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applied statistics and probability for engineers

Questions and Answers of Applied Statistics And Probability For Engineers

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
Let \(p\) be an arbitrary quaternion and let \(q\) be a unit quaternion. Show that the real part of the product \(q p \bar{q}\) is equal to \(\Re(p)\). Show also that the imaginary part of the
Maximum-likelihood for mixtures: Let \(\psi_{0}(\cdot), \ldots, \psi_{k}(\cdot)\) be given probability density functions on \(\mathbb{R}\), and let\[m_{\theta}(y)=\theta_{0}
If the claim made in the last paragraph of section 17.5.2 is to be believed, the re-scaled limit distribution of \(\bar{X}_{n}\) does not have a mean. Discuss this apparent contradiction.
Consider the two-component mixture with \(\psi_{0}\) standard normal, and \(\psi_{1}\) standard Cauchy. The null hypothesis is all Gaussian, i.e., \(\theta=(1,0)\). Show that \(\hat{\theta}_{1}>0\)
Explain why the observation \(P_{0}\left(\bar{X}_{n}>1ight) ightarrow 0\) as \(n ightarrow \infty\) deduced from simulations does not conflict with the law of large numbers \(\bar{X}_{n} ightarrow
Consider the two-component mixture with \(\psi_{0}\) standard normal and \(\psi_{1}\) standard Laplace, or double exponential. Investigate the behaviour of \(P_{0}\left(\bar{X}_{n}>1ight)\) as a
Sparse signal detection. Suppose that the observation \(Y=X+\varepsilon\) is the sum of a signal \(X\) plus independent Gaussian noise \(\varepsilon \sim N(0,1)\). For any signal distribution \(X
Suppose that \(Y_{1}, \ldots, Y_{n}\) are independent and identically distributed with density \(m(y)\). Ignoring the error term, show that the maximum-likelihood estimate of the mixture fraction is
A sequence \(\epsilon_{v} ightarrow 0\) such that \(P_{u}\left(|X|\epsilon_{u} \mid Yight)=\frac{ho \zeta(y)}{1-ho+ho \zeta(y)}+o(1)\]which implies that the 'true discovery rate' is essentially
Let \(\kappa_{0}=ho \zeta(y) /(1-ho+ho \zeta(y))\) be the exceedance probability, and let \(\kappa_{r}\) be the \(r\) th derivative of \(\log (1-ho+ho \zeta(y))\). For \(\zeta(y) \simeq e^{y^{2} / 2}
For \(1 \leq i \leq k\), suppose \(Y_{i}=\alpha_{i}+\epsilon_{i}\), where \(\epsilon_{1}, \ldots, \epsilon_{k}\) are independent standard normal variables, and \(\alpha_{1}, \ldots, \alpha_{k}\) are
Consider a balanced block design having \(m\) blocks each consisting of \(b\) observational units, and let \(B\) be the associated block factor as a Boolean matrix of order \(n=m b\). The
For the balanced block design, show that the log determinant is\[\log \operatorname{det}(\Sigma)=n \log \left(\sigma^{2}ight)+m \log (1+b \theta)\]Show that the ML estimate satisfies \(1+b
The following exercise is concerned with the distribution of the likelihoodratio statistic in a 'fixed-effects' model for a balanced design, where Σ=σ2InΣ=σ2In, and either μ∈1nμ∈1n under
The null hypothesis being tested in Exercise 18.5 is the same as that in Exercise 18.3, but the alternatives are different: one implies exchangeability of block effects, the other does not. Discuss
Positive definiteness of a function \(\mathbb{R}^{d} \times \mathbb{R}^{d} ightarrow \mathbb{R}\) means that, for integer \(n \geq 1\) and each finite collection of points \(\mathbf{x}=\left\{x_{1},
Wilson-Hilferty transformation: Show that the \(r\) th cumulant of the exponential distribution is \(\kappa_{r}=\kappa_{1}(r-1)\) !, and hence that \(Y^{1 / 3}\) is approximately symmetrically
Show that the \(r\) th cumulant of the Poisson distribution is \(\kappa_{r}=\kappa_{1}\), and hence that \(Y^{2 / 3}\) is approximately symmetrically distributed.
For \(\tau>0\), show that the modified transformation \(\mathbb{R}_{+}^{n} ightarrow \mathbb{R}^{n}\)\[(g y)_{i}=\dot{y}+\frac{y_{i}^{\lambda}-\dot{y}^{\lambda}}{\lambda \dot{y}^{\lambda-1}}\]is
For which values of \(\alpha\) is the transformation\[g_{\alpha}(x)=\frac{(-\log (1-x))^{\alpha}}{\alpha}-\frac{(-\log (x))^{\alpha}}{\alpha}\]differentiable and strictly monotone \((0,1) ightarrow
The verb to write has a subject, a direct object and an indirect object. Some of the parts may be empty or missing. Identify the three parts in the following sentences (i) Joe wrote a letter to Anne;
An involution is a transformation \(f\) that is self-inverse, i.e. \(f(f(x))=x\). Show that the verbs to substitute and to replace are both transformations, and that the relation between them is an
Let \(A, B\) be two groups. What are the relations between \(x, x^{\prime}, x^{\prime \prime}\) in \(A\) that are preserved by a group homomorphism \(h: A ightarrow B\) ?
Let \(A, B\) be two vector spaces, i.e., commutative groups with additional structure. What is the additional structure? What are the additional relations between \(x, x^{\prime}, x^{\prime \prime}\)
Discuss whether or not the mapping \(x \mapsto\|x\|^{2}\) (or its inverse image) is a homomorphism\[\left(\mathbb{R}^{d}, \mathcal{B}\left(\mathbb{R}^{d}ight), N\left(0, I_{d}ight)ight)
To compute the control and treatment averages, the \(\mathrm{R}\) command tapply( \(\log (y)\), treat, mean)returns the pair of averages 3.360, 3.085, which is not the pair reported in the text. The
For the model (1.2), verify that the \(\mathrm{R}\) commandsreturn the same parameter estimates and the same standard errors in a slightly different format. reg_fit
The sub-model with zero rat variance can be fitted by omitting the relevant term from the model formula in either syntax. The conventional log likelihood ratio statistic is twice the increase in log
In the balanced case with no missing cells, show that the REML likelihood-ratio statistic for treatment effects is\[\operatorname{LLR}=(n-1) \log (1+F /(n-2)) \text {, }\]where \(n=24\) is the number
Use polynomials up to degree four to re-parameterize the site effects, and repeat the fitting procedure for (1.2) using the unnormalized orthogonal polynomial basis. Check that the treatment effect
How can we be assured that the log transformation is really needed or substantially beneficial? (Chap. 19).
Extend the model (1.2) so that it contains one variance component for treated rats and another for untreated rats. Show your code for fitting the extended model, report the two fitted variance
It is mathematically possible that treatment could have a positive effect at some sites and a negative effect at other sites, so that the average over sites is negligible. Investigate this
Let YY be an n×mn×m array of real-valued random variables with zero mean and covariance matrixfor some non-negative coefficients σ20,…,σ23σ02,…,σ32. Show that the covariance matrix is
For an \(n \times m\) array of real numbers, show that the four quadratic forms, \(m n \bar{Y}_{. .}^{2}\),are invariant with respect to row and column permutations. Here, \(Y_{i \text {. }}\) is the
Each of these quadratic forms is non-negative definite. In each case, the expected value is a non-negative linear combination of the four variance components, in which the coefficient of
Let 1n1n be the vector in RnRn whose components are all one. Show that Jn=Jn= 1n1′n/n1n1n′/n is a projection matrix, i.e., that J2n=JnJn2=Jn, and that it has rank one: tr(Jn) = 1Extra \left or
The space of quadratic forms in the \(n \times m\) array \(Y\) is a vector space of dimension \(m n(m n-1) / 2\). Exhibit a basis. Show that the four quadratic forms in Exercise 1.3 are invariant
Suppose that intact logs are numbered 1:36, and that species is the species factor. Write code in \(\mathrm{R}\) that picks uniformly at random a subset of six logs of each species for debarking, and
The R commands are designed to decompose the total sum of squares additively into components associated with certain subspaces, which are mutually orthogonal for this design. Explain how to compute
Use the method described by Welham and Thompson (1997) to compute the REML likelihood-ratio statistic for comparing the two linear models\[\mathcal{X}_{0}=\text { species }+ \text { bark }+ \text {
In the simple linear model setting with \(\mu \in \mathcal{X}\) and \(\Sigma \propto I_{n}\), show that the maximum value of the \(\log\) likelihood is const \(-n \log \|Q Y\|\), where \(Q=I-P\) is
Check that the \(F\)-ratio for brand differences is in approximate agreement with the Welham-Thompson REML statistic computed in Exercise 2.4. Explain why you need \(m=6-1\) rather than \(m=36-9\) in
Express the random-effects models from the previous section in lmer() syntax, and check that the parameter estimates agree with regress ( ) output.
For \(y \in \mathbb{R}^{n}\), a decomposition of the total sum of squares\[\|y\|^{2}=y^{\prime} y=y^{\prime} A_{1} y+\cdots+y^{\prime} A_{k} y\]is called orthogonal if each \(A_{r}\) is an orthogonal
For a \(m \times n\) array, i.e., for \(y \in \mathbb{R}^{m n}\), show that the Kronecker-product matrices\[J_{m} \otimes J_{n}, \quad J_{m} \otimes\left(I_{n}-J_{n}ight), \quad\left(I_{m}-J_{m}ight)
Use the normal approximation to the binomial to compute the probability that the horizontal line in Fig. 3.1 intersects all 18 whiskers at \(\pm 1\) standard deviations. Devise a better approximation
Is the total number of matings in Table 3.1 related to the number of mating wells? Is the pattern of variation different for the experiments reported in the last three rows? Explain how you address
For the experiment giving rise to the data in Table 3.2, an algebraically natural assumption is that the allowable double matings occur as a Poisson process at a rate proportional to the product of
The file ...birth-death.R contains the data compiled by Phillips and Feldman (1973) on the month of birth and the month of death of 348 'famous Americans'. Investigate whether the month of death is
What was the matter that Lord Denning refused to accept in his 1980 appealscourt judgement when he referred so melodramatically to the 'appalling vista that every sensible person would reject'? Why
In the inverse quadratic model, the height of plant \(i\) at age \(t\) is Gaussian with mean \(\beta_{s(i)} h(t)\) whose limit as \(t ightarrow \infty\) is \(\beta_{s(i)}\). What is the variance of
For the inverse linear model in which brearding is deemed to have occurred two days prior to the first positive measurement, estimate \(\tau\) together with the plateau coefficients. Obtain the
The Brownian motion component of the model can be replaced with fractional Brownian motion with parameter \(0
In the fractional Brownian model with \(v
For 1000 equally spaced \(t\)-values in \((0,10]\) compute the FBM covariance matrix \(K\) and its Choleski factorization \(K=L^{\prime} L\). (If \(t=0\) is included, \(K\) is rank deficient, and the
Several plants reach their plateau well before the end of the observation period. How is the analysis affected if repeated values are removed from the end of each series?
Explain the purpose and the implementation of the Box-Tidwell method. Why must the unmodified REML criterion be avoided?
Investigate the relation between brearding date and ultimate plant height. Is it the case that early-sprouting plants tend to be taller than late-sprouting plants?

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