- In Table S2 of their Appendix, Villa et al. fit the eight-dimensional factorial model host:sex:time to the first principal component values on 3096 lice. Show that this is equivalent to fitting four
- The sex coefficient in Table \(\mathrm{S} 2\) is -2.437 . Which combination of the four \(\alpha\)-values in the previous exercise does this correspond to? MS Host 12 Aviary 290 Lineage 83 Residual
- The host coefficient in Table S2 is 0.449 with standard error 0.159 . What does this imply about the average or expected baseline values for the four subgroups? MS Host 12 Aviary 290 Lineage 83
- A variety of other smoothing techniques can be employed to illustrate longterm secular trends. Pick your favourite kernel density smoother, apply it to the temperature series, and compare the fitted
- Use the profile log likelihood plot in the previous exercise to obtain a nominal 95\% confidence interval for \(\tau\).Data From previous exercise Section 5.4.2 For the model with persistent aviary
- Virtual randomization requires the timezero average for feral hosts to be the same as that for giant runts, but the temporal trends are otherwise unconstrained. It appears that the model matrix
- Use the fitted model from the previous exercise to compute the linear trend coefficient\[\frac{\sum t\left(\hat{\gamma}_{1}(t)-\hat{\gamma}_{0}(t)ight)}{\sum t^{2}}\]and its standard error. You
- For the non-seasonal frequencies, use \(g \operatorname{lm}()\) to fit the additive exponential model\[E\left(|\hat{Y}(\omega)|^{2}ight)=\beta_{0}+\beta_{1} \exp \left(-|2 \pi \lambda \omega|^{1 /
- To the order of approximation used in Sect. 11.6, show that the maximumlikelihood estimate, \(\hat{\alpha}\left(T_{n}ight)\), of the species-diversity parameter as a function of the cumulative
- Download the data, compute the averages at each time point for the two pigeon breeds, and reconstruct the plots in Figs. 5.1 and 5.2. 7.82 7.81 7.80 7.79 7.78 5.69 5.67 5.65 5.63 0 Log body length
- The coefficient of variation is the standard-deviation-to-mean ratio, which is often reported as a percentage. For body length or other size variables, the coefficient of variation is essentially the
- Use anova (. . . ) to re-compute the mean squares in Table 5.2. Use Bartlett's statistic (Exercise 18.9) to test the hypothesis that the residual mean squares have the same expected value at all time
- For the model (5.3), what is the expected value of the within-lineage mean square at time \(t\) ? For the Brownian-motion model (5.4), show that the variance of \(Y_{u}\) increases linearly with
- Use lmer (. . . ) to fit the variance-components model (5.3) to the log body length with (5.2) as the mean-value subspace. Report the two slopes, the slope difference, and the three standard errors.
- Compute the four covariance matrices \(V_{0}, \ldots, V_{3}\) that occur in (5.4). Let \(Q\) be the ordinary least-squares projection with kernel (5.2). Compute the four quadratic forms \(Y^{\prime}
- For \(n=100\) points \(t_{1}, \ldots, t_{n}\) equally spaced in the interval \((0,48)\), compute the matrix\[\Sigma_{i j}=\delta_{i j}+\theta\left(t_{i} \wedge t_{j}ight)\]for small values of
- Distributional invariance. Consider a simplified version of the louse model in which there are 16 feral and 16 giant runt pigeons, no sex differences between lice, and no correlations among
- The model in the previous two exercises has a baseline variance that is larger than the non-baseline residual variance. What is the ratio of fitted variances?
- The fact that measured lice were not returned to their hosts is an interference in the system that may reduce or eliminate temporal correlations that would otherwise be expected. One mathematically
- Construct two versions of Table 5.4, one based on the modified block-factor model, and one based on the combined variance model that includes both. Comment on any major discrepancy or difference in
- Let \(0
- A vector \(x \in \mathbb{R}^{N}\) may be regarded as a function \([N] ightarrow \mathbb{R}\), in which case the composition \(x \varphi\) is a function \([n] ightarrow \mathbb{R}\) or a vector in
- The U.K. Met Office maintains a longer record of monthly average and annual average temperatures for Central England from 1659 onwards in the
- The U.K. Met Office site https://www.metoffice.gov.uk/ keeps long-term weather records-temperature, rainfall, and so on-for a range of stations in Great Britain and Northern Ireland. Monthly rainfall
- Let \(\left(\varepsilon_{k}, \varepsilon_{k}^{\prime}ight)_{k \geq 0}\) be independent and identically distributed standard Gaussian variables. For real coefficients \(\sigma_{k}\), show that the
- Verify the following trigonometric integral for integer \(k\) :\[\int_{0}^{2 \pi} \sin (x / 2) \cos (k x) d x=\frac{-4}{4 k^{2}-1} .\]Hence find the coefficients \(\lambda_{k}\) in the Fourier
- Simulate and plot a random function \(\eta(\cdot)\) on \((0,2 \pi)\) whose covariance is \(\pi / 2-\ell\left(t, t^{\prime}ight)\), where \(\ell(\cdot)\) is the arc-length metric. This function is
- A real-valued process \(Y(\cdot)\) is called a Gaussian random affine function if the differences \(Y(t)-Y\left(t^{\prime}ight)\) are Gaussian with covariances
- Let \(K\) be the covariance function of a stationary process on the real line such that \(K\) is twice differentiable on the diagonal, i.e.,\[K\left(t,
- Let \(K\left(t, t^{\prime}ight)=\sigma^{2} \exp \left(-\left|t-t^{\prime}ight| / \lambdaight)\) be the scaled exponential covariance, and let \(Y\) be a zero-mean Gaussian process with covariance
- Let \(Y_{1}, \ldots, Y_{n}\) be independent and identically distributed standard exponential variables, and let \(0 \leq Y_{(1)} \leq Y_{(2)} \leq \cdots \leq Y_{(n)}\) be the order statistics. Show
- Use fft () to compute the Fourier coefficients for the temperature series on a whole number of years, identify and remove the frequencies that are seasonal, average the power-spectrum values in
- Include the inverse-square frequency as an additional covariate in the exponential model for the power spectrum. In principle, this means re-computing \(\hat{\lambda}\). Compute the Wilks statistic,
- Ordinarily, Wald's likelihood ratio statistic is essentially the same as Wilks's statistic, which in one-parameter problems, is the squared ratio of the estimate to its standard error. But there are
- For \(0
- Use the function \(\mathrm{COv}(\mathrm{cbind}(\mathrm{S} 1[\ldots])\) ) to compute the sample covariance matrix of the four phoneme inventory variables. What does this tell you?
- Use the function \(q r(\operatorname{cbind}(S 1[\ldots]))\) \$rank to deduce that total phoneme diversity is a linear combination of the three constituents. Find the coefficient vector.
- The function vdist \((x 0,1)\) returns a list of linguistic distances from the designated point \(\mathrm{x} 0\) in linguistic region 1 to each of the 504 language locations. Show that Atkinson's
- Assuming the Out-of-Africa hypothesis, total phoneme inventory necessarily depends not just on distance to the origin but also on the speaker population size. By minimizing the residual sum of
- Which language has the greatest vowel inventory in the Santoso compilation, and which has the least? Which language has the greatest consonant inventory, and which has the least?
- Expand the spreadsheet so that it is indexed by bees rather than by colonies. Check that the two versions of the lmer () code produce the same output for the expanded spreadsheet as they do for the
- Plot the average reproductive score against calendar year. Is the range of annual averages high or low in relation to the reproductive scale \(0-4\) ? Does this plot suggest serial correlation?
- Each bird in this study has a sequence length in the range \(0-\mathrm{xx}\). Compute the histogram of sequence lengths. How many sequences are empty? Report the average and the maximum length? What
- Show that the model (10.1) implies exchangeability of initial values \(Y_{i, 1} \sim\) \(Y_{j, 1}\) for every pair of birds, whether recorded in the same year or in different years.
- What evidence is there in the data suggesting serial correlation in the year effects? Can the fit be improved using a model containing non-trivial serial correlation? Extend the model and report a
- What evidence is there in the data suggesting serial correlation in the year effects? Can the fit be improved using a model containing non-trivial serial correlation? Extend the model and report a
- The black-footed ferret is an endangered species; it belongs to the weasel family. A ferret-breeding program has been established by various zoos throughout the United States to study the factors
- For integer \(n \geq 1\), a partition \(B\) of the set \([n]=\{1, \ldots, n\}\) is a set of disjoint non-empty subsets called blocks whose union is \([n]\). A partition into \(k\) blocks, can be
- One of the simplest static versions of the Ewens sampling formula is stated as a probability distribution on the set of partitions of the finite set \([n]\) as follows:\[P_{n,
- By direct calculation, show that the Ewens distributions satisfy the following conditions:Show that \(P_{4, \alpha}\) is the marginal distribution of \(P_{5, \alpha}\) when the element 5 is removed
- Let \(B \sim P_{n, \alpha}\) be the partition after \(n\) customers in the Chinese restaurant process with parameter \(\alpha\), and let \(\hat{\alpha}(B)\) be the maximum-likelihood estimate. One
- In the non-parametric bootstrap, the configuration \(B\) is regarded as a list of \(n\) tables in order of occupation. Each non-parametric bootstrap sample is a sequence of \(n\) tables drawn with
- Each land parcel belongs to the first or inner ring, the second ring, the third ring, or beyond. To be clear, the rings are disjoint, so the phrase 'second ring' excludes the first. A district may
- For non-commercial sales, average sale price per square metre is recorded quarterly for each parcel. Discuss briefly how you might go about constructing a sampling-consistent Gaussian model that
- Consider a statistical model for a competition experiment in which each observational unit is an ordered pair \((i, j)\) of distinct competitors (chess players). The state space consists of three
- For the Ewens distribution (12.5), show that the conditional distribution given \(\# B=k\) is\[p_{\theta}(B \mid \# B=k)=\frac{\prod_{b \in B} \Gamma(\# b)}{s_{n, k}}\]where \(s_{n, k}\) is
- Let \(n=12\), and let \(B\) have the Ewens distribution with parameter \(\theta>0\). Suppose \(B\) has six blocks. Which is more likely: (a) that \(B\) has all blocks of size two; (b) that \(B\) has
- Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density\[\frac{1}{2 \pi}(1+\psi \cos y)\]on the interval \(-\pi
- Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density\[\frac{1}{2 \pi}(1+\psi \cos y)(1+\sin y / 2)\]on the interval \(-\pi
- Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density\[\frac{1}{2 \pi}(1+\psi \cos y)(1+\lambda \sin y)\]on the interval \(-\pi
- By definition, the randomization protocol is a known distribution on treatment assignments. In this context, 'known' means declared at baseline and \(P_{\theta}(\mathbf{t})=\)
- Let \(X_{0}, X_{1}\) be given matrices of order \(100 \times 5\) and \(110 \times 5\) such that \(X_{0}^{\prime} X_{0}=\) \(X_{1}^{\prime} X_{1}=F\), and let \(P_{\beta}\) be the Gaussian mixture
- This exercise is concerned with a possible action of the additive group of real numbers on the space of positive definite matrices of order \(n\). Let \(\mathcal{X} \subset \mathbb{R}^{n}\) be a
- Show that the same triple is sufficient for the six-parameter random coefficient model (14.1) with one block. Deduce that the likelihood is maximized at the boundary point
- Let \(\Theta\) be the extended complex plane. For each \(\theta=\theta_{0}+i \theta_{1}\) let \(P_{\theta}\) be the distribution on the extended real line with density\[P_{\theta}(d
- Let \(\Theta=\mathbb{R}^{2}\), and let \(P_{\theta}\) be the von Mises-Fisher distribution on the unit circle with density\[P_{\theta}(d \phi)=\frac{e^{\theta^{\prime} y} d
- In Exercise 14.10, the null hypothesis of no treatment effect \(H_{0}: \tau=0\) is the left endpoint of the parameter interval \(\tau \in[0,2 \pi)\). Explain why this is not a boundary point in the
- Find the survival distribution \(P\) associated with the hazard measure\[\theta(d t)=\left\{\begin{array}{cc}d t /(1-t) & 0
- Show that the set of \(2 n \times 2 n\) real matrices of the form\[\left(\begin{array}{rr}A & B \\-B & A\end{array}ight)\]is closed under matrix addition and multiplication. Show also that the
- Let \(A+i B\) be a full-rank Hermitian matrix of order \(n\). Show that the inverse matrix \(C+i D\) is also Hermitian and satisfies the pair of equations\[A D+B C=0 ; \quad A C-B D=I_{n} .\]Deduce
- Deduce that the linear transformation \(Y \mapsto L Y=\hat{\mu}=X \hat{\beta}\) is a projection \(\mathcal{H} ightarrow \mathcal{H}\), but not an orthogonal projection unless \(Q
- For the complementary projection \(P=I_{n}-Q\) whose image is \(\mathcal{K}\), deduce that the composite linear transformation \(Y \mapsto L_{0} Y=P \hat{\mu}\) is nilpotent, i.e., that
- Show that the least-squares estimate of the conditional distribution of \(Y\) given \(Z\) is\[N_{n}\left(Q Y+P \hat{\mu}, s^{2} P Vight)\]for some scalar \(s^{2}\). Show that the least-squares
- Suppose that \(X\) is uniformly distributed on the surface of the unit sphere in \(\mathbb{R}^{d}\), and that \(Y \sim N\left(X, \sigma^{2} I_{d}ight)\) is observed. Show that Eddington's formula
- Suppose that \(X\) is uniformly distributed on the interior of the unit sphere in \(\mathbb{R}^{d}\), and that \(Y \sim N\left(X, \sigma^{2} I_{d}ight)\) is observed. Show that Eddington's formula is
- Show that the \(3 \times 3\) Hermitian matrix\[\left(\begin{array}{lll}1 & ho & \bar{ho} \\\bar{ho} & 1 & ho \\ho & \bar{ho} & 1\end{array}ight)\]has determinant \(1-3|ho|^{2}+2
- By making the transformation \(u=1 /\left(1+x^{2}ight)\) and converting to a beta-type integral on \((0,1)\), show that\[2 \int_{0}^{\infty} \frac{x^{d-1} d x}{\left(1+x^{2}ight)^{v+d / 2}}=B(v, d /
- The Matérn spectral measure on the real line is proportional to the symmetric type IV distribution in the Pearson class, which is also equivalent to the Student \(t\) family (Pearson type VII). For
- By transforming to spherical polar coordinates in \(\mathbb{R}^{d}\), show that\[\int_{\mathbb{R}^{d}} \frac{d \omega}{\left(1+\|\omega\|^{2}ight)^{v+d / 2}}=A_{d-1} \int_{0}^{\infty} \frac{x^{d-1} d
- For any linear functional x:Rd→Rx:Rd→R, show
- Use integration by parts to show that\[\begin{aligned}\int_{-\infty}^{\infty} \frac{\omega \sin (t \omega) d \omega}{\left(1+\omega^{2}ight)^{v+3 / 2}} & =\frac{t}{2 v+1} \int_{-\infty}^{\infty}
- Let \(Z\) be a real Gaussian space-time process with zero mean and full-rank separable covariance function:\[\operatorname{cov}\left(Z(x, t), Z\left(x^{\prime}, t^{\prime}ight)ight)=K\left(x,
- This exercise is concerned with stereographic projection from the unit sphere in \(\mathbb{R}^{d+1}\) onto the equatorial plane \(\mathbb{R}^{d}\). Latitude on the sphere is measured by the polar
- Points near the north pole are transformed stereographically to high frequencies, and points near the south pole to low frequencies. For \(v>d / 2\), the weighted distribution with density
- For the special case \(d=2\), we may regard \(\mathbb{R}^{2} \cong \mathbb{C}\), so that \(\omega\) is a complex number and \(e\) is a unit complex number. Show that the weighted spherical
- Consider a fixed tessellation of the plane into a countable set of polygonal cells \(A_{1}, \ldots\), and let \(0 \leq \ell_{i j}
- Let \(Y\) be a stationary real-valued Gaussian process on \(\mathbb{R}^{d}\) with isotropic covariance function \(\exp \left(-\left\|x-x^{\prime}ight\|^{2} / 2ight)\). Show that the gradient field
- Under what conditions on \(ho\) is the special Gaussian process stationary on \(\mathbb{Z}\) \((\bmod 3) ?\)
- For each \(v>0\) and \(\omega \in \mathbb{R}\), the Matérn function \(M_{v}\left(\left\|t-t^{\prime}ight\|ight) e^{i \omega\left(t-t^{\prime}ight)}\) defines a stationary complex Gaussian process on
- For each \(ho \in \mathbb{R}^{3}\) such that \(\|ho\| \leq 1\), deduce that the following symmetric functions are positive definite on \(\mathbb{R} \times[3]\) :\[\begin{aligned}&
- For each \(v>0\) and \(\omega \in \mathbb{R}^{3}\), the Matérn function \(M_{v}(\|x-x\|) e^{i \omega^{\prime}\left(x-x^{\prime}ight)}\) defines a stationary complex Gaussian process on
- For each \((\omega, ho)\), deduce that the matrix-valued function\[M_{v}\left(\left\|x-x^{\prime}ight\|ight)\left(\cos \left(\omega^{\prime}\left(x-x^{\prime}ight)ight) I_{3}-\sin
- The parameters \(\omega, ho\) of the Gaussian process are two points in \(\mathbb{R}^{3}\), which determine the frequency and direction of spatial anisotropies in the given frame of reference. In the
- For each \(\alpha \in \mathbb{R}\) and \(\omega, ho \in \mathbb{R}^{3}\) with \(\|ho\| \leq 1\), deduce that the matrix-valued function\[\begin{aligned}M_{v}\left(\left\|x-x^{\prime}ight\|ight)
- Let \(x=\left(x_{0}, x_{1}, x_{2}, x_{3}ight)\) be a unit vector in \(\mathbb{R}^{4}\), let \(v=\left(x_{1}, x_{2}, x_{3}ight)\) and let \(\chi(v)\) be the \(3 \times 3\) matrix in (16.25). Show that
- A quaternion is a formal linear combination \(q=q_{0}+q_{1} \mathbf{i}+q_{2} \mathbf{j}+q_{3} \mathbf{k}\) of four basis elements \(\{1, \mathbf{i}, \mathbf{j}, \mathbf{k}\}\) with real coefficients,
- The conjugate quaternion is \(\bar{q}=q_{0}-q_{1} \mathbf{i}-q_{2} \mathbf{j}-q_{3} \mathbf{k}\), so \(q=\bar{q}\) means that \(q\) is real, and \(q=-\bar{q}\) means that \(q\) is purely imaginary.
- Show that \(|p q|=|p| \times|q|\), i.e., that the modulus of a product is the product of the moduli.
- A quaternion of modulus one is called a unit quaternion. Show that the set of unit quaternions is a group containing the finite sub-group \(\{ \pm 1, \pm \mathbf{i}, \pm \mathbf{j}, \pm
- Show that the \(4 \times 4\) matrices \(e_{0}=I_{4}\),\[e_{1}=\left(\begin{array}{rrrr}0 & -1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 0 & -1 \\0 & 0 & 1 & 0\end{array}ight), \quad