why or why not?
16. Assume that the yield curve is YT = 0.04 + 0.001 T.(a) What is the price of a par-$1,000 zero-coupon bond with a maturity of 10 years?(b) Suppose you buy this bond. If 1 year later the yield
15. Suppose that a bond pays a cash flow Ci at time Ti for i = 1,...,N. Then the net present value (NPV) of cash flow Ci is NPVi = Ci exp(−Ti yTi ).Define the weightsωi = NPViN j=1 NPVj and
14. An investor is considering the purchase of zero-coupon bonds with maturities of one, three, or 5 years. Currently the spot rates for 1-, 2-, 3-, 4-, and 5-year zero-coupon bonds are,
13. Suppose that the continuous forward rate is r(t)=0.03 + 0.001t −0.00021(t − 10)+. What is the yield to maturity on a 20-year zero-coupon bond? Here x+ is the positive part function defined by
12. Suppose the continuous forward rate is r(t)=0.04 + 0.001t when a 8-year zero coupon bond is purchased. Six months later the forward rate is r(t) =0.03 + 0.0013t and bond is sold. What is the
11. Suppose that the continuous forward rate is r(t)=0.033 + 0.0012t. What is the current value of a par $100 zero-coupon bond with a maturity of 15 years?
10. Suppose that a coupon bond with a par value of $1,000 and a maturity of 7 years is selling for $1,040. The semiannual coupon payments are $23.(a) Find the yield to maturity of this bond.(b) What
9. A coupon bond with a par value of $1,000 and a 10-year maturity pays semiannual coupons of $21.(a) Suppose the yield for this bond is 4 % per year compounded semiannually. What is the price of the
8. A par $1,000 zero-coupon bond that matures in 5 years sells for $828.Assume that there is a constant continuously compounded forward rate r.(a) What is r?(b) Suppose that 1 year later the forward
7. One year ago a par $1,000 20-year coupon bond with semiannual coupon payments was issued. The annual interest rate (that is, the coupon rate)at that time was 8.5 %. Now, a year later, the annual
6. Verify the following equality:2T t=1 C(1 + r)t +PAR(1 + r)2T = C r+PAR − C r(1 + r)−2T .
5. The 1/2-, 1-, 1.5-, and 2-year semiannually compounded spot rates are 0.025, 0.028, 0.032, and 0.033, respectively. A par $1,000 coupon bond matures in 2 years and has semiannual coupon payments
4. Suppose that the forward rate is r(t)=0.032 + 0.001t + 0.0002t 2.(a) What is the 5-year continuously compounded spot rate?(b) What is the price of a zero-coupon bond that matures in 5 years?
3. A coupon bond has a coupon rate of 3 % and a current yield of 2.8 %.(a) Is the bond selling above or below par? Why or why not?(b) Is the yield to maturity above or below 2.8 %? Why or why not?
2. Suppose that the forward rate is r(t)=0.04 + 0.0002t − 0.00003t 2.(a) What is the yield to maturity of a bond maturing in 8 years?(b) What is the price of a par $1,000 zero-coupon bond maturing
1. Suppose that the forward rate is r(t)=0.028 + 0.00042t.(a) What is the yield to maturity of a bond maturing in 20 years?(b) What is the price of a par $1,000 zero-coupon bond maturing in 15 years?
11. Suppose that daily log-returns are N(0.0005, 0.012). Find the smallest value of t such that P(Pt/P0 ≥ 2) ≥ 0.9, that is, that after t days the probability the price has doubled is at least 90
10. The daily log returns on a stock are normally distributed with mean 0.0002 and standard deviation 0.03. The stock price is now $97. What is the probability that it will exceed $100 after 20
9. Suppose that X1, X2,... is a lognormal geometric random walk with parameters μ = 0.1, σ = 0.2.(a) Find P(X3 > 1.2X0).(b) Find the conditional variance of Xk/k given X0 for any k.(c) Find the
8. Suppose that X1, X2,... is a lognormal geometric random walk with parameters (μ, σ2). More specifically, suppose that Xk = X0 exp(r1 + ··· +rk), where X0 is a fixed constant and r1, r2,...
7. Let rt be a log return. Suppose that r1, r2,... are i.i.d. N(0.06, 0.47).(a) What is the distribution of rt(4) = rt + rt−1 + rt−2 + rt−3?(b) What is P{r1(4) < 2}?(c) What is the covariance
6. The prices and dividends of a stock are given in the table below.(a) Find R3(2),(b) Find r4(3).t Pt Dt 1 82 0.1 2 85 0.1 3 83 0.1 4 87 0.125
5. The prices and dividends of a stock are given in the table below.(a) What is R2?(b) What is R4(3)?(c) What is r3?
4. Suppose the prices of a stock at times 1, 2, and 3 are P1 = 95, P2 = 103, and P3 = 98. Find r3(2).
3. The yearly log returns on a stock are normally distributed with mean 0.08 and standard deviation 0.15. The stock is selling at $80 today. What is the probability that 2 years from now it is
2. The yearly log returns on a stock are normally distributed with mean 0.1 and standard deviation 0.2. The stock is selling at $100 today. What is the probability that 1 year from now it is selling
1. Suppose that the daily log returns on a stock are independent and normally distributed with mean 0.001 and standard deviation 0.015. Suppose you buy $1,000 worth of this stock.(a) What is the
=+Task 9: Analyse the data in Task 7 with a robust model. Do children take longer to put their hands in a box that they believe contains an 901 animal about which they have been told nasty things?
=+Task 8: Log-transform the scores in Task 7 and repeat the normality tests.
=+Task 7: Early in my career I looked at the effect of giving children information about animals. In one study (Field, 2006), I used three novel animals (the quoll, quokka and cuscus), and children
=+Task 6: Using SPSS Tip 15.3, change the syntax in SimpleEffectsAttitude.sps to look at the effect of drink at different levels of imagery.
=+Task 5: In the previous chapter we came across the beer-goggles effect. In that chapter, we saw that the beer-goggles effect was stronger for unattractive faces. We took a follow-up sample of 26
=+Task 4: The ‘roving eye’ effect is the propensity of people in relationships to ‘eye up’ people other than their current partner. I fitted 20 people with incredibly sophisticated glasses
=+Task 3: Calculate the effect sizes for the analysis in Task 1.
=+Task 2: Repeat the analysis for Task 1 using SPSS Statistics and interpret the results.
=+Task 1: It is common that lecturers obtain reputations for being ‘hard’ or‘light’ markers (or, to use the students’ terminology, ‘evil manifestations from Beelzebub’s bowels’ and
=+6 A less well-known country musician not to be confused with anyone who has a similar name and produces music that makes you want to barf.
=+ranging from 0 (no injury) to 20 (severe injury)). Fit a model to see whether injury severity is significantly predicted from the type of game, the type of console and their interaction.
=+Task 10: A researcher was interested in what factors contributed to injuries resulting from game console use. She tested 40 participants who were randomly assigned to either an active or static
=+Task 9: There are reports of increases in injuries related to playing Nintendo Wii (http://ow.ly/ceWPj). These injuries were attributed mainly to muscle and tendon strains. A researcher
=+Task 8: Using SPSS Tip 14.1, change the syntax in GogglesSimpleEffects.sps to look at the effect of alcohol at different 836 levels of type of face.
=+Task 7: Compute omega squared for the effects in Task 6 and report the results of the analysis.
=+Task 6: At the start of this chapter I described a way of empirically researching whether I wrote better songs than my old bandmate Malcolm, and whether this depended on the type of song (a
=+Task 5: In Chapter 4 we used some data that related to learning in men and women when either reinforcement or punishment was used in teaching (Method Of Teaching.sav). Analyse these data to see
=+Task 4: Compute omega squared for the effects in Task 3 and report the results of the analysis.
=+Task 3: In Chapter 5 we used some data that related to male and female arousal levels when watching The Notebook or a documentary about notebooks (Notebook.sav). Fit a model to test whether men and
=+Task 2: Compute omega squared for the effects in Task 1 and report the results of the analysis.
=+Task 1: I’ve wondered whether musical taste changes as you get older:my parents, for example, after years of listening to relatively cool music when I was a kid, hit their mid-forties and
=+Task 8: In Chapter 10 we compared the number of mischievous acts(mischief2) in people who had invisibility cloaks to those without(cloak). Imagine we also had information about the baseline number
=+Task 7: Compare your results for Task 6 to those for the corresponding task in Chapter 11. What differences do you notice and why?
=+Task 6: In Chapter 4 (Task 6) we looked at data from people who had been forced to marry goats and dogs and measured their life satisfaction and also how much they like animals (Goat or Dog.sav).
=+Task 5: The highlight of the elephant calendar is the annual elephant soccer event in Nepal (google search it). A heated argument burns between the African and Asian elephants. In 2010, the
=+Task 4: Compute effect sizes for Task 3 and report the results.
=+I feel really full of beans and healthy) two hours later (this variable is called well). He measured how drunk the person got the night before on a scale of 0 = as sober as a nun to 10 = flapping

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