Applied Stochastic Modelling 2nd Edition Byron J.T. Morgan - Solutions

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1.19 One urn contains w1 white balls and b1 black balls. Another urn contains w2 white balls and b2 black balls. A ball is drawn at random from each urn, then one of the two such chosen is selected at random.a) What is the probability that the final ball selected is white?b) Given that the final
1.18 Calculating Odds. If we know the probabilities of a certain event, another way we can express these probabilities is through the odds – that is, the return given to betting on the event. To give an example, we can easily calculate the probability of the next card in a deck of 52 cards being
1.17 Generalize the previous problem to n sets using induction.
1.16 Prove that, if two events A and B are independent, then so are their complements.
1.15 Show that P(A|B) = P(A) is the same as independence of the events A and B.
1.14 If two events are such that A ∩ B = ∅, are A and B independent? Justify.
1.13 Prove the total probability formula equation (1.8) and the Bayes Formula equation 1.9.
1.12 A monotone algebra is a σ-algebra Let ℱ be an algebra that is also a monotone class. Show that ℱ is a σ-algebra.
1.11 If ℱ0 is an algebra, m(ℱ0) is the minimal monotone class over ℱ0, and ????2 is defined as????2 = {B : A ∪ B ∈ m(ℱ0), ∀A ∈ m(ℱ0)}Then show that ????2 is a monotone class.Hint: Look at the proof of theorem 1.38 on page 39, and repeat the arguments therein.
1.10 No matter how many zeros do not add to more than zero Prove the Lemma 1.26 on page 32.Hint: You may use countable subadditivity.
1.9 Properties of probability measures Prove properties 1–4 in the Proposition 1.20 on page 21.Hint: You only have to use the definition of probability. The only thing nontrivial in the definition is the countable additivity property.
1.8 Show that the following classes all generate the Borel σ-algebra, or, put differently, show the equality of the following collections of sets:σ ((a,b) : a
1.7 Show that the Borel sets of R ℬ = σ ({(−∞, x]| x ∈ R}).Hint: show that the generating set is the same, that is, show that any set of the form (−∞, x] can be written as countable union (or intersection) of open intervals, and, vice versa, that any open interval in R can be written
1.6 A σ-Algebra does not necessarily contain all the events in ΩLet ℱ = {A ⊆ Ω| A countable or Ac is countable}. Show that ℱ is a σ-algebra.Note that, if Ω is uncountable, it implies that it contains a set A such that both A and Ac are uncountable and thus A /∈ ℱ .
1.5 Let ℱ = {A ⊆ Ω| A finite or Ac is finite}.a) Show that ℱ is an algebra.b) Show that, if Ω is finite, then ℱ is a σ-algebra.c) Show that, if Ω is infinite, then ℱ is not a σ-algebra.
1.4 An algebra which is not a σ-algebra Let ℬ0 be the collection of sets of the form (a1, a1] ∪ (a2, a2] ∪···∪ (am, am], for any m ∈ N∗ = {1, 2 ... } and all a1 < a1 < a2 < a2 < ··· < am < am inΩ = (0, 1].Verify that ℬ0 is an algebra. Show that ℬ0 is not a σ-algebra.
1.3 Suppose two events A and B are in some space Ω. List the elements of the generated σ algebra σ(A, B) in the following cases:a) A ∩ B = ∅b) A ⊂ B ,c) A ∩ B = ∅; A \ B = ∅ and B \ A = ∅.
1.2 Suppose Ω = {a,b, c} is a probability space, and a discrete probability function is introduced such that P({a})=1/3 and P({b})=1/2. List all events in the maximal sigma algebra P(Ω) and calculate the probability of each such event.
1.1 Roll a die. Then Ω = {1, 2, 3, 4, 5, 6}. An example of an event is A = { Roll an even number} = {2, 4, 6}. Find the cardinality (number of elements)of ???? (Ω) in this case.
8.24 Write down the link function for the model of Example 8.6, for fixed λ. If
8.23 Use the Matlab program of Figure 8.4(a) to smooth the approximate likelihoods from Monte Carlo inference, and displayed in Figure 6.7.
8.22 Discuss the results of Figure 8.6, in which the same GAM has been fitted to the data from each state, but the state responses differ by a constant term. Compare the results with those of Figure 8.1. Suggest alternative breakdowns of the data which may be important.
8.21∗ In Example 7.2, the annual reporting probabilities of dead birds are assumed to vary from year-to-year. The model adopted there has a separate reporting probability, λi, for the ith year of the study. An alternative approach to describing variation in the reporting probability is to treat
8.20∗ Consider how you would use numerical integration in Matlab in order to evaluate the unconditional GLMM log-likelihood of Equation (8.9).
8.19∗ Explain why the constraints of Equation (8.12) hold.
8.18∗ Show, using integration by parts, that the penalty term of Equation(8.11) can be written as: b a(f(x))2dx = fKf.Show further that the function S(f) of Equation (8.11) can be written as:S(f)=(y − f)(y −f) + νfKf, and deduce that the minimising value of F is F = (I + νK)−1y.
8.17 A kernel smoother uses a kernel function to weight local information in a scatter plot. Suppose we have a set of data, (xi, yi, 1 ≤ i ≤ n). For any
8.16∗ Show that when a canonical link function is used in a GLM, then the method of scoring is equivalent to the Newton-Raphson method for obtaining maximum-likelihood estimates.
8.15 As in Section 5.3.2, the exponential family is often presented as:f(w; θ) = exp{(wθ − b(θ))/a(φ) + c(w, φ)}, for specific functions, a( ), b( ) and c( ), and known parameter φ. Verify that this is identical to the canonical form of the expression of (8.4).
8.14 Discuss the power logit model for quantal response data, in which the probability of response to dose d is given by:P(d) = (1 + e−β(d−θ))−m, where β, θ and m are parameters.
8.13 In GLMs, verify that for a binomial distribution the canonical link function is logistic, while for a Poisson distribution the canonical link function is log.
8.12 Discuss whether the model of Exercise 5.10 is a GLM.
8.11∗ In the Cox proportional hazards model (Cox, 1972), the multiplicative function λ(t) in the hazard:h(t|x) = λ(t)eβx is not specified. Consider how you might fit such a model to data. Might a GLM approach be feasible? See Aitkin et al. (1989, Section 6.15) for detailed discussion.
8.10 A group of 97 patients with inoperable lung cancer were given chemotherapy, resulting in observed survival times ranging from 1 to 587 days. On each patient the covariates recorded were: tumour type (adeno, small, large or squamous) and a measure of fitness. This is a subset of a data set
8.9∗ If T has the Weibull distribution of Exercise 8.8, then Y = log T has the extreme value distribution. Derive the form of the pdf of Y . Consider how the extreme value distribution may be fitted to survival data as a GLM, when the covariates enter the model in the same way as in Exercise 8.8.
8.8 In the Weibull model for survival data, the time to response, T , has pdf f(t) = αλtα−1 e−λtαfor t > 0.Derive the forms for the hazard function and the survivor function. Investigate how changing the shape parameter α affects the shape of the hazard function. Suppose all individuals
8.7 The zero-truncated Poisson distribution has probabilities:pr(Y = j) = e−λλj j!(1 − e−λ) = λj j!(eλ − 1), for j = 1, 2.....Show that Y is a member of the exponential family distribution with canonical parameter log(λ). Discuss how you would fit this model to data when there is a
8.6 The Common Birds Census data are described in Example 8.4, where a GLM is proposed for the analysis. The Independent newspaper published in Britain on 12 August 1999 comments on the dramatic decline in many farmland bird species over the past 30 years. Discuss how you would use the results of
8.5 (Dobson, 1990, p.33.) The data below are the numbers of deaths from AIDS in Australia per quarter for the period, 1983–1986. Plot the data and describe the salient features. Explain how you would analyse these data using a GLM.Quarter 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No. dead 0 1 2 3 1 4 9 18
8.4 Assume that random variables Yi are independent, and have the distribution, Yi ∼ Po(α + βxi), where the {xi} are known covariates and α, βare parameters to be estimated, 1 ≤ i ≤ n. Write down the likelihood, and show that the maximum-likelihood estimates of α and β satisfy the
8.3 The contingency table below presents data from Roberts et al. (1981).Individuals with malignant melanoma, which is a skin cancer, are classified according to tumour type and site.
8.2 Write a Matlab program to fit a logit model to the age-of-menarche data of Exercise 4.14, by iterating generalised least squares.
8.1 Verify that the rth iterative estimate of b in GLMs, using the method of scoring, is given asβˆ(r)= (X W X)−1 X W z, in the notation of Section 8.2.3.
7.35 Experiment with the program of Figure 7.10.
7.34 Modify and run the program of Figure 7.2, so that it corresponds to the trivariate distribution of Example 7.1.
7.33* Devise a RJMCMC procedure for estimating model probabilities for Example 7.3 on the survival of dippers. Compare the results with those of Table 7.2.
7.32 Write a Matlab program in order to derive model probabilities for the geometric and beta-geometric models of Example 7.4, applied to the smokers data of Table 1.1. Interpret the results in the light of a likelihoodratio test comparison of the two models.
7.31 (Millar, 2004.) Consider the case of a random variable X which has a N(μ, σ2) distribution, where σ2 is known, and the parameter μ has a N(ν, τ 2) distribution.An important issue in Bayesian analysis is the sensitivity of posterior distributions to assumptions made regarding prior
7.30 In the genetic linkage study of Example 5.6, 197 animals are divided into four categories with frequencies of 125, 18, 20, 34, with corresponding probabilities given below 12 + π4, 1 4(1 − π), 1 4(1 − π),π4.With a uniform prior density for π, write down the posterior density forπ.
7.29 The Hardy-Weinberg law predicts that the proportions of genotypes AA, Aa and aa in a population are θ2, 2θ(1 − θ) and (1 − θ)2, respectively,
7.28 A simple probability model for the data of Table 7.1 has two probabilities as parameters, p, the probability of capture in any year and φ, the annual dipper survival probability. For the first five rows of the table the data are multinomial, with first two multinomial probabilities,
7.27 Devise a Metropolis method to simulate random variables from the Cauchy pdf, f(x) = 1π(1 + x2), −∞
7.26 For the model of Exercise 7.25, suppose that the total number infected, i, occurs with frequency Xi, i = 1, 2, 3. Obtain the expected Fisher information. Write down the asymptotic distribution of the maximum-likelihood estimator ˆθ, and use it to obtain an approximate 99.5% confidence
7.25 (Examination question, University of Nottingham.)A model for the spread of a measles epidemic initiated by one infective in a household of size three predicts that the total number of people infected in the household has the following distribution Total number infected 1 2 3 Probability (1 −
7.24* (Examination question, University of Nottingham.)Consider a long document that consists of n(> 2) pages. It is known that one secretary started typing the document but it was completed by another.However, exactly where in the document the second secretary took over from the first is unknown.
7.23 Data, x = (x1, x2, x3), arise with the multinomial probability, f(x|θ, η) ∝ θx1 ηx2 (1 − θ − η)x3 ,
7.22∗ (Brooks et al., 2000b.) In Examples 4.6 and 5.5 we have discussed a model for bird survival in which the likelihood surface has a completely flat ridge. Consider whether a Bayesian analysis with independent U(0, 1)priors for all of the model probabilities is likely to produce useful results.
7.21 Consider how you would provide a Bayesian formulation and a Bayesian analysis of the fecundability and polyspermy models of Chapter 2.
7.20 The data of Exercise 3.3 describe the number of dead fetuses in litters of mice. This is one of six data sets considered by Brooks et al. (1997). Brooks(2001) has analysed these data from a Bayesian perspective and considered five alternative models for each data set, viz., a beta-binomial
7.19 Shown below are the posterior means and standard deviations (SD)under independent U(0, 1) priors, for survival probabilities of four models fitted to the teal data set of Exercise 7.5, from a sample of 10,000 observations from the posterior distribution, together with the Bayesian p-values and
7.18 (Link et al., 2002.) A Metropolis-Hastings method rejects candidate values in order to obtain N(0, 1) random variables. Two sequences of candidate values are illustrated in Figure 7.11. The lighter line results from adding N(0, 100) deviates to the current values; the darker line results from
7.17 Why is the output from the Metropolis-Hastings method a Markov chain? How can the autocorrelation function for such output reveal appreciable correlations between values several steps apart?
7.16 When a prior distribution is completely flat, what is the relationship between the likelihood and the posterior distribution?
7.15 In the Matlab program of Figure 7.7 to carry out a Metropolis-Hastings analysis for Example 7.1, discuss why you think potential transitions were made from setting q(x, y) = g(y), rather than an alternative form for q(x, y).
7.14 In Table 7.2, use the posterior model probabilities for Prior 1 to calculate the model-averaged variance of φn.
7.13 Consider how you would use data augmentation for the problem of Example 5.1, for the censored survival data problem in Figure 5.2(b), and for fitting mixtures of distributions, as in Example 5.9.
7.12 Experiment with running the Matlab programs of Figure 7.7 for different values of m and k.
7.11 Verify that when the transition matrix P is constructed from the transition Q in the Metropolis-Hastings method, then P is irreducible if and only if Q is.
7.10 Show that the Hastings acceptance probability of Equation (7.10) includes the Barker acceptance probability of Equation (7.13).
7.9 (Tanner, 1993, p.138.) Consider again the genetics model of Example 5.7. Assume a uniform prior for the single model parameter. Suggest a suitable transition probability function, and simulate a random sample from the posterior distribution for the model parameter using the MetropolisHastings
7.8 (Chib and Greenberg, 1995.) We saw in Exercise 6.21, how to simulate from a particular bivariate normal pdf. Consider how the MetropolisHastings method can be extended to deal with continuous random variables, using this bivariate distribution as an illustration. Suggest a suitable transition
7.7 Verify that the Metropolis acceptance function of Equation (7.14) results in a reversible Markov chain. Verify that the Hastings acceptance function of Equation (7.10) also results in a reversible Markov chain.
7.6 Use the Matlab program of Figure 7.2 to construct a sequence of values from the Be(1,2) pdf. Use the fact that f(y|x) ∼ Be(x + 1, n − x + 2), to construct the Rao-Blackwell estimate of f(y), for the case n = 20.
7.5 Shown below are the reported recoveries of dead blue-winged teal marked as young from 1961 to 1973. The data are taken from Brownie et al. (1985).
7.4 Verify the expression given for the equilibrium distribution specified in Equation (7.8).
7.3 Verify in Example 7.1 that for fixed n, f(y) ∼ Be(α, β).
7.2 Verify the expression for the beta posterior distribution in Example (ii)of Section 7.2.
7.1 Verify the expression for the normal posterior distribution, given by Equations (7.3).
6.34 Experiment with the Matlab program of Figure 6.6. For example, examine the effect of increasing s, and consider whether deterministic and/or stochastic optimisation procedures might deliver an optimum, after the script file is re-written as a function file.
6.32 The random variable X has an extreme-value distribution, with cumulative distribution function:P(X ≤ x)=1−k xa, with parameters a and k, for a > 0, x ≥ k > 0.Explain how to simulate X using the inversion method.6.33 The random variable X has a logistic distribution, with pdf given by
6.31 One way to choose the smoothing parameter, h, in kernel density estimation is to minimise the AMISE criterion, given by :AMISE =R(K)nh +1 4η4h4R(f).Here f(x) denotes the pdf being estimated, and K denotes the kernel being used (itself a pdf with mean zero and variance η2) and R(g) is
6.30* Edit and run the bootstrap program of Figure 6.12, so that the smoothed bootstrap is used.
6.29 A Γ(n, 1) random variable has the pdf f(x) = xn−1e−xΓ(n) for x ≥ 0, n> 1.Describe how to simulate such random variables, using the rejection method with envelope g(x) = ke−x/n n , for x ≥ 0 and suitable k > 1.
6.28 Outline how the rejection method for simulating random variables works;use as an illustration an enveloping pdf which is exponential of mean unity and a target distribution (the one we wish to simulate from) which is halfnormal, with pdff(x) = . 2π e− 1 2 x2, for x ≥ 0.(i) Show that the
6.27 The rejection method of Example 6.2 simulates Γ(r, 1) random variables.Explain how to simulate Γ(r, λ) random variables.
6.26 Compare bootstrap and parametric bootstrap results for the beta-geometric model fitted to a data set other than the ‘smokers’ data set from Table 1.1.
6.25 Modify the program of Figure 6.6 so that it becomes a function producing the Monte Carlo estimate of the log-likelihood at a single value of θ.Use fminbnd to try to obtain a Monte Carlo maximum-likelihood estimate of θ. Compare the performance of fminbnd with simulated annealing.
6.24 Experiment with alternative smoothing for the Monte Carlo inference program in Figure 6.6. Also try using an alternative kernel.
6.23 Suggest a way to simulate data from a beta-binomial distribution.
6.22 Suggest verification procedures to check that the simulation functions of this book work correctly.
6.21 Suppose we want to simulate from the bivariate normal distribution N2(μ, Σ), where μ = (1, 2) and Σ = 1 0.9 0.9 1 .The Cholesky approach is to obtain a matrix A such that AA = Σ, and then set X = μ+AY, where Y ∼ N2(0, I). Find A in this case, and hence simulate from N2(μ, Σ).
6.20 In the general rejection method of Example 6.2, why does it suffice only to specify the pdf up to a multiplicative constant term?
6.19 Show that the probability of rejection in the general rejection method of Example 6.2 is 1 − k−1.
6.17 For the geyser data of Exercise 6.13, construct 10 bootstrap samples, and provide a kernel probability density estimate for each. Comment on the poor results.6.18 Simulate data from an exponential distribution, Ex(2), and construct a standard kernel density estimate from the data. Comment on
6.16 As an exercise in verification, simulate appropriate multinomial data using the simulation program, ecosim, below, and then obtain the maximum likelihood estimates of the parameters using the function ecolik.(a)% Program to simulate multinomial data corresponding to% a simple bird survival
6.15 As an exercise in verification, simulate 200 samples from a zero-inflated Poisson distribution, using the function zipsim of Exercise 4.9. Fit the model to each of the simulated data sets, and construct histograms of theλ estimates and of the w estimates. Smooth the histograms using kernel
6.14 Suppose we were not supplied with the Matlab routine randn for simulating normal random variables. A simple way to simulate approximately standard normal random variables is to make use of the Central Limit Theorem of probability theory, and set Y = 12 i=1 Ui − 6, where the {Ui} are
6.13 For the geyser data below, (a) experiment in Matlab with different histogram representations (try: help hist); (b) experiment with different kernels, in forming a kernel density estimate.Durations (in minutes) of 107 eruptions of Old Faithful geyser (Weisberg, 1980)
6.12 Check that the ‘naive’ density estimate of Equation (C.1) in Appendix C is different from the traditional histogram.
6.11 For the data sets of Table 1.1, conduct a sensitivity analysis of the parametric bootstrap estimate of error for the parameters θ and μ in the beta-geometric fecundability model, by (a) changing both θ and μ by an amount equal to estimated asymptotic standard errors, and (b) obtaining

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Applied Stochastic Modelling 2nd Edition Byron J.T. Morgan - Solutions

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