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nonparametric statistical inference

Questions and Answers of Nonparametric Statistical Inference

If {Xj} ∼ i.i.d. Exp(1/θ, 0) (exponential distribution), then find the UMVU estimator of θ.
If {Xj} ∼ i.i.d. Po(λ) (Poisson distribution), then find the UMVU estimator of λ.
Let {Xj} ∼ i.i.d. N(μ, σ). Show that³Pn j=1 Xj , Pn j=1 X2 j´is a sufficient and complete statistic for θ = (μ, σ).
Verify (3.50).
Find an orthogonal matrix T whose first row is t1 =¡n−1/2, . . . , n−1/2¢.
(i) Verify (3.44) and (3.46).(ii) Show (3.57).
Verify the equivalent relation (3.37)-(3.39).
Minimize Eθ {T(X) − a(θ)(∂/∂θ) log fθ(X) − b(θ)}2 with respect to a(θ)and b(θ), where a(θ) and b(θ) are independent of X.
Verify (3.23).
If {Xj} ∼ i.i.d. N(θ, 1), then show that the distribution of n−1Pn j=1 Xj is N(θ, n−1).
Prove the statement (3.10).
Prove Theorem 3.1 in the case that X is continuous.
Let {Xn, n = 0, 1, 2 . . . } be a sequence of random variables. If Xn d−→ c for some constantc, then show that Xn p−→ c.
(Slutsky’s lemma) Let {Xn, n = 0, 1, 2 . . . } and {Yn, n = 0, 1, 2 . . . } be sequences of random variables. If Xn d−→ X0 and Yn p−→ c for a constant c as n → ∞, then show that(1) Xn +
Suppose that X0,X1, . . . is a sequence of random variables defined on the probability space (Ω,A, P). If g : (R, B1) → (R, B1) is a continuous function, then show that(1) If Xn−a−.s→. X0
Prove the statements of (i)- (v) of Theorem 2.6 for the continuous case.
Prove the statements of (i)- (iii) of Theorem 2.6 for the discrete case.
Let E|X|n < ∞ for some positive integer n. Show that the nth derivative of ϕX(t) exists and is continuous on R, andϕ(n)X (t) ≡ dn dtn ϕX(t) = E©(iX)neitXª, (2.75)and in particular, E(Xn) =1
Let X1, . . . ,Xn be independent random variables, and let Sn = X1 +· · ·+Xn. Show that the characteristic function of Sn is the product of the characteristic functions of the Xi.
Compute the expectation, variance and characteristic function of each distribution in Table 2.1.
Prove the statements (i)-(v) of Theorem 2.4.
Let X and Y be independent random variables with E|X| < ∞ and E|Y |
Show that the density (2.25) is symmetric about zero and unimodal.
(i) Let X1, X2 be independent, each distributed according to the Cauchy distribution (2.24). Find the distribution of a1X1 + a2X2. (ii) Use (i) to prove (by induction) that if X1, . . . ,Xn are
Let the distribution of X be N(μ, σ2) and Y = aX +b. Show that the distribution of Y is N(aμ +b, a2σ2).
Show that we can replace intervals of the form (a, b] in the definition of the family of Borel sets by other classes of intervals, for instance, all closed intervals, all intervals [a, b),a, b ∈ R,
Suppose that B is a σ-field of Ω and A is a nonempty subset of Ω. Show that the collection of events B ∩ A ≡ {B ∩ A : B ∈ B} (2.73)is a σ-field of A.
Let A ⊂ Ω. Show that σ[A] ≡ {∅, A,Ac,Ω} is a σ-field of Ω.
We denote the collection of all subsets of Ω by S(Ω). Show that S(Ω) is aσ-field of Ω.
Let A be a σ-field, and An ∈ A for all n ∈ N. Show that the following statements hold:(B4) ∅ ∈ A.(B5)Sn i=1 Ai ∈ A.(B6)Tn i=1 Ai ∈ A.(B7)T∞i=1 Ai ∈ A.(B8) lim supk→∞ Ak ∈
Show that A and ()-(4)-04 (2.72)
13.32. Suppose we use a centered variable for the covariate and express the interaction model when the categorical predictor has two categories as E{y) = a + piix - ixx) + P2Z + /^(x - ^)Xz.Explain
13.30. In the United States, the mean annual income for blacks (p.]) issmallerthan for whites {p.2), the mean number of years of education is smaller for blacks than for whites, and annual income is
13.29. In the model E{y) = a + /3ix + /ibz, where z is a dummy variable,(a) The categorical predictor has two categories.(b) One line has slope /3i and the other has slope Pi-(c) (S2 is the
13.28. A regression model is fitted to annual income(thousands of dollars), using predictors age and marital status. Table 13.20 shows the sample mean incomes and the adjusted means for the model.How
13.27. Draw a scatterplot with sets of points representing two groups such that Htf. equal means would be rejected in a one-way ANOVA but not in an analysis of covariance.
13.26. Let y = death rate and x = mean age of residents. measured for each county in Louisiana and in Florida. Sketch a hypothetical scattcrplot, identifying points for each state, when the mean
13.24. You have two groups, and you want to compare their regressions of y on x, in order to test the hypothesis that the true slopes are identical for the two groups. Explain how you can do this
13.23. For the "2005 statewide crime" data file at the text Web site let z be a dummy variable for whether a state is in the South, with z = 1 for AL, AR, FL, GA, KY, LA, MD. MS, NC, OK, SC, TN, TX,
13.22. Analyze the "house selling price 2" data file at the text Web site by modeling selling price in terms of size of house and whether it is new.(a) Fit the model allowing interaction, and test
13.20. Table 13.18 is a printout based on GSS data.The response variable is an index of attitudes toward premarital, extramarital, and homosexual sex. Higher scores represent more permissive
13.19. Table 13.17 shows results of fitting a regression model to data on salaries (in dollars) of about 35,000 college professors. Four predictors are categorical (binary), with dummy variable
13.18. An article2 on predicting attitudes toward homosexuality used GSS data to model a response variable with a four-point scale in which homosexual relations were scaled from 1 = always wrong to 4
13.17. Referto the "OECD data" file atthe text Web site, shown in Table 3.11 on page 62. Pose a research question about how GDPand whether a nation isin Europe relates to carbon dioxide emissions.
13.13. Refer to the previous exercise. Conduct the analyses for the interaction model and for comparing that model to the no-interaction model, as shown in Sections 13.3 and 13.4, after adding these
13.12. Table 13.1 did not report the observations for ten Asian Americans. Their (x._y) values were Subject 123456789 10 Education 16 14 12 18 13 12 16 16 14 10 Income 70 42 24 56 32 38 58 82 36 20
13.11. Refer to the previous exercise. The means of percentage registered for the three categories are x\ - 76.2, *2 - 49.5, J3 = 39.7. The overall mean x = 60.4.(a) Find the adjusted mean on
13.9. Replicate all the analyses shown in Sections 13.2-13.4 using the "Income, education, and racial-ethnic status" data file at the text Web site.
13.8. Refer to the previous exercise. Table 13.15 shows results of fitting the model allowing interaction, where new^size refers to the cross-product term.(a) Report the lines relating predicted
13.7. For the "house selling price" data file at the text Web site, Table 13.14 shows results of modeling y = selling price (in dollars) in terms of size of home (in square feet) and whether the home
What would you need to do to test the effect of marital status (all categories at once), controlling for the other variables?
What hypothesis does it test?
13.6. For the survey in the previous exercise, the sample size was 1417.(a) Test the null hypothesis that sex has no effect on the response, controlling for the other predictors. Interpret.(b)
13.5. Based on a national survey, Table 13.13 shows results of a prediction equation reported for y = alcohol consumption, measured as the number of alcoholic drinks the subject drank during the past
13.4. For data1 from 27 automotive plants on y =number of assembly defects per 100 cars and x = time to assemble each vehicle (in hours, falling between 12 and 54), y = 61.3 + 0.35x. Adding z =
13.3. A regression analysis for the 100th Congress predicted the proportion of each representative's votes on abortion issues thattook the "pro-choice"position (R. Tatalovich and D. Schier, American
13.2. Table 9.13 on page 294 showed data for several nations on various indices. Let y = percentage who use the Internet, x = per capita GDP (in thousands of dollars), and z = whether the nation is a
13.1. The regression equation relating y = education(number of years completed) to race (z = 1 for whites, z = 0 for nonwhitcs) in a certain country is E{y) = 11 + 2z. The regression equation
• ^asg observations having sample mean y, explain why E(« - y)7(s -1)estimates the variance (T2 jn of the sampling distribution of the y,-values.(b) Using (a), explain why 2«(y/ - y)2/(g -
12.57. This exercise motivates the formula for the between-groups variance estimate in one-way ANOVA. Suppose the sample sizes all equal n and the population means all equaljx. The sampling
12.55. You know the sample mean, standard deviation. and sample size for each of three groups.Can you conduct an ANOVA F test comparing the population means, or would you need more information?
12.54. Interaction terms are needed in a two-way ANOVA model when(a) Each pair of variables is associated.(b) Both explanatory variables have significant effects in the model without interaction
• = /Xg is false(a) The smaller the between-groups variation and the larger the within-groups variation?
12.52. One-way ANOVA provides relatively more evidence that Hq: /X| =
12.50. True or false? Suppose that forsubjects aged under 50, there is little difference in mean annual medical expenses forsmokers and nonsmokcrs, but forsubjects aged over 50 there is a large
12.49. A random sample of 26 female graduate students at the University of Florida were surveyed about their attitudes toward abortion. Each received a score on abortion attitude according to how
12.48. The 25 women faculty in the humanities division of a college have a mean salary of $66,000, whereas the five in the science division have a mean salary of $80,000. On the other hand, the 20
12.46. The null hypothesis of equality of means for a factor is rejected in a two-way ANOVA. Docs this imply that the hypothesis will be rejected in a oneway ANOVA Ftest if the data are collapsed
11.63. The numerator R 2 — r-2 VV| of the squared partial correlation rjX2.Xl gives the increase in the proportion of explained variation from adding X2 to the model. This increment, denoted by is
11.62. Let R 2, r, denote R 2for the multiple regression model with k explanatory variables. Explain why
11.61. The sample value of R 2tends to overestimate the population value, because the sample data fall closer to the sample prediction equation than to the true population regression equation. This
11.60. Software reports four types of sums of squares in multiple regression models. The Type I (sometimes called sequential) sum ofsquares represents the variability explained by a variable,
11.59. Suppose the correlation between y and X| equals the multiple correlation between y and xi and X2.What docs this imply about the partial correlation fyxyxil Interpret.
11.58. Which of the following sets of correlations would you expect to yield the highest R 2value? Why?(a) r yx\ = 0.4, r yx2 = 0.4, r X IXl = 0.0(b) r yx\ = 0.4, r yx2 = 0.4, r X\X2 = 0.5(e) r yx\ =
11.57. Whenever xi and X2 are uncorrelated, then R 2for the model ^(y) = a + 0ixi + 02X2 satisfies='yxj + r yx2' t^'s casc' draw a figure that portrays the variability in y, the part of that
11.56. For the models E{y) = a + 0x and E{y) =a + 0ixi + 02X2, express null hypotheses in terms of correlations that are equivalentto the following:(a) Hq: 0 = 0(b) Ho: 0i = @2 = 0(c) Ho: 132 = 0
11.55. Give an example of three variables for which you expect 0 # 0 in the model E{y) = a + 0xi but 01 = 0 in the model ^(y) = a + 0ixi + 02-V2.
11.54. Let y = height, xi = length of right leg. X2 =length ofleft leg. Describe what you expect for the relative sizes of rXiX2, R. and ryX2.Xl.
11.53. Explain the difference in the purposes of the correlation. the multiple correlation, and the partial correlation.
11.52. The F test for comparing a complete model to a reduced model(a) Can be used to test the significance of a single regression parameter in a multiple regression model(b) Can be used to test Ho\
11.51. Ify = 2 + 3xi + 5x2 - 8x3,(a) ryXi < 0(b) ryXyXi < 0(c)r yxyX\JC2 ^ ^(d) Insufficient information to answer.(e) Answers (a), (b), and (c) are all correct.
11.49. Ify = 2 + 3x| + 5x2 - 8x3, then controlling for X2 and X3, the predicted mean change in y when xi is increased from 10 to 20 equals(a) 3 (b)30 (c)0.3 (d) Cannot be given—depends on specific
11.48. In regression analysis, which ofthe following statements must be false? Why?(a) ryXl = 0.01, ryX2 = -0.75,7? = 0.2(b) The value ofthe residual sum ofsquares, SSE, can increase as we add
11.47. Table 11.22 shows results of fitting various regression modelsto data on _v = college GPA,xi = high school GPA, X2 = mathematics entrance exam score, and X3 = verbal entrance exam score.
11.46. In Exercise 11.1 on y = college GPA, X] = high school GPA, and X2 = college board score, E{y) = 0.20 + 0.50xi + 0.002x2. True or false:Since /3i = 0.50 is larger than fa = 0.002, this implies
11.45. In the study mentioned in the previous exercise, a separate model did not contain interaction terms.The best predictor of attitudes toward homosexuality was educational level, with an
11.44. A recent article6 used multiple regression to predict attitudes toward homosexuality. The researchers found thatthe effect ofnumber ofyears of education on a measure of tolerance toward
11.43. The Economist magazine5 developed a qualityof-lifc index for nations as the predicted value obtained by regressing an average of lifesatisfaction scores from several surveys on gross domestic
11.42. A 2002 study4 relating the percentage of a child's life spent in poverty to number of years of education completed by the mother and the percentage of a child's life spent in a single parent
11.41. A study3 of mortality rates found in the U.S. that states with higher income inequality tended to have higher mortality rates. The effect of income inequality disappeared after controlling for
11.40. For Example 11.2 on mental impairment. Table 11.19 shows the result of adding religious attendance as a predictor, measured as the approximate number oftimes the subject attends a religious
11.39. Analyze the "house selling price" data file at the text Web site (which were introduced in Example 9.10 on page 278), using selling price of home, size of home, number of bedrooms, and taxes.
11.38. In about 200 words, explain to someone who has never studied statistics what multiple regression does and how it can be useful.
11.37. Table 9.13 on page 294 in Chapter 9 is the "UN data" data file at the text Web site. Construct a multiple regression model containing two explanatory variables that provide good predictions
11.36. For the previous exercise, repeat the analysis, excluding the observation for D.C. Describe the effect on the various analyses of this observation.
11.35. Using software with the "2005 statewide crime"data file at the text Web site, conduct a regression analysis of violent crime rate with predictors poverty rate, the percent living in urban
11.34. Refer to the "OECD data" file at the text Web site, shown in Table 3.11 on page 62 of Chapter 3.Pose a research question about how at least two of the variables shown in that table relate to
11.33. Refer to the student data file you created in Exercise 1.12. For variables chosen by yourinstructor, fit a multiple regression model and conduct descriptive and inferential statistical
11.32. Repeat the previous exercise using y = college CPA with predictors high school GPA and number of weekly hours of physical exercise.
11.30. Exercise 11.11 showed a regression of violent crime rate on poverty rate and percent living in metropolitan areas. The estimated standardized regression coefficients are 0.473 for 11.31. Refer

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