Robust Statistics Theory And Methods 2nd Edition Ricardo A. Maronna, R. Douglas Martin, Victor J. Yohai, Matías Salibián-Barrera - Solutions

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The study of psychological issues to seek knowledge for its own sake is to as the study of psychological issues that have practical significance and potential solutions is to .a. basic; appliedb. applied; basicc. naturalistic; laboratoryd. laboratory; naturalistic
involves making claims that appear to be scientific but that actually violate the criteria of science.a. The principle of falsifiabilityb. Systemic empiricismc. Being a skepticd. Pseudoscience
A prediction regarding the outcome of a study is to as an organized system of assumptions and principles that attempts to explain certain phenomena and how they are related is to .a. theory; hypothesisb. hypothesis; theoryc. independent variable; dependent variabled. dependent variable; independent
Tom did really well on his psychology exam last week, and he believes that it is because he used his lucky pen. He has now decided that he must use this pen for every exam that he writes because he believes that it will make him lucky.This belief is based on:a. superstition.b. rationalism.c.
A belief that is based on subjective feelings is to knowing via as stubbornly clinging to knowledge gained from repeated ideas is to knowledge via .a. authority; superstitionb. superstition; intuitionc. tenacity; intuitiond. superstition; tenacity
The group is the group of participants that serves as the baseline in a study.They do not receive any level of the independent variable.questions, re-study the relevant material.
The variable in a study that is manipulated is the variable.
A characteristic inherent in the participants that cannot be changed is known as a variable.
The method is a method in which the degree of relationship between at least two variables is assessed.
All of the people about whom a study is meant to generalize are the .
A is an in depth study of one or more individuals.
research involves the study of psychological issues that have practical significance and potential solutions.
The three goals of science are ,, and .
involves making claims that appear to be scientific but that actually violate the criteria of science.
are questions that are potentially answerable by means of currently available research techniques.
A person who questions the validity, authenticity, or truth of something purporting to be factual is a .
A is a prediction regarding the outcome of a study that often involves a prediction regarding the relationship between two variables in a study.
To gain knowledge from repeated ideas and to cling stubbornly to them despite evidence to the contrary exemplifies gaining knowledge via.
To gain knowledge without being consciously aware of where the knowledge was gained exemplifies gaining knowledge via .
The time it takes to make the response is measured.a. What is the independent variable in this study?b. What is the dependent variable in this study?c. Identify the control and experimental groups in this study.d. Is the independent variable manipulated, or is it a participant variable?
Gerontologists interested in the effects of age on reaction time have two groups of participants take a test in which they must indicate as quickly as possible whether a probe word was a member of a previous set of words. One group of participants is between the ages of 25 and 45, and the other
Researchers interested in the effects of caffeine on anxiety have randomly assigned participants to one of two conditions in a study—the nocaffeine condition or the caffeine condition. After drinking two cups of either regular or decaffeinated coffee, participants will take an anxiety
In a study of the effects of type of study on exam performance, participants are randomly assigned to one of two conditions. In one condition, participants study in a traditional manner—alone using notes they took during class lectures. In a second condition, participants study in interactive
Identify a piece of information that you have gained through each of the sources of knowledge discussed in the chapter (superstition and intuition, authority, tenacity, rationalism, empiricism, and science).
Your mother claims that she has found a wonderful new treatment for her arthritis. She read “somewhere” that rubbing vinegar into the affected area for 10 minutes twice a day would help. She tried this and is convinced that her arthritis has been lessened. She now thinks that the medical
What type of method would you recommend researchers use to answer the following questions?a. What percentage of cars run red lights?b. Do student athletes spend as much time studying as student nonathletes?c. Is there a relationship between type of punishment used by parents and aggressiveness in
In a study designed to assess the effects of smoking on life satisfaction, participants were assigned to groups based on whether or not they reported smoking. All participants then completed a life satisfaction inventory.a. What is the independent variable?b. What is the dependent variable?c. Is
In a recent study, researchers found a negative correlation between income level and incidence of psychological disorders. Jim thinks this means that being poor leads to psychological disorders. Is he correct in his conclusion? Why or why not?
Identify a currently popular pseudoscience, and explain how it might violate each of the criteria identified previously.
Explain how a theory such as Freud’s, which attributes much of personality and psychological disorders to unconscious drives, violates the principle of falsifiability.
A friend tells you that she is not sure why but, because she has a feeling of dread, she thinks that you should not take the plane trip you were planning for next week.
Based on several observations that Pam has made, she feels sure that cell phone use does not adversely affect driving ability.
A celebrity is endorsing a new diet program, noting that she lost weight on the program and so will you.
Show that the location estimator with Huber’s ????k and previous dispersion ????̂ is uniquely defined unless there exists a solution ????̂ of ∑n i=1 ????k((xi − ????̂)∕????̂) = 0 such that |xi − ????̂| > k????̂ for all i.
Show that scale M-estimators are uniquely defined if ????(x) is strictly increasing for all x such that ????(x) < ????(∞). (To make things easier, assume ???? differentiable).
Verify that if ???? is odd, ???? is even, and F symmetric, then V in (10.18) is diagonal;and compute the asymptotic variances of ????̂ and ????. ̂
Verify that if x is symmetric about ????, then (10.23) is equal to ????.
Let Fn = (1 − n−1)N(0, 1) + n−1????n2 , where ????x is the point-mass at x. Verify that Fn → N(0, 1), but its mean and variance tend to infinity.
Let F be such that F(a) = F(b) = 0.5 for some a
Let x1, ..., xn be i.i.d. with continuous distribution function F. Show that the distribution function of the order statistic x(m) is G(t) = ∑n k=m(n k)F(t)k(1 − F(t))n−k(Hint: for each t, the variable Nt = #{xi ≤ t} is binomial and verifies x(m) ≤t ⇐⇒ Nt ≥ m).
Calculate the spectral density for the case that xt0 and xt0+k are replaced by A and −A respectively
Verify (8.13) using (8.134).
Prove (8.169).
Verify (8.160)–(8.162).
Prove the orthogonality condition (8.37).
Show that the LS estimator of the AR(p) parameters given by (8.26) is equivalent to solving the Yule–Walker equation(s) (8.28) with the true covariances replaced by the sample ones (8.30).
Verify that for an AR(1) model with parameter ???? we have ????(1) = ????.
Verify the expression for the Yule–Walker equations given by (8.28).
Construct a probability model for additive outliers ????t that has non-overlapping patches of length k > 0, such that ????t = A within each patch and ????t = 0 otherwise, and with P(????t ≠ 0) = ????.
Show that the limiting value as A → ∞ of ????̂(1) defined in (8.4), when there is an isolated outlier of size A, is −1∕T + O(1∕T2).
Show that for a “doublet” outlier at t0 (i.e., yt0 = A = −yt0+1) with t0 ∈ (1, T), the limiting value as A → ∞ of ????̂(1) in (8.3) is −0.5.
Show that |????̂(1)| ≤ 1 for ????̂(1) in (8.3). Also show that if the summation in the denominator in (8.3) ranges only from 1 to T − 1, then |????̂(1)| can be larger than one.
Explain why this may happen.
Consider the CUBIF estimator defined in Section 7.3.(a) Show that he correction term c(a,b) defined above (7.26) is a solution of the equation Em−1(a)(????H b (y − g(a) − c(a, b))) = 0.(b) In the case of the logistic model for the Bernoulli family put g(a) = ea∕(1 + ea). Then prove that
For the model in Problem 7.3, define an estimator by the equation∑n i=1(yi − p(xi, ????))sgn(xi) = 0.Since deleting all xi = 0 yields the same estimate, it will be henceforth assumed that xi ≠ 0 for all i.(a) Show that this estimator is Fisher-consistent.(b) Show that the estimator is a
Let Zn = {(x1, y1),...,(xn, yn)}, be a sample for the logistic model, where the first coordinate of each xi is 1 if the model contains an intercept. Consider a new sample Z∗n = {(−x1, 1 − y1),...,(−xn, 1 − yn)}.(a) Explain why is desirable that an estimator ????̂ satisfies the
Consider the univariate logistic model without intercept for the sample(x1, y1),...,(xn, yn) with xi ∈ R; that is, ????′x = ????x. Let p(x, ????) = e????x∕(1 + e????x) = P(y = 1).(a) Show that An(????) = ∑n i=1(yi − p(xi, ????))xi is decreasing in ????.(b) Call ????̂n the ML estimator.
Prove (7.10).
The dataset neuralgia (Piegorsch, 1992) contains the values of four predictors for 18 patients, the outcome being whether the patient experimented pain relief after a treatment. The data are described in Chapter 11. Compare the fits given by the different logistic regression estimators discussed in
The first principal component is often used to represent multispectral images.The dataset image (Frery, 2005) contains the values corresponding to three frequency bands for each of the 1573 pixels of a radar image. Compute the classical and robust principal components and compare the directions of
The dataset glass from (Hettich and Bay, 1999) contains measurements of the presence of 7 chemical constituents in 76 pieces of glass from nonfloat windows. Compute the classical and robust estimators of location and scatter and the respective squared distances. For both, make the QQ plots of
Let (x, y) be bivariate normal with zero means, unit variances and correlation ????, and let ???? be a monotone ???? -function. Show that E(????(x)????(y)) is an increasing function of ???? (Hint: y = ????x +√1 − ????2z with z ∼ (0, 1) independent of x).
Prove (6.30).
Let ????̂ and ????̂ be the MVE estimators of location and scatter. Let ????̂∗, ????̂∗ be the values of ???? and ???? minimizing |????| under the constraint that the ellipsoid{x ∈ Rp ∶ (x − ????)′????−1(x − ????) ≤ 1}of volume |????| contains at least n∕2 sample points. Show
Let ????̂ and ????̂ be the MCD estimators of location and scatter, which minimize the scale ????̂(d1, ...dn) = ∑h i=1 d(i). For each subsample A = {xi1, .., , xih} of size h call xA and CA the sample mean and covariance matrix corresponding to A. Let A∗ be a subsample of size h that
Show that x and Var(X) are the values of ????̂ and Σ̂ minimizing |????| subject to(1∕n)∑n i=1 d(xi, ????, ????) = p
Let ????̂ and Σ̂ be S-estimators of location and scatter based on the scale ????̂, and let ????̂0 = ????̂(d(X, ̂????, Σ)) ̂ . Given a constant ????0, define ????̂∗ and Σ̂∗ as the values ????and Σ that minimize |????| subject to ????̂(d(x, ????, Σ)) = ????0. Prove that ????̂∗ =
Prove that if ????(t) is a bounded nondecreasing function, then t????′(t) cannot be nondecreasing.
Show that the result of the first part of Section 6.17.4.1 is valid for any ????.
Show that the asymptotic value of an equivariant ????̂ for a spherical distribution is a scalar multiple of I.
Prove that M-estimators (6.14)–(6.15) are affine equivariant.
Show that if x has distribution (6.7), then Ex = ???? and Var(x) = c????.
Show that the adaptive MM-lasso can be computed by transforming the regressors as x∗ij = |????j|0xij.
Let {(x1, y1), . . . .,(xn, yn)} be a regression dataset, and ????̂ an S-estimator with finite BP equal to ????∗. Let D ⊂ (1, .., n) with #(D) < n????∗. Show that there exists K such that:(a) ????̂ as a function of the yi is constant if the yi with i ∉ D remain fixed and those with i ∈
For each ???? let c(????) the minimum c such that#{i ∶ ????′xi − c ≤ yi ≤ ????′xi + c} ≥ n∕2.Show that the LMS estimator minimizes c(????).
The minimum ????-quantile regression estimator is defined for ???? ∈ (0, 1) as the value of ???? minimizing the ????−quantile of |y − x′????|. Show that this estimator is an S-estimator for the scale given by ????(u) = I(|u| > 1) and ???? = (1 − ????). Find its asymptotic breakdown point.
Show that when using the shortcut (5.44), the number of times that the M-scale is computed has expectation ∑N i=1(1∕i) ≤ lnN, where N is the number of subsamples.
For the “median of slopes” estimator and the model yi = ????xi + ui, calculate the following, assuming that P(x = 0) = 0:(a) the asymptotic BP(b) the influence function and the gross-error sensitivity(c) the maximum asymptotic bias (hint: use (3.68)).
Show that the “median of slopes” estimator (5.93) is a GM-estimator (5.85).
The dataset waste (Chatterjee and Hadi, 1988) contains, for 40 regions, the solid waste and five variables on land use. Fit a linear model to these data using the LS, L1, MM and DCML estimators. Draw the respective Q–Q plots of residuals, and the plots of residuals against fitted values, and
The dataset alcohol (Romanelli et al., 2001) gives, for 44 aliphatic alcohols, the logarithm of their solubility together with six physicochemical characteristics.The interest is in predicting the solubility. Compare the results of using the LS, MM and DCML estimators to fit the log-solubility as a
The stack loss dataset (Brownlee, 1965, p. 454) given in Table 5.12 are observations from 21 days’ operation of a plant for the oxidation of ammonia, a stage in the production of nitric acid. The predictors X1, X2, X3 are respectively the air flow, the cooling water inlet temperature, and the
Show that S-estimators are regression, affine and scale equivariant.
Table 4,5 (Roberts and Cohrssen, 1968) gives prevalence rates in percentage terms for men aged 55–64 with hearing levels 16 dB or more above the audiometric zero, at different frequencies (hertz) and for normal speech. The columns classify the data into seven occupational groups:
Show that for the fit of yi = ????xi + ui with design (4.61), the FBP given by (4.80)is zero.
Show that, for large n, the FBP given by (4.80) for fitting yi = ????xk i + ui with a uniform design of n points is approximately 1 − 0.51∕(k+1).
Write computer code for the median polish algorithm and apply it to the original and modified oats data of Example 4.2 and to the data of Problem 4.9.
Verify (4.53) and show that, at each step of the median polish algorithm, the sum∑i∑j|yij − ????̂ − ????̂i − ̂????j| does not increase.
Show that the L1 estimator for the model of regression through the origin yi = ????xi + ui is the median of zi = yi∕xi, where zi has probability proportional to |xi|.
Show that the solution ????̂ of (4.37) is the LS estimator of the regression of y∗i on xi, where y∗i = ???? (yi, x′i????̂, ̂????), with ???? being “pseudo-observations”, defined as in (2.95). Use this fact to define an iterative procedure to compute a regression M-estimator.
Let ????̂ be a solution of (4.39) where ????̂ verifies (4.48). Then ????̂ is regression, affine and scale equivariant.
Let ????̂ be a solution of (4.39) with fixed ????. Show that:(a) if yi is replaced by yi + x′i????, then ????̂ + ???? is a solution(b) if xi is replaced by Axi, then A′−1????̂ is a solution.
Consider the estimator ????̂1 defined by the one-step Newton–Raphson procedure defined in Section 2.10.5.1. Assume that the underlying distribution is symmetric about ????, that ???? is odd and differentiable, and that the initial estimator ????̂0 is consistent for ????.(a) Show that ????̂1 is
For the exponential family f????(x) = I(x ≥ 0) exp(−x∕????)∕????:(a) Show that the estimator with smallest GES is Med (x)∕ ln 2.(b) Find the asymptotic distribution of this estimator and its efficiency with respect to the MLE.(c) Find the form of the Hampel-optimal estimator for this
Show the equivalence of the direct and dual Hampel problems (3.39) and (3.40)
Show that the BP of the estimator Qn in (2.64) is 0.5. Calculate the BP for the estimator defined as the median of the differences; that is, with k = m∕2 in(2.64).
Let the density f(x) be a decreasing function of |x|. Show that the shortest interval covering a given probability is symmetric about 0. Use this result to calculate the IF of the Shorth estimator (Problem 2.16a) for data with distribution f .
Consider the location M-estimator with Huber function ????k and the MADN as previously computed dispersion. Recall that it has BP = 1/2 for all k. Show however that for each given ???? < 0.5, its maximum bias MB(????) at a given distribution is an unbounded function of k.
Show that the asymptotic BP of sample ????-quantiles is min(????, 1 − ????) (recall Problem 2.13)
Show that the BP of equivariant dispersion estimators is ≤ 0.5.

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Robust Statistics Theory And Methods 2nd Edition Ricardo A. Maronna, R. Douglas Martin, Victor J. Yohai, Matías Salibián-Barrera - Solutions

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