Questions and Answers of Applied Statistics And Probability For Engineers

Repeat the preceding exercise taking $\mathcal{X}=\mathbf{1}$, and $\Sigma$ a linear combination of the block matrices $I_{n}$, row and col.
In the balanced case with no missing cells, the standard analysis first reduces the data to 24 rat averages $\bar{Y}_{i .}$, the treatment and control averages $\bar{Y}_{T}, \bar{Y}_{C}$, and the
The $F$-ratio reported by anova (...) for treatment effects is not the ratio shown above. At least one is misleading for this design. Which one? Explain your reasoning.
REML, or residual maximum likelihood, is the standard method for the estimation of variance components: see Chap. 18 for details. Both regress () and lmer () allow other options, but both use REML as
A quantitative factor $x$ with four equally-spaced levels $0,1,2,3$ may be coded using either the indicator basis $e_{0}, e_{1}, e_{2}, e_{3}$ (such that $e_{r}(i)=I\left(x_{i}=right)$ ) or
Parameter estimates reported in Sect. 1.5 were computed using the code in Exercise 1.4. Following recommendations in Sect. 18.5, the likelihood ratio statistic for treatment effects was computed
It is possible that treatment could have an effect on variances in addition to its effect on the mean. Investigate this possibility by replacing the identity matrix with two diagonal matrices
The set of linear functionals $\mathbb{R}^{n m} ightarrow \mathbb{R}$ is called the dual vector space; it has dimension $m n$. Show that the column and row totals $Y \mapsto Y_{. r}$ and \(Y
Use the averages for the six saws A-F\[2.122,2.060,1.975,2.070,2.156,1.920\]to compute the brand sum of squares on two degrees of freedom, the saw replicate sum of squares on three degrees of
In the simple linear model setting, the $F$-ratio for testing the hypothesis $\mu \in \mathcal{X}_{0}$ versus $\mu \in \mathcal{X}_{1}$ is the ratio of mean squares\[F=\frac{\left\|Q_{0}
For a Latin-square design of order $m$, show that the last term in the preceding decomposition can be split into two parts associated with letters. Show also that the five-part decomposition is
Hypergeometric simulation in Sect. 3.3.5 implies a symmetric null distribution with standard deviation 0.23 for the weighted sample correlation of homogamic pairs. One suggested alternative to random
The advice sometimes given for the validity of the $\chi^{2}$ approximation to the null distribution of Pearson's statistic is that the minimum expected value should exceed a suitable threshold,
Check the calculations reported in the penultimate paragraph of Sect. 3.4.2 for Bortkewitsch's horsekick data. Compute the row and column totals, and simulate the null distribution of X2X2 by random
Explain where the factor $1-r$ comes from in the penultimate paragraph of Sect. 3.3.5.
Bearing in mind that the heights are measured to the nearest millimetre, comment briefly on the magnitude of the estimated variance components for the FBM model.
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

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