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statistical sampling to auditing

Questions and Answers of Statistical Sampling To Auditing

“Asymptotic Normality of an estimator is an extension of the central limit theorem for functions of the sample beyond the sample mean.” Discuss.
(a) Compare and contrast full efficiency with relative efficiency.(b) Explain the difference between full efficiency and asymptotic efficiency.
Discuss the key differences between finite sample and asymptotic properties of estimators, and why these differences matter for the reliability of inference with x0.
Explain the difference between the Cramer–Rao and Bhattacharya lower bounds.
(a) Explain the notion of sufficiency.(b) Explain the notion of a minimal sufficient statistic and how it relates to the best unbiased estimator.
(a) Discuss the difference between the following two definitions of the notions of the bias and MSE: (i) E()=0,MSE) = E-0), for all 0 ; == (ii) E()=0*, MSE;0*) = E 0*), where * is the true 0 in .
Consider the Normal (two-parameter) statistical model.(a) Derive (not guess!) the sampling distributions of the following estimators: (i) = Xn, (ii) = (x1+x2+X3), (iii) 3=(X-X), (iv) n = Xi (HINT:
Consider the simple Poisson model based on f (x; θ) = θxeθ /x!, θ>0, x = 0, 1, 2, . . .(a) Derive the Cramer–Rao lower bound for unbiased estimators of θ.(b) Explain why X = 1nni=1 Xi is an
(a) Explain what the gold standard for the optimality of estimators amounts to in terms of a combination of properties; give examples if it helps.(b) Explain why the property of admissibility is
(a) Explain the concept of the likelihood function as it relates to the distribution of the sample.(b) Explain why the likelihood function does not assign probabilities to the unknown parameter(s)
(a) Explain how the likelihood function ensures learning from data as n→∞.(b) Explain why the identity below is mathematically correct but probabilistically questionable:explain how one can
(a) Define the concept of the score function and explain its connection to Fisher’s information.(b) Explain how the score function relates to both sufficiency and the Cramer–Rao lower bound.
(a) State and explain the finite sample properties of MLEs under the usual regularity conditions.(b) State and explain the asymptotic properties of MLEs under the usual regularity conditions.(c)
Consider the simple Normal statistical model (Table 12.9).(a) Derive the MLEs of (μ, σ2) and state their sampling distributions.(b) Derive the least-squares estimators of (μ, σ2) without the
(a) The maximum likelihood method is often criticized for the fact that for a very small sample size, say n = 5, MLEs are not very reliable. Discuss.(b) Explain why the fact that the ML method gives
(a) Explain why least squares as a mathematical approximation method provides no basis for statistical inference.(b) Explain how Gauss has transformed the least-squares method into a statistical
(a) State and explain the Gauss–Markov theorem, emphasizing its scope.(b) Discuss how one can use the Gauss–Markov theorem to test the significance of the slope coefficient.(c) Explain why the
(a) Explain the moment matching principle as an estimation method.(b) Use the MM principle to derive the MM estimator of θ in the case of the simple Laplace statistical model with density function f
(a) Explain how the sample raw moments provide consistent estimators for the distribution moments, but their finite sample properties invoke the existence of much higher moments that (i) are often
(a) Explain why it is anachronistic to compare the maximum likelihood method to the parametric method of moments.(b) Compare and contrast Pearson’s method of moments with the parametric method of
(a) Explain why the notion of nuisance vs. parameters of interest is problematic empirical modeling.(b) Using your answer in (a), explain why eliminating nuisance parameters distorts the original
What are the similarities and differences between Karl Pearson’s chi-square test and Fisher’s t-test for μ = μ0.
In the case of the simple Normal model, Gosset has showed that for any n > 1:(a) Explain why the result in (13.78), as it stands, is meaningless unless it is supplemented with the reasoning
(a) Define and explain the key components introduced by Fisher in specifying his significance testing.(b) Apply a Fisher significance test for H0: θ = .5, in the context of a simple Bernoulli model,
(a) Define and explain the notion of the p-value.(b) Using your answer in (a) explain why each of the following interpretations of the p-value are erroneous: (i) The p-value is the probability that
(a) Explain the new features the N-P framing introduced into Fisher’s testing and for what purpose. In what respects it modified Fisher’s testing.(b) Compare and contrast the N-P significance
(a) Explain the notions of a type I and type II error. Why does one increase when the other decreases?(b) How does the Neyman-Pearson procedure solve the problem of a trade off between the type I and
(a) In the case of the simple (one parameter) Normal model, explain how the sampling distribution of the test statistic √n(Xn−μ0)σ changes when evaluated under the null and under the
(a) State and explain the N-P lemma, paying particular attention to the framing of the null and alternative hypotheses.(b) State and explain the two conditions needed to extend the N-P lemma to more
(a) In the context of the simple Bernoulli model, explain how you would reformulate the null and alternative hypotheses when the substantive hypotheses of interest are: H0: θ = .5, vs. H1: θ =
(a) State and explain the fallacies of acceptance and rejection and relate your answer to the distinction between statistical and substantive significance.(b) Explain how the post-data severity
(a) Explain the concept of a post-data severity assessment and use it – in the context of a simple Bernoulli model– to evaluate the severity of the N-P decision for the hypotheses: H0: θ ≤ .5,
(a) Explain the notions testing within and testing without (outside) the boundaries of a statistical model in relation to Neyman–Pearson (N-P) and mispecification (M-S)testing.(b) Specify the
(a) Explain the likelihood ratio test procedure and comment on its relationship to the Neyman–Pearson lemma.(b) Explain why when the postulated statistical model is misspecified all Neyman-Pearson
(a) “Confidence intervals are more reliable than p-values and are not vulnerable to the large n problem.” Discuss.(b) “The middle of an observed CI is more probable than any other part of the
Explain why an equation expressed in terms of variables and parameters with a stochastic error term attached does not necessarily constitute a proper statistical model.
Compare and contrast the specification of the LR model in Tables 14.1 and 14.2 from the statistical modeling perspective that includes specification, estimation, misspecification testing and
Explain how the distinction between reduction and model assumptions (Table 14.3) can be useful for statistical modeling purposes.
Compare and contrast the sampling distributions of the ML estimators of the LR model and the ML estimators of the simple Normal model: Me(x): Xk NIID(,), xR, R, o > 0, KEN.
(a) Explain why the R2 as a goodness-of-fit measure for the LR model is highly vulnerable to any mean-heterogeneity in data Z0:=(y,X).(b) Explain the relationship between the R2 and the F-test for
Plot the residuals from the four estimated LR models in Example 14.3, and discuss the modeling strategy to avoid such problems.
(a) Test the hypotheses H0: σ2 = σ20, H1: σ2=σ20 for σ20= .2 at α = .05 using the estimated LR model in Example 14.2.(b) Using the estimated LR model in Example 14.2, derive the .95 two-sided
(a) Using the data in Table 1, Appendix 5A repeat example 14.8 by replacing the Intel log-returns with the CITI log-returns and compare your results with those in the example.(b) Explain intuitively
(a) Using the data in 1, Appendix 5.A, estimate the following LR model for the Intel log-returns (yt):Yt = β0 + β1x1t + β1x1t + ut, (14.91)where x1t= log-returns of the market (SP500) and x2t=
Explain why adding an additional explanatory variable in the auction estimated LR model in Example 14.9 has nothing to do with statistical misspecification; it is a case of substantive
Compare and contrast the LR with the Gauss linear (GL) model in terms of their specifications in Tables 14.1 and 14.5, respectively.
Discuss the assumptions and the conclusions of the Gauss–Markov theorem and explain why it provides a very poor basis for statistical inference purposes.
Explain why the error term distributions (ii)–(iv) in Table 14.8 raise questions about the appropriateness of least-squares as the relevant estimation method.
Explain the connection between the Normality of the joint distribution f (x, y) and the LR assumptions [1]–[5].
Compare and contrast the asymptotic properties of the OLS estimators of the LR parameters with those of the ML estimators under the Normality assumption.
Explain why the specification of the LR model in Table 14.10 can be misleading for modeling purposes.
Explain the relationship between the statistical GM: Yt = β0 + β1 xt + ut, t ∈ N and the estimated orthogonal decomposition: Yt = β 0 + β1 xt + ut.
Explain the variance decomposition in Table 14.11 and its importance for testing purposes.
Explain why relying exclusively on the asymptotic sampling distributions of the OLS estimators of the LR model parameters can cause problems for the reliability and precision of inference.
(a) Explain the problem of near-collinearity of the (X X) matrix and how it might affect the ML estimators of the LR model parameters.(b) Explain why near-collinearity of the (X X) matrix is neither
(a) The Norm condition number κ2(X X) provides a reliable measure of the illconditioning of the (X X) matrix, but det(X X) 0 does not.(b) Explain why κ2(X X) invokes no probabilistic assumptions
(a) Explain why the concept of a variance inflation factor makes no sense for trend terms (1, t, t2, . . . , tm).(b) How should one handle the terms (1, t, t2, . . . , tm) in the context of an LR
Using the data in Example 14.11, evaluate the volatility upper bounds (i)-(ii) in (14.73), and explain why these bounds indicate that any inference concerning the regression coefficients is likely to
(a) Explain the notion of a non-typical observation and how one can detect such observations.(b) Explain the notion of a high-leverage observation and discuss how it should be treated if it affects
(a) Explain the role of the distinction between the modeling and inference facets in empirical modeling.(b) Using the differences between the LR and GL models compare the modeling with the
Explain how statistical misspecification might undermine the consistency of an estimator or induce sizeable discrepancies between the nominal and actual type I and II errors. Use the LR model as an
(a) Explain how the following confusions contribute to the neglect of misspecification(M-S) testing: empirical modeling vs. curve fitting, statistical vs. substantive misspecification, the modeling
(a) Explain the notions of testing within and testing without (outside) the boundaries of a statistical model in relation to Neyman–Pearson (N-P) and misspecification (M-S)testing.(b) Specify the
(a) Explain how one can secure statistical adequacy in practice by employing particular strategies in misspecification testing.(b) Explain why in practice one should combine non-parametric (omnibus)
(a) Describe the Durbin–Watson test as it is applied in traditional textbook econometrics, and explain why it might not be an effective M-S test of assumption [4] in the linear regression model
Using the monthly data for the period August 2000 to October 2005 (n = 64) in Appendix 5.A, for yit=log returns of a particular firm i = 1, 2, . . . , 6, x1t=monthly log-returns of the 3-month
Explain where auxiliary regressions for M-S testing come from, and provide an intuitive rationale pertaining to the objective of the joint M-S tests.
(a) For the Yule (1926) data in Table 15.A.1 (Appendix 15.A), test whether the two data series {(xt, yt), t = 1, 2, . . . , n} can be viewed as typical realizations of a simple Normal model (Table
(a) For the spurious correlation data in Table 15.A.2 (Appendix 15.A), xt=honeyproducing bee colonies (US, USDA), 1000s of colonies, yt-Juvenile arrests for possession of marijuana (US, DEA), for the
1. Refer to Table 16.1.a) Interpret the estimated effects of race and gender on the hazard rate.b) Show how to test the effect of race. and interpret
2. In studying the effect of race on job dismissals in the federal bureaucracy. C. Zwerling and H. Silver (American Sociological Review. Vol. 57, 1992, p. 651) used event history analysis to model
3. Explain what is meant by a censored observation. and give an example in which most observations would be censored.
4. Consider the variables / annual income, E = attained educational level, Y = number of years experience in job. Af = motivation. A = age. G = gender. and P = parents' attained educational levela)
5 Refer to Table 9.13 and Problem 9.17. Draw a path diagram relating B burth rate, G = GNP, L = literacy. 7 = television ownership, and C = contraception. Specify the models you would need to fit to
6. In Table 9.1, consider the variables murder rate, percentage metropolitan. percentage high school graduates, and percentage in poverty. Do not use the observation for D Ca) Construct a reahstic
7. Refer to Table 9.16 in Problem 9.24. Consider the spurious causal inodel for the associa- tion between crime rate and percentage high school graduates, controlling for percentage urban. Analyze
8. Refer to Table 9.1. Using software as shown by your instructor. conduct a factor analy- sis How many factors scem appropriate? Interpret the factors, using the estimated factor loadings.
9. Refer to the previous problen Remove the observation for D.C.. and repeat. How sensi- tive are the estimated factor loadings and your identification of factors to that one obser- vation?
10. Construct a diagram representing the following covariance structure model, three ob- served response vanables are described by a single latent variable, and that latent variable is regressed on
11. Construct a diagram representing the following covariance structure inodel. for vanables measured for each state. The latent response variable is based on two observed indicators, violent crime
12. Refer to the previous problem. Using software that your instructor introduced. fit this model to the data in Table 9.1 and interpret results
13. Construct a diagram representing a covariance structure model in which (i) in the meas- urement part of the model. a single factor represents violent crime rate and murder rate and a single
14. A variable is measured at three times, Y, at time 1. Y at tume 2, and Y at time 3 Sup- pose the chain relationship holds, with Y, affecting Y, which in turn affects Ya Does this sequence of
12.1. For GSS data comparing the reported number of good friends for those who are (married, widowed, divorced, separated, never married), an ANOVA table reports F = 0.80.(a) Specify the null and
12.2. A General Social Survey asked subjects how many good friends they have. Is this associated with the respondent’s astrological sign (the 12 symbols of the zodiac)?The ANOVA table for the GSS
12.3. A recent General Social Survey asked, “What is the ideal number of kids for a family?” Show how to define dummy variables, and formulate a model for this response with explanatory variable
12.4. Refer to the previous exercise. Table 12.22 shows an ANOVA table for the model.(a) Specify the hypotheses tested in this table.(b) Report the F test statistic value and the P-value. Interpret
12.5. A recent GSS asked, “How often do you go to a bar or tavern?” Table 12.23 shows descriptive statistics and an ANOVA table for comparing the mean reported number of good friends at three
12.6. Table 12.24 shows scores on the first quiz (maximum score 10 points) in a beginning French course. Students in the course are grouped as follows:Group A: Never studied foreign language before,
12.7. Refer to the previous exercise.(a) Suppose that the first observation in the second group was actually 9, not 1. Then, the standard deviations are the same, but the sample means are 6, 7, and 8
12.8. In a study to compare customer satisfaction at service centers for PC technical support in San Jose(California), Toronto (Canada), and Bangalore (India), each center randomly sampled 100 people
12.9. For g groups with n = 100 each, we plan to compare all pairs of population means.We want the probability to equal at least 0.80 that the entire set of confidence intervals contains the true
12.10. A recent GSS asked, “Would you say that you are very happy, pretty happy, or not too happy?” and “About how many good friends do you have?” Table 12.26 summarizes results, with number
12.11. When we use the GSS to evaluate how the mean number of hours a day watching TV depends on sex and race, for subjects of age 18–25, we get the results shown in Table 12.27. The sample means
12.12. A recent GSS asked, “What is the ideal number of kids for a family?” Table 12.28 shows results of evaluating the effects of gender and race.(a) Explain how to interpret the results of the
12.13. Table 12.13 on page 380 gave the prediction equationˆy = 5.23 − 1.77p1 − 1.24p2 − 0.01s relating political ideology to political party ID and to sex. Find the estimated means for the
12.14. Using software with the Housesdata set at the text website, conduct an ANOVA for y = house selling price with factors whether the house is new and whether number of bathrooms exceeds two.(a)
12.15. For the 2014 GSS, when we regress y = number of hours per day watching TV on s = sex (1 = male, 0 = female)and religious affiliation (r1 =1 for Protestant, r2 = 1 for Catholic, r3 = 1 for
12.16. In the United States, the Bureau of Labor Statistics recently reported that for males the current population mean hourly wage is $22 for white-collar jobs, $11 for service jobs, and $14 for
12.17. In 2013, the U.S. Census Bureau reported that the population median income was $29,127 for white females,$26,006 for black females, $41,086 for white males, and$30,394 for black males.(a)

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