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Questions and Answers of Statistics

Suppose that we have two independent random samples: X1,..., Xn are exponential(Î¸), and Y1,..., Ym are exponential(Î¼).(a) Find the LRT of H0: Î¸ =
We have already seen the usefulness of the LRT in dealing with problems with nuisance parameters. We now look at some other nuisance parameter problems.(a) Find the LRT ofH0: Î¸
A special case of a normal family is one in which the mean and the variance are related, the n(θ, αθ) family. If we are interested in testing this relationship, regardless of the value of θ, we
Stefanski (1996) establishes the arithmetic-geometric-harmonic mean inequality (see Example 4.7.8 and Miscellanea 4.9.2) using a proof based on likelihood ratio tests. Suppose that Y1,...,Yn are
If L(x) and U(x) satisfy Pθ(L(X) ≤ θ) = 1 - α1 and Pθ(U(X) ≥ θ) = 1 - α2, and L(x) ≤ U(x) for all z, show that Pθ(L(X) ≤ θ ≤ U(X)) = 1 - α1 - α2.
If T is a continuous random variable with cdf FT(t|θ) and α1 + α2 = a, show that an a level acceptance region of the hypothesis H0: θ = θ0 is {t: α1 ≤ FT(t|θ0) ≤ 1 - α2}, with associated
Find a pivotal quantity based on a random sample of size n from a n(θ, θ) population, where θ > 0. Use the pivotal quantity to set up a 1 - α confidence interval for θ.
Let X be a single observation from the beta(θ, 1) pdf. a. Let Y = - (logX)-1. Evaluate the confidence coefficient of the set [y/2, y]. b. Find a pivotal quantity and use it to set up a confidence
Let X1,...,Xn be iid n(Î¼, Ïƒ2), where both parameters are unknown. Simultaneous inference on both Î¼ and Ïƒ can be made using the Bonferroni Inequality
Solve for the roots of the quadratic equation that defines Fieller's confidence set for the ratio of normal means (see Miscellanea 9.5.3). Find conditions on the random variables for which a. The
Let X1,..., Xn be iid n(μ, σ2), where σ2 is known. For each of the following hypotheses, write out the acceptance region of a level a test and the 1 - α confidence interval that results from
Find a 1 - α confidence interval for θ, given X1,..., Xn iid with pdf a. f(x|θ) = 1, θ -1/2 < x < θ + 1/2. b. f(x|θ) = 2x/θ2, 0 < x < θ, θ > 0.
In this exercise we will investigate some more properties of binomial confidence sets and the Sterne (1954) construction in particular. As in Example 9.2.11, we will again consider the binomial(3,p)
Prove part (b) of Theorem 9.2.12. Theorem 9.2.12. If FT(t|θ) is an increasing function of θ for each t, define θL(t) and θU(t) by FT(t|θU(t)) = 1 - α2, FT(t|θL(t)) = α1. Then the random
In Example 9.2.15 it was shown that a confidence interval for a Poisson parameter can be expressed in terms of chi squared cutoff points. Use a similar technique to show that if X ~ binomial(n, p),
a. Let X1,..., Xn be a random sample from a Poisson population with parameter A and define Y = ˆ‘Xi. In Example 9.2.15 a confidence interval for Î» was found using the method of
If X1,... ,Xn are iid with pdf f(x|μ) = e-(x-μ)I[μ,∞) (x), then Y = min{X1,... ,Xn} is sufficient for μ with pdf fY(y|μ) = ne-n(y-μ) I[μ,∞)(y) In Example 9.2.13 a 1 - α confidence
a. Let X1,... ,Xn be iid observations from an exponential (Î») pdf, where Î» has the conjugate IG(a, b) prior, an inverted gamma with pdfShow how to find a 1 - Î±
Let X1,..., Xn are a sequence of n Bernoulli (p) trials. a. Calculate a 1 - α credible set for p using the conjugate beta(a, b) prior. b. Using the relationship between the beta and F distributions,
The independent random variables X1,..., Xn have the common distributiona. In Exercise 7.10 the MLEs of α and β were found. If α is a known constant, αo, find an upper confidence limit for β
Complete the coverage probability calculation needed in Example 9.2.17.a. If X22Y is a chi squared random variable with Y ~ Poisson(A), show that E(x22y) = 2Î», Var(x22Y) =
Let X ~ n(p, 1) and consider the confidence interval Ca(x) = {μ: min{0, (x - a)} ≤ μ. ≤ max{0, (x + a)}}. a. For a = 1.645, prove that the coverage probability of Ca{x) is exactly .95 for all
Suppose that X1,..., Xn is a random sample from a n(μ, σ2) population. a. If σ2 is known, find a minimum value for n to guarantee that a .95 confidence interval for n will have length no more than
Let X1,..., Xn be a random sample from a n((μ, σ2) population. Compare expected lengths of 1 - α confidence intervals for μ that are computed assuming σ2 is known. σ2 is unknown.
Let X1,..., Xn be independent with pdfs fx, (x|θ) = eiθ-x I(iθ,∞)(x). Prove that T = mini(Xi/i) is a sufficient statistic for 6. Based on T, find the 1 - α confidence interval for θ of the
Let X1,...,Xn be iid uniform(0, Î¸), Let Y be the largest order statistic. Prove that Y/Î¸ is a pivotal quantity and show that the intervalis the shortest 1 - Î±
Prove a special case of Theorem 9.3.2. Let X ~ f{x), where / is a symmetric unimodal pdf. For a fixed value of 1 - α, of all intervals [a, b] that satisfy ∫ba f(x)dx= 1 - α, the shortest is
Let X1,..., Xn be a random sample from a n(0, σ2X), and let Y1,..., Ym be a random sample from a n(0, σ2Y), independent of the As. Define λ = σ2Y/ σ2X. a. Find the level a LRT of H0: λ = λ0
a. Prove the following, which is related to Theorem 9.3.2. Let X ~ f(x), where f is a strictly decreasing pdf on [0, ∞). For a fixed value of 1 - α, of all intervals [a, b] that satisfy ∫ba f(x)
Juola (1993) makes the following observation. If we have a pivot Q(X, $), a 1 - Î± confidence interval involves finding a and b so that P(aor, more generally, a. Prove that the solution
Let X1,..., Xn be iid exponential A).a. Find a UMP size a hypothesis test of H0: Î» = Î»0 versus H1: Î» b. Find a UMA 1 - Î± confidence interval based
Show that if λ(θ0) is an unbiased level α acceptance region of a test of H0: θ = θ0 versus H1: θ ≠ θ0 and C(x) is the 1 - α confidence set formed by inverting the acceptance regions, then
Let X1,..., Xn be a random sample from a n(θ, σ2) population, where σ2 is known. Show that the usual one-sided 1 - α upper confidence bound {θ: θ < + zασ /√n} is unbiased, and so is the
Let X1...,Xn be a random sample from a n(θ, σ2) population, where σ2 is unknown. a. Show that the interval θ < + tn-1,α/√n can be derived by inverting the acceptance region of an LRT. b. Show
We are to testH0: Î¸ = Î¸0 versus H1: Î¸ > Î¸0,where Î¸ is the mean of one of two normal distributions and Î¸0 is a fixed but
In Example 9.2.5 a lower confidence bound was put on p, the success probability from a sequence of Bernoulli trials. This exercise will derive an upper confidence bound. That is, observing X1,...,
If X1..., Xn are iid from a location pdf f (x - θ), show that the confidence set C(x1, ... xn) = {θ: - k1 ≤ θ ≤ + k2}, where k1 and k2 sire constants, has constant coverage probability.
Let X1,..., Xn be a random sample from a n(Î¼, Ïƒ2) population, where both Î¼ and Ïƒ2 are unknown. Each of the following methods of finding confidence
Let X ~ n(Î¼, Ïƒ2), Ïƒ2 known. For each c ‰¥ 0, define an interval estimator for Î¼ by C(x) = [x - cÏƒ, x + cÏƒ] and
The decision theoretic approach to set estimation can be quite useful (see Exercise 9.56) but it can also give some unsettling results, showing the need for thoughtful implementation. Consider again
Let X1,..., Xn be iid n(μ, σ2), where σ2 is known. We know that a 1 - α confidence interval for μ is ± zα/2 σ/√n. a. Show that a 1 - α prediction interval for Xn+1 is ± zα/2 σ √1
a. Derive a confidence interval for a binomial p by inverting the LRT of H0: p = p0 versus H1: p ≠ p0. b. Show that the interval is a highest density region from pv (1 - p)n-y and is not equal to
a. Find the 1 - α confidence set for a that is obtained by inverting the LRT of H0: α = a0 versus H1: a ≠ a0 based on a sample X1,..., Xn from a n(θ, aθ) family, where θ is unknown. b. A
Show that each of the three quantities listed in Example 9.2.7 is a pivot.Example 9.2.7
A random sample X1,..., Xn is drawn from a population with pdfFind a consistent estimator of Î¸ and show that it is consistent.
This problem will look at some details and extensions of the calculation in Example 10.1.18.(a) Reproduce Figure 10.1.1, calculating the ARE for known Î². (You can follow the calculations
Refer to Example 10.1.19.(a) Verify that the bootstrap mean and variance of the sample 2, 4, 9, 12 are 6.75 and 3.94, respectively.(b) Verify that 6.75 is the mean of the original sample.(c) Verify
Use the Law of Large Numbers to show that Var*B(θ) of (10.1.11) converges to Var* (θ) of (10.1.10) as B → ∞.
Efron (1982) analyzes data on law school admission, with the object being to examine the correlation between the LSAT (Law School Admission Test) score and the first-year GPA (grade point average).
Another way in which underlying assumptions can be violated is if there is correlation in the sampling, which can seriously affect the properties of the sample mean. Suppose we introduce correlation
The breakdown performance of the mean and the median continues with their scale estimate counterparts. For a sample X1,..., Xn: (a) Show that the breakdown value of the sample variance S2 = ∑(Xi) -
In this exercise we will further explore the ARE of the median to the mean, ARE(Mn, ).(a) Verify the three AREs given in Example 10.2.4.(b) Show that ARE(Mn, ) is unaffected by scale changes. That
If f(x) is a pdf symmetric around 0 and p is a symmetric function, show that ∫
Consider the situation of Example 10.6.2.(a) Verify that IF(, x) = x - Î¼.(b) For the median we have T(F) = m if P(X ‰¤ m) = 1/2 or m = F-1(1/2). If X ~ F6, show thatand
From (10.2.9) we know that an M-estimator can never be more efficient than a maximum likelihood estimator. However, we also know when it can be as efficient. (a) Show that (10.2.9) is an equality if
A random sample X1,..., Xn is drawn from a population that is n(Î¸, Î¸), where Î¸ > 0.(a) Show that the MLE of Î¸, Î¸, is a root of the
Binomial data gathered from more than one population are often presented in a contingency table. For the case of two populations, the table might look like this:where Population 1 is binomial(n1,
(a) Let (X1,... , Xn) ~ multinomial (m, pi,..., pn). Consider testing H0: p1 = p2 versus H1: p1 ‰ p2. A test that is often used, called McNemar's Test, rejects H0 ifShow that this test
Fill in the gap in Theorem 10.3.1. Use Theorem 10.1.12 and Slutsky's Theorem (Theorem 5.5.17) to show that (θ - θ) / √-l′′(θ|x) → n(0, 1), and therefore -2log λ(X) → X21.
Let X1,..., Xn be a random sample from a n(μ, σ2) population.(a) If μ is unknown and σ2 is known, show that Z = √n( - μ0)/σ is a Wald statistic for testing H0: μ = μ0.(b) If σ2 is unknown
Let X1,..., Xn be a random sample from a n(μ, σ2) population. (a) If μ is unknown and σ2 is known, show that Z = √n( - μ0)/σ is a score statistic for testing H0: μ = μ0. (b) If σ2 is
Expand the comparisons made in Example 10.3.7. (a) Another test based on Huber's M-estimator would be one that used a variance estimate, based on (10.3.6). Examine the performance of such a test
A variation of the model in Exercise 7.19 is to let the random variables Y1,..., Yn satisfy Yi = βXi + εi, i = 1,... , n, where X1,... , Xn are independent n(μ, τ2) random variables, εi,...,
Let X1,..., Xn be iid negative binomial(r, p). We want to construct some approximate confidence intervals for the negative binomial parameters. (a) Calculate Wilks' approximation (10.4.3) and show
In Example 10.4.7, two modifications were made to the Wald interval. (a) At y = 0 the upper interval endpoint was changed to 1 - (α/2)1/n, and at y = n the lower interval endpoint was changed to
Solve for the endpoints of the approximate binomial confidence interval, with continuity correction, given in Example 10.4.6. Show that this interval is wider than the corresponding interval without
Let X1,..., Xn be iid negative binomial(r, p).(a) Complete the details of Example 10.4.9; that is, show that for small p, the intervalis an approximate 1 - Î± confidence interval. (b)
For the situation of Example 10.1.8 show that for Tn = √n/n: (a) Var (Tn) = ∞. (b) If μ ≠ 0 and we delete the interval (-δ, δ) from the sample space, then Var (Tn) < ∞. (c) If μ ≠ 0,
Suppose that X1,..., Xn are iid Poisson(λ). Find the best unbiased estimator of (a) e-λ, the probability that X = 0. (b) λe-λ, the probability that X = 1. (c) For the best unbiased estimators of
An ANOVA variance-stabilizing transformation stabilizes variances in the following approximate way. Let Y have mean θ and variance v(θ). (a) Use arguments as in Section 10.1.3 to show that a
Suppose we have a oneway ANOVA with five treatments. Denote the treatment means by Î¸1,..., Î¸5, where Î¸1 is a control and Î¸2, ..., Î¸5
Suppose that we have a oneway ANOVA with equal numbers of observations on each treatment, that is, ni = n, i = 1,..., k. In this case the F test can be considered an average t test.(a) Show that a t
Under the oneway ANOVA assumptions, show that the likelihood ratio test of H0: Î¸1 = Î¸2 = ... = 0k is given by the F test of (11.2.14).
The Scheffe simultaneous interval procedure actually works for all linear combinations, not just contrasts. Show that under the oneway ANOVA assumptions, if M = ˆškFk,N-k,a (note the change
(a) Show that for the t and F distributions, for any v, α, and k, tv,α/2 ≤ √(k - 1)Fk-l,v,α. (Recall the relationship between the t and the F. This inequality is a consequence of the fact that
In Theorem 11.2.5 we saw that the ANOVA null is equivalent to all contrasts being 0. We can also write the ANOVA null as the intersection over another set of hypotheses. (a) Show that the
A multiple comparison procedure called the Protected LSD (Protected Least Significant Difference) is performed as follows. If the ANOVA F test rejects H0 at level Î±, then for each pair
Demonstrate that "data snooping," that is, testing hypotheses that are suggested by the data, is generally not a good practice. (a) Show that, for any random variable Y and constants a and b with a >
Let Xi ~ gamma(Î»i, 1) independently for i = 1,..., n. Definei = 1,..., n - 1, and (a) Find the joint and marginal distributions of Yi, i = 1,..., n. (b) Connect your results to any
Verify that the following transformations are approximately variance-stabilizing in the sense of Exercise 11.1. (a) Y ~ Poisson, g*(y) = √y (b) y ~ binomial(n, p), g*(y) = sin-1(√y/n) (c) y has
(a) Illustrate the partitioning of the sums of squares in the ANOVA by calculating the complete ANOVA table for the following data. To determine diet quality, male weanling rats were fed diets with
Use the model in Miscellanea 11.5.3. (a) Show that the mean and variance of Yij are EYij = μ + τi and Var Yij = σ2B + σ2. (b) If ∑ai = 0, show that the unconditional variance of ∑ai i. is
In Section 11.3.1, we found the least squares estimators of Î± and Î² by a two-stage minimization. This minimization can also be done using partial derivatives.(a) Compute and
Observations (xi, Yi), i = 1,... ,n, follow the model Yi = α + βxi + εi, where E εi = 0, Var εi = σ2, and Cov(εi, εj) = 0 if i ≠ j. Find the best linear unbiased estimator of α.
Show that in the conditional normal model for simple linear regression, the MLE of Ïƒ2 is given by
Consider the residuals 1,..., n defined in Section 11.3.4 by i = Yi - - xi.(a) Show that Ei = 0.(b) Verify that
The Box-Cox family of power transformations (Box and Cox 1964) is defined bywhere Î» is a free parameter. (a) Show that, for each y, g*Î»(y) is continuous in A. In particular,
Fill in the details about the distribution of d left out of the proof of Theorem 11.3.3.(a) Show that the estimator = - can be expressed aswhere (b) Verify that (c) Verify that
Verify the claim in Theorem 11.3.3, that i is uncorrelated with and . (Show that i = ∑ejYj, where the ejs are given by (11.3.30). Then, using the facts that we can write = ∑cjYj and =
Observations (xi, Yi), i = 1,..., n, are made according to the model Yi = α + βxi + εi, where x1,...,xn are fixed constants and ε1,...,εn are iid n(0, σ2). The model is then reparameterized
Observations (Xi, Yi), i = 1,...,n, are made from a bivariate normal population with parameters (Î¼X, Î¼Y, Ïƒ2X, Ïƒ2Y, p), and the model Yi =
(a) Illustrate the partitioning of the sum of squares for simple linear regression by calculating the regression ANOVA table for the following data. Parents are often interested in predicting the
Observations Y1,...,Yn are described by the relationship Yi = θx2i + εi, where x1,... ,xn are fixed constants and ε1,..., εn are iid n(0, σ2). (a) Find the least squares estimator of θ. (b)
Observations Y1,... ,Yn are made according to the model Yi = α + βxi + εi, where x1,...,xn are fixed constants and ε1,...,εn are iid n(0, σ2). Let and denote MLEs of α and β. (a) Assume
We observe random variables Y1,...,Yn that are mutually independent, each with a normal distribution with variance σ2. Furthermore, EYi = βxi, where β is an unknown parameter and x1,... ,xn are
An ecologist takes data (xi, yi), i = 1,...,n, where xi is the size of an area and Yi is the number of moss plants in the area. We model the data by Yi ~ Poisson(θxi), Yis independent. (a) Show that
Verify that the simultaneous confidence intervals in (11.3.42) have the claimed coverage probability.
In the discussion in Example 12.4.2, note that there was one observation from the potoroo data that had a missing value. Suppose that on the 24th animal it was observed that O2 = 16.3. (a) Write down
Suppose that random variables Yij are observed according to the overparameterized oneway ANOVA model in (11.2.2). Show that, without some restriction on the parameters, this model is not identifiable
Under the oneway ANOVA assumptions:(a) Show that the set of statistics (1., 2.,..., k., S2p) is sufficient for (Î¸1,Î¸2, ..., Î¸k, Ïƒ2).(b) Show thatis

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