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probability and stochastic modeling

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

6.10 Use a Monte Carlo test to compare the beta-binomial model with a mixture of two binomial distributions for the ant-lion data of Example 1.10.
6.9 Check the goodness-of-fit of the model of Exercise 6.8 to the data of Table 1.5, using a Monte Carlo test.
6.8 Use the bootstrap to simulate replicate data sets to that of Table 1.5.Use these replicates to estimate the standard errors of the parameters in the model (φ1, φa, λ), fitted to these data.
6.7∗ Use Monte Carlo inference to estimate the parameter in an exponential Ex(λ) distribution. Experiment with alternative kernels.
6.6 (Morgan, 1984, p.169.) Explain how you would use importance sampling to estimate the standard normal cdf, making use of the logistic pdf, f(y) = π exp(−πy/√3)√3{1 + exp(−πy/√3)}2 , −∞
6.5∗ Prove that the ratio-of-uniforms method specified in Example 6.3 works as stated. If the constant κ could be chosen, how would it be selected?
6.4 Verify the optimum value of k, the expansion factor for the enveloping function in Equation (6.4).
6.3 Show that the rejection algorithm for normal random variables, given in Example 6.1, is a consequence of enveloping the half-normal pdf with a multiple of an exponential Ex(1) pdf. The procedure of using an exponential envelope in this way is outlined in Example 6.2 for simulating gamma random
6.2 Derive algorithms for simulating Weibull and logistic random variables by the inversion method. Provide Matlab functions for these algorithms.
6.1 Verify that the function of Figure 6.3(b) that simulates Poisson random variables does so through the mechanism of the Poisson Process, described in Appendix A.
5.45 Write a Matlab program to display the contours of the log-likelihood surface for the model and data of Example 5.1, in the parameterisation of Equation (5.2).
5.44 The following data are a random sample from an exponential distribution, with all observations that are greater than 4 censored at 4, and denoted by a ∗.
5.43 Continuation of Exercise 4.35. If p denotes the probability of an egg escaping infection, write down an expression for the variance of the maximumlikelihood estimator ˆp. Use the δ-method to approximate the variance of ˆμ, where p = e−μ, and compare the result with that given in
5.42 In a study of the failure times of transplanted kidneys, m > 0 individuals have failure times that are observed directly, leading to the data: {xi}. A further (n − m) individuals have failure times that are right-censored at time τ. Explain how you would model these data using an
5.41 On each of T separate visits to N sites it is recorded whether or not a particular bird species is present or absent. An illustration of the data that may result from such a study is given below, for the American blue jay, Cyanocitta cristata. Here T = 11 and N = 50.Number of detections
5.40 Timed Species Counts are used to estimate the abundance of wild birds.An observer spends one hour in an area of interest and records in which of 6 consecutive 10-minute periods (if any) a bird of a particular species is first heard. This experiment is repeated on 15 separate occasions, and
5.39 (Examination question, University of Nottingham.)The EM algorithm is to be used as an alternative to the Newton-Raphson method for maximising the likelihood of Exercise 3.21 by imputing the values of the censored observations.(i) Show that the E step of the algorithm involves the term E(Ti|Ti
5.38 In the ABO blood group system there are four blood groups, A, B, AB and O, occurring with respective relative frequencies, (p2 + 2pr), (q2 + 2qr), 2pq and r2, where p, q and r are probabilities and p + q + r = 1.(i) Write down the likelihood, L, when the different groups are observed with
5.37 In the re-parameterisation of Equation (5.1), the new parameter, θ, is defined asθ = (1 − w)(1 − e−λ).The maximum-likelihood estimator, ˆθ, is given by ˆθ = 1− (proportion of litters with Y = 0). The δ-method for approximating the variance of ˆθspecifies that Var(ˆθ) ≈
5.36 In an experiment to estimate the annual survival probabilities of a particular species of bird, a single cohort of n birds ringed as nestlings is released, and the numbers r1, r2, r3, r4 of rings recovered from birds reported dead during the first year, second year, third year and fourth year
5.35 Discuss how the δ-method may be used(a) for the transformation of Section 5.4, and(b) in the re-parameterisation of Example 5.1.
5.34 Suppose that data y1,...,yn came from a multinomial distribution with probabilities {pi}, where Σpi = 1 and Σyi = m. Show that the score vector is given by U = DΠ−1y, where Π is a suitable diagonal matrix.Note the correspondence between the expression for the expected information matrix,
5.33 Experiment with SEM for the application of Example 5.1. Produce a sequence of estimates of π, and try to ascertain when the sequence has reached equilibrium.
5.32 Experiment with GEM, applied to the data of Example 5.1, by modifying one step of a Newton-Raphson iteration, as suggested by Rai and Matthews (1993).
5.31∗ Biologists are interested in the distribution of the cytoplasmic heritable determinant of the yeast, Saccharomyces cerevisiae, which exhibits prion-like properties. In an experiment, samples of yeast cells each have effectively, at time zero, an unknown number, m, of inanimate markers,
5.30∗ (Catchpole and Morgan, 1998.) Compartment models are used to describe the movement of material through different compartments, which may for example be parts of an organism. An example is illustrated in Figure 5.4.
5.29∗ The estimator, α(s) of Example 5.10 is not unique. Consider how you would obtain a unique estimate of α. Discuss generally how you would select the variable, s, of the characteristic function.
5.28 For the negative binomial distribution of Exercise 5.23, write down pr(X = 0), and hence derive a method of ‘mean and zero-frequency’ for estimating n and p. Derive a mean and zero-frequency method for estimating the parameters of the Poisson process with the two rates model of Section
5.27 In radioligand binding, two alternative models relating y to x are specified as follows, where α, κ, κ1 and κ2 are parameters.(i) y = p 1 + 10x−κ1 + (1 − p)1 + 10(αx−κ2) ,(ii) y = 1 1 + 10x−κ .Identify which non-regular problem results when we compare these two models.
5.26 In enzyme kinetics, two alternative models relating y to x1 and x2 are specified as follows, where V , κ1, κ2 and κ3 are parameters.(i)y = V x1κ11 + x2κ2+ x11 + x2κ3,(ii)y = V x1κ11 + x2κ2+ x1.Identify which non-regular problem results when we compare these two models.
5.25∗ (Quandt and Ramsey, 1978.) The random sample (y1,...,yn) is observed for the random variable Y which has a pdf which is a mixture o?
5.24 Suppose that Y has a multivariate normal distribution with mean, E[Y] = Xθ and V (Y) = σ2I, where θ is a vector of parameters to be estimated, and I is the identity matrix. Write down the likelihood and show that the maximum-likelihood estimate of θ results from minimising with respect to
5.23 A random variable X has the negative binomial distribution, pr(X = k) = n + k − 1 n − 1 p qk 1 − p qn, for k ≥ 0, in which q = 1+ p. Show that the expectation and variance of X are given by E[X] = np, Var(X) = np(1 + p). Use these expressions to derive estimates of n and p using
5.22∗ Ridout et al. (1999) develop the following model for the strawberry branching structure of Exercise 2.14. Let X denote the number of flowers at rank r, given the number of flowers, m, at the previous rank. The branches at the previous rank are called parent branches. If Yi denotes the
5.21 In quantal response data we observe ri individuals responding out of ni exposed to dose di, 1 ≤ i ≤ k. The logit model specifies the probability of response to dose di as:
5.20 We have already seen that the annual survival rates of wild animals may be estimated by means of ring-recovery experiments, resulting in data of the kind displayed in Table 1.5, and using models such as that of Exercise 5.8.An alternative approach is to make observations of previously marked
5.19 One way of reparameterising to obtain uncorrelated parameter estimators is to use principal component analysis (Jolliffe, 2002). Apply principal component analysis to the correlation matrices of Example 4.1, by using the Matlab command eig. Discuss the results and indicate whether you think
5.18 In the zero-truncated Poisson distribution of Example 5.8, if n0 denotes the estimated missing number of zeros, verify thatλˆ = log 1 +∞i=1 ni/n0=∞i=1 ini3∞i=0 ni.
5.17 In Example 5.10, we encountered the empirical characteristic function.Write down an expression for the empirical cumulative distribution function.
5.16 Adapt the Matlab program of Figure 5.2(b) to fit the gamma, Γ(2, λ), distribution using the EM algorithm, and compare the approach with that?
5.15 Run the Matlab program of Figure 5.2(b), and comment on the speed of convergence of the EM algorithm.
5.14 The log-likelihood for a model which is a mixture of two univariate normal pdfs has the form:(ψ) = n i=1 log{αφ(xi|μ1, σ1) + (1 − α)φ(xi|μ2, σ2)}, using a self-explanatory notation. We wish to fit this model using the EM algorithm.Let n(m)j = n i=1 wij (ψ(m)) for j = 1, 2.Show
5.13 (Morgan and Titterington, 1977.) The mover-stayer model has state transition probabilities of the form:qij = (1 − si)pj (i = j = 1, 2,...,m), qii = (1 − si)pi + si (i = 1, 2,...,m), where {pk} is a probability distribution and 1 − si ≥ 0 ,(1 − si)pi + si ≥ 0 , for i = 1, 2,...,m
5.12 Use the EM algorithm to fit a π : (1 − π) mixture of two Poisson distributions, P0(θ1) and P0(θ2), to the data:No. 0 1 2 3 4 5 6 7 8 9 frequency 162 267 271 185 111 61 27 8 3 1
5.11 (Tanner, 1993, p.41.) Ten motorettes were tested at each of four temperatures: 150◦, 170◦, 190◦, and 220◦C. The failure times, in hours, are given below, with a ‘*’ indicating a right-censored value.
5.10 Abbott’s formula for quantal dose-response data gives the probability of response at dose d as, P(d) = λ + (1 − λ)F(d), where λ is the probability of natural mortality, and F(d) is a customary cdf, as used when there is no natural mortality. For this question, set F(d) =1/[1 + exp{a + b
5.9 Check, using both direct optimisation and the EM algorithm, that μ =0.627 in the multinomial model with data: [125 18 20 34] and cell probabilities 1 1 2 + π4 , 1 4 (1 − π), 1 4 (1 − π), π4 2.Obtain the quadratic equation satisfied by the maximum-likelihood estimate π. As an
5.8 Study the Matlab program given below for fitting the animal survival model, (λ, φ1, φa), to a set of simulated data, using the EM algorithm.
5.7 Use the Matlab function fmax, of Figure 4.1, to obtain the maximumlikelihood estimates of the parameters a and b in the logit model fitted to the data of Exercise 3.10. Use the δ-method and the estimated covariance matrix for the estimators a and b to estimate the variance of the
5.6∗ If X is an exponential random variable with mean unity, E[X log X] =1 − γ, where γ is Euler’s constant (γ ≈ 0.577215). Use this result to derive an orthogonal parameterisation for the Weibull distribution.
5.5 Provide an example of nested optimisation.
5.4 Show that when the random variable X has a Weibull, Wei (ρ, κ) distribution, the random variable Xκ has an exponential distribution and give its mean value. What is the distribution of (ρX)κ?
5.3 For the logit model and data of Exercise 3.10, reparameterise the model so that the probability of response to dose di is given by: P(di)=1/(1 +e{a+b(di−d)}), where d is the sample mean of the {di}. Use fmax to estimate the correlation between a and b and discuss your findings. Draw
5.2 In the Matlab program of Exercise 4.1, experiment with an alternative procedure for ensuring that the parameters which are probabilities stay within range during the iterations leading to the maximum-likelihood estimate.
5.1 Obtain the maximum-likelihood estimate of λ in Example 5.1.
4.36 With reference to the fecundity illustration of Section 2.2, show that if the geometric model is appropriate, and that there are n couples, with conception times censored at r cycles, then E"d2 dp2#= −n{1 − (1 − p)r}p2(1 − p) , where denotes the log-likelihood. Compare the asymptotic
4.35 A batch of n eggs is infected by bacteria, and the number of bacteria per egg is thought to be described by a Poisson distribution, with meanμ. Write down the likelihood if it is observed that r eggs are free from infection. Obtain the maximum-likelihood estimator for μ, and show that its
4.34 Show that the score function for comparing a Weibull distribution with an exponential distribution has the form u = d +u logxi − dxilogxi xi, where d is the number of uncensored terms in the sample and u denotes summation for only uncensored terms.Complete the score test statistic.
4.33 (Examination question, University of Glasgow.)During an investigation into the possible effect of the chemical DES on the development of human cells, a geneticist grew independent samples of human cells in tissue culture at five different dose levels and counted the number of chromosomes in
4.32 The pdf of a stable distribution with fixed parameter α = 1/2 exists in closed form, and is given by f(x;c) = c(2πx3)1/2 exp −c2 2x, for x > 0,c> 0, where c is the only free parameter of the distribution.Write down the likelihood corresponding to a random sample, {xi, i =1,...,n}. If
4.31 (Continuation of Exercise 2.26)An experiment took place in St. Andrews University, in which groups of golf tees were placed randomly in a field, and records were kept of which groups were observed on each of T sampling occasions; see Borchers et al.(2002, p.13). If fi denotes the number of
4.30 (Examination question, University of Glasgow.)An investigation was carried out into the effect of artificial playing pitches on the results of certain professional English soccer teams. In particular, the results for Luton Town on their own pitch over the periods 1980–1985(grass pitch) and
4.29 The two-parameter Cauchy distribution has pdf given by f(x) = βπ{β2 + (x − α)2}, for − ∞
4.28 (Examination question, University of Glasgow.)In an industrial bottling plant, labels are glued to the bottles at the final stage of the process. Unfortunately, the shape of the bottles used in the plant sometimes leads to bottles not being held completely straight by the labelling machine,
4.27 Explain how you would use geometric and beta-geometric models for the time to conception in order to produce statistical tests of the effect of a woman smoking on the probability of conception.
4.26* The data below are taken from Pierce et al. (1979), and summarise the daily mortality in groups of fish subjected to three levels of zinc concentration. Half the fish at each concentration level received one week’s acclimatisation to the test aquaria, and half received two weeks’
4.25 Let x1,...,xn denote a random sample from the Cauchy distribution, with single unknown parameter θ, which has pdf given by f(x) = 1π{1+(x − θ)2}, −∞
4.24 (Examination question, University of Glasgow.)Sharon Kennedy conducted a research project in Trinidad in July 1999 on the occurrence and distribution of larvae of the hoverfly Diptera on the flowering bracts of Heliconia plants. In a small part of her study she observed that on 20 randomly
4.23 Fit a beta-geometric model to the data of Exercise 2.1 on numbers of tests needed by car drivers.
4.22∗ (Morgan and North, 1985.) When quail eggs start to hatch they move through a number of distinct stages in sequence. Consider how you might devise stochastic models for such a system, and use them to analyse the data of Example 1.6.
4.21 Consider how the model for misreporting, in Example 4.2, may be used to estimate the proportion of individuals who do not report the correct time to conception.
4.20 A bivariate parameter vector θ can be assumed to have a known bivariate normal sampling distribution. Explain how you would construct appropriate confidence ellipsoids.
4.18 Consider a stochastic model for avian survival, involving the parameter set (λ, φ1, φa). This model is obtained from setting φ2 = ... = φk = φa in the model of Example 4.6. Construct the cell recovery probabilities for the data from 1955–1958 inclusive from Table 1.5. Deduce the form
4.17 Compare the estimated variance of the estimator of the parameter p in Example 1.1 with the estimated variance of the maximum-likelihood estimator of p.
4.16 (Examination question, Sheffield University.) A random variable X has probability function, pr(X = k) = 1 k!exp(θk − eθ), for k = 0, 1, 2,....Derive the maximum-likelihood estimator of θ in terms of the members of a random sample, (x1,...,xn), of values for X.
4.15 (Examination question, Sheffield University.) Suppose that X1 and X2 are independent random variables with respective exponential distributions, Ex(aλ) and Ex(bλ). Here a = b are known positive constants. Find the maximum-likelihood estimators of λ and λ−1. Compare and contrast the
4.14 The data following are taken from Milicer and Szczotka (1966). For a sample of 3918 Warsaw girls they record whether or not that the girls had reached menarche, which is the start of menstruation. Analyse these data by means of a logit model.
4.13 We have stated that, subject to regularity conditions, evaluated at the maximum-likelihood estimator θˆ, the scores vector is identically zero, that is, U(θˆ) = 0.Provide an example of when this equation does not hold.
4.12 Verify in the model for polyspermy of Chapter 2 that if secondary fertilisations take place at rate μ(t), then the probabilities of any egg having n sperm at time t have the form:p0(t) = exp(−λt), pn(t) = λ t 0 exp(−λx) exp{−M(t − x)}M(t − x)n−1dx(n − 1)! for n ≥ 1, where
4.11 (Examination question, St. Andrews University.) The random variable X has the inverse Gaussian distribution, with pdf:f(x) = λ2πx31 2exp −λ2 xμ2 − 2μ+1 x , 0 ≤ x < ∞, with λ > 0, μ> 0.For the case of fixed λ, show, by calculating the first-order derivative of
4.10∗ Suppose that the probabilities {πi, 1 ≤ i ≤ t} of a multinomial distribution depend on the parameter vector θ = (θ1,...,θd). Show that the expected information matrix for θ is given by:J = mDr−1D, where m is the index of the multinomial distribution, D is a d × t matrix with
4.9∗ A zero-inflated Poisson distribution has the probability function, pr(X = 0) = w + (1 − w)e−λpr(X = k) = (1 − w)e−λλk k! , k = 1, 2,....
4.8 Complete the algebra for the score test of Example 4.10. Apply this test to the data of Exercise 3.10.
4.7 Repeat the model-selection of Table 4.3, using the BIC, rather than AIC.
4.6 Discuss the following extension to the first part of the right hand side of Equation (4.5):a(di) = 1 − [1 + λ2 exp{η(di)}]−1/λ2 , λ2 = 0 1 − exp[− exp{η(di)}] , λ2 = 0, where η(di) = α1+α2 log(di). Show how to obtain the logit model as a special case. Note the relationship
4.5 Produce Matlab code for obtaining the parameter space path of Figure 4.7.
4.4 Read the paper by Venzon and Moolgavkar (1988).
4.3 Explain why the projections of the two-dimensional confidence region of Figure 4.6 are wider than the marginal confidence regions with the same confidence coefficient.
4.3, estimate the correlations between the parameter estimators in each case for the data of Exercise 3.10. Comment on the results and how they tie-in with the contour plots of Figure 4.2.
4.2 For the two alternative parameterisations of the logit model of Example
4.1 Study the Matlab program given below for fitting a digit confusion model to fecundability data. Discuss the logistic transformation of parameters used in the program. Use the fmax function to obtain maximumlikelihood parameter estimators and obtain estimates of standard error and correlation
3.22 Fit appropriate models to the data of Table 1.15.
3.21 (Examination question, University of Nottingham.) An electrical component has failure time given by the gamma, Γ(2, λ), pdf:f(t) = λ2te−λt for t > 0, f(t) = 0 for t ≤ 0, where λ > 0.The corresponding cdf is F(t)=1 − e−λt(1 + λt) for t > 0,= 0 for t ≤ 0.For m components the
3.20 Run the program of Figure 3.2(b), but replacing contour by mesh. Obtain different views of the surface from rotating it, and present the results by using the subplot command.
3.19 Study the Matlab program below, which produces Figure 3.4.% Program to produce the isometric projection and contours% of the Cauchy log-likelihood surface%_________________________________________________________ data=[1.09,-0.23,0.79,2.31,-0.81]; n=length(data);[a,b]=meshgrid(-1:0.01:2.5,
3.18 Run the Matlab program of Figure 3.7, and record the maximumlikelihood estimates of the two parameters. By changing the starting value, verify that the Newton-Raphson method may diverge.
3.17 Verify the expression for the likelihood in Equation (2.5) and use the Matlab routine fminsearch to obtain the maximum-likelihood estimates of φ and p.
3.16∗ Study the Matlab code below, provided by Paul Terrill, to calculate an approximate covariance matrix, following use of the Matlab routine fminsearch.% Code to be added to ‘fminsearch.m’ in order to also% calculate an approximate variance-covariance matrix after minimising% minus a
3.15 Suppose data arose from an exponential distribution, with pdf f(x) = λe−λx, for x ≥ 0 .We have a random sample of values: x1,...,xm, and further (n−m) values(with n>m) which are right-censored at value τ. Thus the censored values are not known exactly — all we know is that they
3.14 Set up m-files to produce the geometric model likelihoods as explained in Section 2.2 for the two sets of fecundability data. Obtain the maximum likelihood estimates of p numerically, using the Matlab routine fminbnd as follows. Suppose one of the negative log-likelihoods is specified in the

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