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statistical techniques in business
Questions and Answers of
Statistical Techniques in Business
Consider the ellipsoid \(E=\left\{\boldsymbol{x} \in \mathbb{R}^{d}: x \boldsymbol{\Sigma}^{-1} \boldsymbol{x}=1\right\}\) in (4.42). Let \(\mathbf{U D}^{2} \mathbf{U}^{\top}\) be an SVD of
Figure 4.13 shows how the centered "surfboard" data are projected onto the first column of the principal component matrix \(\mathbf{U}\). Suppose we project the data instead onto the plane spanned by
Figure 4.14 suggests that we can assign each feature vector \(\boldsymbol{x}\) in the iris data set to one of two clusters, based on the value of \(\boldsymbol{u}_{1}^{\top} \boldsymbol{x}\), where
We can modify the Box-Muller method in Example 3.1 to draw \(X\) and \(Y\) uniformly on the unit disc, \(\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leqslant 1\right\}\), in the following way:
A simple acceptance-rejection method to simulate a vector \(\boldsymbol{X}\) in the unit \(d\)-ball \(\left\{\boldsymbol{x} \in \mathbb{R}^{d}\right.\) : \(\|x\| \leqslant 1\}\) is to first generate
Let the random variable \(X\) have pdf\[ f(x)= \begin{cases}\frac{1}{2} x, & 0 \leqslant x
Construct simulation algorithms for the following distributions:(a) The weib(a, \(\lambda\) ) distribution, with cdf \(F(x)=1-\mathrm{e}^{-(\lambda x) \alpha}, x \geqslant 0\), where \(\lambda>0\)
We wish to sample from the pdf\[ f(x)=x \mathrm{e}^{-x}, \quad x \geqslant 0 \]using acceptance-rejection with the proposal pdf \(g(x)=e^{-x / 2} / 2, x \geqslant 0\).(a) Find the smallest \(C\)
Let \([X, Y]^{\top}\) be uniformly distributed on the triangle with corners \((0,0),(1,2)\), and \((-1,1)\). Give the distribution of \([U, V]^{\top}\) defined by the linear transformation\[
Explain how to generate a random variable from the extreme value distribution, which has cdf\[ F(x)=1-\mathrm{e}^{-\exp \left(\frac{x-\mu}{\sigma}\right)}, \quad-\infty
Write a program that generates and displays 100 random vectors that are uniformly distributed within the ellipse\[ 5 x^{2}+21 x y+25 y^{2}=9 \][Hint: Consider generating uniformly distributed
Suppose that \(X_{i} \sim \operatorname{Exp}\left(\lambda_{i}\right)\), independently, for all \(i=1, \ldots, n\). Let \(\boldsymbol{\Pi}=\left[\Pi_{1}, \ldots, \Pi_{n}\right]^{\top}\) be the random
Consider the Markov chain with transition graph given in Figure 3.17, starting in state 1.(a) Construct a computer program to simulate the Markov chain, and show a realization for \(N=100\) steps.(b)
As a generalization of Example C.9, consider a random walk on an arbitrary undirected connected graph with a finite vertex set \(\mathscr{V}\). For any vertex \(v \in \mathscr{V}\), let \(d(v)\) be
Let \(U, V \sim_{\text {iid }} \mathscr{U}(0,1)\). The reason why in Example 3.7 the sample mean and sample median behave very differently is that \(\mathbb{E}[U / V]=\infty\), while the median of
Consider the problem of generating samples from \(Y \sim \operatorname{Gamma}(2,10)\).(a) Direct simulation: Let \(U_{1}, U_{2} \sim\) idd \(\mathscr{U}(0,1)\). Show that \(-\ln \left(U_{1}\right) /
Let \(\boldsymbol{X}=[X, Y]^{\top}\) be a random column vector with a bivariate normal distribution with expectation vector \(\boldsymbol{\mu}=[1,2]^{\top}\) and covariance matrix\[
Here the objective is to sample from the 2-dimensional pdf\[ f(x, y)=c \mathrm{e}^{-(x y+x+y)}, \quad x \geqslant 0, \quad y \geqslant 0 \]for some normalization constant \(c\), using a Gibbs
We wish to estimate \(\mu=\int_{-2}^{2} \mathrm{e}^{-x^{2} / 2} \mathrm{~d} x=\int H(x) f(x) \mathrm{d} x\) via Monte Carlo simulation using two different approaches: (1) defining \(H(x)=4
Consider estimation of the tail probability \(\mu=\mathbb{P}[X \geqslant \gamma]\) of some random variable \(X\), where \(\gamma\) is large. The crude Monte Carlo estimator of \(\mu\) is\[
One of the test cases in [70] involves the minimization of the Hougen function. Implement a cross-entropy and a simulated annealing algorithm to carry out this optimization task.
In the binary knapsack problem, the goal is to solve the optimization problem:\[ \max _{\boldsymbol{x} \in\{0,1\}^{n}} \boldsymbol{p}^{\top} \boldsymbol{x} \]subject to the constraints \[
Let \(\left(C_{1}, R_{1}\right),\left(C_{2}, R_{2}\right), \ldots\) be a renewal reward process, with \(\mathbb{E} R_{1}
Prove Theorem 3.3
Prove that if \(H(\mathbf{x}) \geqslant 0\) the importance sampling pdf \(g^{*}\) in (3.22) gives the zero-variance importance sampling estimator \(\widehat{\mu}=\mu\).
Let \(X\) and \(Y\) be random variables (not necessarily independent) and suppose we wish to estimate the expected difference \(\mu=\mathbb{E}[X-Y]=\mathbb{E} X-\mathbb{E} Y\).(a) Show that if \(X\)
Suppose that the loss function is the piecewise linear function\[ \operatorname{Loss}(y, \hat{y})=\alpha(\hat{y}-y)_{+}+\beta(y-\hat{y})_{+}, \quad \alpha, \beta>0 \]where \(c_{+}\)is equal to
Show that, for the squared-error loss, the approximation error \(\ell\left(g^{\mathscr{C}}\right)-\ell\left(g^{*}\right)\) in (2.16), \(\begin{array}{llllll}\text { is } \quad \text { equal } & \text
Suppose \(\mathscr{G}\) is the class of linear functions. A linear function evaluated at a feature \(\boldsymbol{x}\) can be described as \(g(\boldsymbol{x})=\boldsymbol{\beta}^{\top}
Show that formula (2.24) holds for the \(0-1\) loss with \(0-1\) response.
Let \(\mathbf{X}\) be an \(n\)-dimensional normal random vector with mean vector \(\boldsymbol{\mu}\) and covariance matrix \(\boldsymbol{\Sigma}\), where the determinant of \(\boldsymbol{\Sigma}\)
Let \(\widehat{\boldsymbol{\beta}}=\boldsymbol{A}^{+} \boldsymbol{y}\). Using the defining properties of the pseudo-inverse, show that for any \(\boldsymbol{\beta}\) \(\in \mathbb{R}^{p}\)\[
Suppose that in the polynomial regression Example 2.1 we select the linear class of functions \(\mathscr{G}_{p}\) with \(p \geqslant 4\). Then, \(g^{*} \in \mathscr{G}_{p}\) and the approximation
Observe that the learner \(g_{\mathscr{T}}\) can be written as a linear combination of the response variable: \(g_{\mathscr{T}}(\boldsymbol{x})=\boldsymbol{x}^{\top} \mathbf{X}^{+} \boldsymbol{Y}\).
Consider again the polynomial regression Example 2.1. Use the fact that \(\mathbb{E}_{\mathbf{X}} \widehat{\boldsymbol{\beta}}=\mathbf{X}^{+} \boldsymbol{h}^{*}(\boldsymbol{u})\), where
Consider the setting of the polynomial regression in Example 2.2. Use Theorem C.19 to prove that\[ \begin{equation*} \sqrt{n}\left(\widehat{\boldsymbol{\beta}_{n}}-\boldsymbol{\beta}_{p}\right)
In Example 2.2 we saw that the statistical error can be expressed (see (2.20)) as\[ \int_{0}^{1}\left(\left[1, \ldots,
Consider again Example 2.2. The result in (2.53) suggests that \(\mathbb{E} \widehat{\boldsymbol{\beta}} \rightarrow \beta_{p}\) as \(n \rightarrow \infty\), where \(\beta_{p}\) is the solution in
For our running Example 2.2 we can use (2.53) to derive a large-sample approximation of the pointwise variance of the learner \(g_{\mathscr{T}}(\boldsymbol{x})=\boldsymbol{x}^{\top}
Let \(h: \boldsymbol{x} \mapsto \mathbb{R}\) be a convex function and let \(\boldsymbol{X}\) be a random variable. Use the subgradient definition of convexity to prove Jensen's inequality:\[
Using Jensen's inequality, show that the Kullback-Leibler divergence between probability densities \(f\) and \(g\) is always positive; that is,\[ \mathbb{E} \ln
The purpose of this exercise is to prove the following Vapnik-Chernovenkis bound: for any finite class \(\mathscr{G}\) (containing only a finite number \(|\mathscr{G}|\) of possible functions) and a
Consider the problem in Exercise 16a above. Show that\[ \left|\ell_{\mathscr{T}}\left(g_{\mathscr{T}}^{\mathscr{G}}\right)-\ell\left(g^{\mathscr{G}}\right)\right| \leqslant 2 \sup _{g \in
Show that for the normal linear model \(\boldsymbol{Y} \sim \mathscr{N}\left(\mathbf{X} \beta, \sigma^{2} \mathbf{I}_{n}\right)\), the maximum likelihood estimator of \(\sigma^{2}\) is identical to
Let \(X \sim \operatorname{Gamma}(\alpha, \lambda)\). Show that the pdf of \(Z=1 / X\) is equal to\[ \frac{\lambda^{\alpha}(z)^{-\alpha-1} \mathrm{e}^{-\lambda(z)-1}}{\Gamma(\alpha)}, \quad z>0 \]
Consider the sequence \(w_{0}, w_{1}, \ldots\),where \(w_{0}=g(\boldsymbol{\theta})\) is a non-degenerate initial guess and \(w_{t}(\boldsymbol{\theta}) \propto w_{t-1}(\boldsymbol{\theta}) g(\tau
Consider the Bayesian model for \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\) with likelihood \(g(\tau \mid \mu)\) such that \(\left(X_{1}, \ldots, X_{n} \mid \mu\right) \sim_{\text {idd }}
Consider again Example 2.8, where we have a normal model with improper prior \(g(\boldsymbol{\theta})\) \(=g\left(\mu, \sigma^{2}\right) \propto 1 / \sigma^{2}\). Show that the prior predictive pdf
Assuming that \(\boldsymbol{X}_{1}, \ldots, \boldsymbol{X}_{n} \stackrel{\text { iid }}{\sim} f\), show that (2.48) holds and that \(\ell_{n}^{*}=-n \mathbb{E} \ln f(\boldsymbol{X})\).
Suppose that \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\) are observations of iid continuous and strictly positive random variables, and that there are two possible models for their pdf. The first
Suppose that we have a total of \(m\) possible models with prior probabilities \(g(p), p=\) \(1, \ldots, m\). Show that the posterior probability of model \(g(p \mid \tau)\) can be expressed in terms
Given the data \(\tau=\left\{x_{1}, \ldots, x_{n}\right\}\), suppose that we use the likelihood \((X \mid \boldsymbol{\theta}) \sim \mathscr{N}\left(\mu, \sigma^{2}\right)\) with parameter
Visit the UCI Repository https://archive.ics.uci.edu/. Read the description of the data and download the Mushroom data set agaricuslepiota.data. Using pandas, read the data into a DataFrame called
Change the type and value of variables in the nutri data set according to Table 1.2 and save the data as a CSV file. The modified data should have eight categorical features, three floats, and two
It frequently happens that a table with data needs to be restructured before the data can be analyzed using standard statistical software. As an example, consider the test scores in Table 1.3 of 5
Create a similar barplot as in Figure 1.5, but now plot the corresponding proportions of males and females in each of the three situation categories.That is, the heights of the bars should sum up to
The iris data set, mentioned in Section1.1, contains various features, including 'Petal.Length' and 'Sepal.Length', of three species of iris:setosa, versicolor, and virginica.(a) Load the data set
Import the data set EuStockMarkets from the same website as the iris data set above. The data set contains the daily closing prices of four European stock indices during the 1990s, for 260 working
Consider the KASANDR data set from the UCI Machine Learning Repository, which can be downloaded from https://archive.ics.uci.edu/ml/machine-learningdatabases/00385/de.tar.bz2.This archive file has a
Visualizing data involving more than two features requires careful design, which is often more of an art than a science.(a) Go to Vincent Arel-Bundocks’s website (URL given in Section1. 1)and read
Prove Eq. (1.16) and discuss the increase in the variance of sample mean with respect to the independent case when the series has a non-zero first-order autocorrelation coefficient and zero
Using the Taiwan AirBox Data in Example 5.5, compare the clustering results obtained using the ACF and using the coefficients of an AR fitting to the series.Data From Example 5.5:We used the mclust
Again, consider the US monthly macroeconomic data set used in Example 8.4, but use the unemployment rate (UNRATE) as the dependent variable. Apply a DL network to obtain forecasts in the testing
Consider the 99 world financial market indexes. Compute the log returns of the indexes. Obtain a time plot of all series and perform a PCA of the log returns. Summarize the results of PCA, including
Consider the clothing data set of Figure 1.8. Perform PCA on the sales data and summarize the results. Obtain time plots of the first 12 PCs with 6 series on one page. Figure 1.8: In (sales) 7 8 11
Consider, again, the clothing data set. Obtain the three summary plots of the sample cross-correlations for lags 1 to 21.
Consider the temperature data of Figure 1.1. (a) Obtain the sample mean and sample covariance matrix of the data. (b) Obtain the lag-1 to lag-10 sample CCMs of the data.Figure 1.1: South Amer North
Consider the hourly \(\mathrm{PM}_{2.5}\) measurements at 15 monitoring stations in the southern Taiwan; columns 4 to 18 of the file TaiwanPM25 . csv. (a) Compute the sample mean and sample
Compute the variance and the ACF (lag-1 to lag-4) of the following ARMA models with \(\operatorname{Var}\left(a_{t}\right)=1\) : (a) \(z_{t}=0.7 z_{t-1}+a_{t}\); (b) \(z_{t}=0.4 a_{t-1}+a_{t}\); (c)
Simulate the three ARIMA models of Exercise 1 with the command arima. sim and compare the theoretical ACFs with the sample ACFs.Data From Exercise 1:Compute the variance and the ACF (lag-1 to lag-4)
Compare the EDQ and the TWQ for probabilities \((0.05,0.5,0.95)\) of the logs of CPI series in file CPIEurope2000-15.csv. Compute these quantiles in levels and first differences. Note the extreme
Compute the ACF of the process \(z_{t}=y_{t}+v_{t}\), where \(y_{t}=0.4 a_{t-1}+a_{t}\) with \(\operatorname{Var}\left(a_{t}\right)=4\) and \(v_{t}\) a white noise process with
Find the roots of the characteristic equation of the following ARMA models: (a) \((1-6 B) z_{t}=a_{t}\); (b) \(\left(1-1.4 B+0.8 B^{2}\right) z_{t}=a_{t}\); (c) \(\left(1-0.6 B+1.2 B^{2}\right)
Compute the periodogram of the first three series identified as seasonal in Exercise 2 (Series 4th, 5th, and 9th) with the transformation \(abla \log \left(\mathrm{CPI}_{t}\right)\) and compare the
Write the Kalman filter equations for an \(\operatorname{AR}(1)\) process written is state space form with \(H_{t}=1, \alpha_{t}=z_{t}\) and \(V_{t}=0, \Omega_{t}=\phi, R_{t}=\sigma_{a}^{2}\). Show
Compare the 1-step and 2-step ahead forecast error variances of the ARMA models of Exercise 1.Data From Exercise 1:Compute the variance and the ACF (lag-1 to lag-4) of the following ARMA models with
Consider the log series of the US monthly exports and imports data in the file m-expimpcnus.csv. (a) Are the two log series unit-root non-stationary? Perform unit-root tests to draw
Consider, again, the US monthly export and import series of Problem 1. (a) Build a bivariate time series model for the two series. Perform model simplification and model checking to justify the
Consider the Taiwan AirBox data in TaiwanAirBox032017.csv. The file contains 514 series with 744 observations. Focus on Series 2, 3, and 4. Build a multivariate time series for the three-dimensional
Consider the November temperatures of Europe, North America, and South America. See the file temperatures.txt. (a) Is there a unit root in the individual time series? Why? (b) Are the three series
Again consider the three temperature series of Europe, North America, and South America. Build a vector ARMA model (AR or MA model is allowed) for the three series. Perform model checking. Obtain
Consider the World Stock Indexes. Apply a hierarchical clustering using as dissimilarity measure the cross linear dependency. Compare the results obtained with different number of lags in the
Apply \(k\)-means and \(k\)-medoids to cluster the World Stock Indexes using the Euclidean distance among the standardized series. Comment on the differences between the clusters found and those
Consider quarterly economic series of European Union from 2000 to 2019 in the file UMEdata2000_2018.Csv. Compare the results of a hierarchical clustering using dissimilarities between univariate
Follow the analysis of Example 5.7 to discriminate between the five files with EEG data. Apply a similar analysis to discriminate between the data in the files EEGsetB.CSV, EEGsetC.CSv and
Apply SVM to discriminate between the data in the files EEGsetA.CSv and EEGsetB. csv, EEGsetC.CSv,EEGsetD.CSv and EEGsetE.csv in order to classify the results from healthy individuals or seizure
Compute the optimal interpolation for the univariate ARMA process \((1-0.6 B-\) \(\left.0.3 B^{2}\right) z_{t}=5+a_{t}\) at time \(h\) as a function of the observations before and after \(t=h\). How
Prove that the optimal interpolation of the vector process \((\boldsymbol{I}-\boldsymbol{\Phi} B) \boldsymbol{z}_{t}=\boldsymbol{a}_{t}\) at time \(t=h\) is given by
Use the package tsoutliers to detect outliers in the 9th and 10th series of the data set TaiwanAirBox032017.csv. Then, use the Lasso approach to detect level shifts and AOs in the same two time
Compare the results of Exercise 3 with those obtained via the program arima.rob.Data From Exercise 3:Use the package tsoutliers to detect outliers in the 9th and 10th series of the data set
With the three series of world temperature (Temperatures.CSv) find outliers using the programs tso, arima.rob and outlierLasso.
Simulate, as in Example 2.2, 100 values of the three series that follow an ARMA model. For instance, an AR(2) or ARMA(1,1). Introduce in the three series an outlier of size 3 and compare the results
In a white noise series \(a_{t}\) of variance \(\sigma^{2}\), an outlier of size \(\omega\) is identified by the ratio \(\omega / \sigma\). In a vector white noise \(\boldsymbol{a}_{t}\) of \(k\)
Suppose that \(k\) is large and that the series follows a VAR(1) with a sparse parameter matrix with many coefficients equal to zero and rank \(r
In the DFM \(\boldsymbol{z}_{t}=\boldsymbol{P} \boldsymbol{f}_{t}+\boldsymbol{a}_{t}\) with \(\boldsymbol{P}=\frac{1}{\sqrt{k}} \mathbf{1}\) is a \(k \times 1\) vector, \(\mathbf{1}=(1, \ldots,
Suppose the DFM \(z_{t}=\boldsymbol{P} \boldsymbol{f}_{t}+\boldsymbol{n}_{t}\), where \(\boldsymbol{n}_{t}\) follows a diagonal VAR model. Under what conditions are the factors \(f_{t}\) linear
Consider the GDFM with one factor and two lags, \(\boldsymbol{z}_{t}=\boldsymbol{P}_{0} f_{t}+\boldsymbol{P}_{1} f_{t-1}+\boldsymbol{P}_{2} f_{t-3}+\boldsymbol{n}_{t}\), where the factor follows
Suppose that the GDFM \(z_{t}=\boldsymbol{P}_{0} f_{t}+\boldsymbol{P}_{1} f_{t-1}+\boldsymbol{a}_{t}\), where \(\boldsymbol{a}_{t}\) is white noise, is estimated by the ODPC with one lag, and
Fit a DFM to the EUUS (CPI) price indexes in file CPIEurope2000-15.CSv. Apply the transformation \(abla abla_{12} \log z_{t}\) and the command dfmpc to fit the model. Analyze the properties of the
Fit a DFM to the data in levels of price indexes EUUS (CPI) in file CPIEurope2000-15.csv. Apply the transformation \(\log \left(z_{t}\right)\) and the command dfmpc to fit the model. Analyze the
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