PART III: PROBLEMS

Section 5.2

5.2.1 Let X1, …, Xn be i.i.d. random variables having a rectangular distribution R(θ1, θ2), -∞ < θ1 < θ2 < ∞.

5.2.2 Let X1, …, Xn be i.i.d. random variables having an exponential distribution, E(λ), 0 < λ < ∞.

5.2.3 Let X1, …, Xn be i.i.d. random variables having a two–parameter exponential distribution, X1 ∼ μ + G(λ, 1). Derive the UMVU estimators of μ and λ and their covariance matrix.

5.2.4 Let X1, …, Xn be i.i.d. N(μ, 1) random variables.

5.2.5 Consider Example 5.4. Find the variances of the UMVU estimators of p(0; λ) and of p(1; λ). [Hint: Use the formula of the p.g.f. of a P(nλ).]

5.2.6 Let X1, …, Xn be i.i.d. random variables having a NB(, ν) distribution; 0 < < ∞ (ν known). Prove that the UMVU estimator of is

5.2.7 Let X1, …, Xn be i.i.d. random variables having a binomial distribution B(N, θ), 0 < θ < 1.

5.2.8 Let X1, …, Xn be i.i.d. N(μ, 1) random variables. Find a constant b(n) so that

is a UMVU estimator of the p.d.f. of X at ξ, i.e., . [Hint: Apply the m.g.f. of ( – ξ)2.]

5.2.9 Let J1, …, Jn be i.i.d. random variables having a binomial distribution B(1, e-Δ/θ), 0 < θ < 1 (Δ known). Let n = . Consider the estimator of θ

Determine the bias of n as a power–series in 1/n.

5.2.10 Let X1, …, Xn be i.i.d. random variables having a binomial distribution B(N, θ), 0 < θ < 1. What is the Cramér – Rao lower bound to the variance of the UMVU estimator of ω = θ (1-θ)?

5.2.11 Let X1, …, Xn be i.i.d. random variables having a negative–binomial distribution NB(, ν). What is the Cramér – Rao lower bound to the variance of the UMVU estimator of ? [See Problem 6.]

5.2.12 Derive the Cramér – Rao lower bound to the variance of the UMVU estimator of δ = e– λ in Problem 2.

5.2.13 Derive the Cramér – Rao lower bound to the variance of the UMVU estimator of Φ(μ) in Problem 4.

5.2.14 Derive the BLBs of the second and third order for the UMVU estimator of Φ(μ) is Problem 4.

5.2.15 Let X1, …, Xn be i.i.d. random variables having a common N(μ, σ2) distribution, -∞ < μ < ∞, 0 < σ2 < ∞.

5.2.16 Let X1, …, Xn be i.i.d. random variables having a common N(μ, σ2) distribution, -∞ < μ < ∞, 0 < σ < ∞. Determine the Cramér – Rao lower bound for the variance of the UMVU estimator of ω = μ + zγ σ, where zγ = Φ−1(γ), 0 < γ < 1.

5.2.17 Let X1, …, Xn be i.i.d. random variables having a G(λ, ν) distribution, 0 < λ < ∞, ν ≥ 3 fixed.

5.2.18 Consider Example 5.8. Show that the Cramér – Rao lower bound for the variance of the MVU estimator of cov(X, Y) = ρ σ2 is .

5.2.19 Let X1, …, Xn be i.i.d. random variables from N(μ1, ) and Y1, …, Yn i.i.d. from N(μ2, ). The random vectors X and Y are independent and n ≥ 3. Let δ = .

5.2.20 Let X1, …, Xn be i.i.d. random variables having a rectangular distribution R(0, θ), 0 < θ < ∞. Derive the Chapman – Robbins inequality for the UMVU of θ.

5.2.21 Let X1, …, Xn be i.i.d. random variables having a Laplace distribution L(μ, σ), -∞ < μ < ∞, 0 < σ < ∞. Derive the Chapman – Robbins inequality for the variances of unbiased estimators of μ.

Section 5.3

5.3.1 Show that if (X) is a biased estimator of θ, having a differentiable bias function B(θ), then the efficiency of (X), when the regularity conditions hold, is

5.3.2 Let X1, …, Xn be i.i.d. random variables having a negative exponential distribution G(λ, 1), 0 < λ < ∞.

5.3.3 Consider Example 5.8.

Section 5.4

5.4.1 Let X1, …, Xn be equicorrelated random variables having a common unknown mean μ. The variance of each variable is σ2 and the correlation between any two variables is ρ = 0.7.

5.4.2 Let X1, X2, X3 be i.i.d. random variables from a rectangular distribution R(μ – σ, μ + σ), -∞ < μ < ∞, 0 < σ < ∞. What is the best linear combination of the order statistics X(i), i = 1, 2, 3, for estimating μ, and what is its variance?

5.4.3 Suppose that X1, …, Xn are i.i.d. from a Laplace distribution with p.d.f. f(x;μ, σ) = , where (z) = . What is the best linear unbiased combination of X(1), Me, and X(n) for estimating μ, when n = 5?

5.4.4 Let k(T) = .

5.4.5 Let Xt = f(t) + et, where t = 1, …, T, where

et are uncorrelated random variables, with E{et} = 0, V{et} = σ2 for all t = 1, …, T.

5.4.6 The annual consumption of meat per capita in the United States during the years 1919 – 1941 (in pounds) is (Anderson, 1971, p. 44)

5.4.7 Let (x1i, Y1i), i = 1, …, n1, and (x2i, Y2i), i = 1, …, n2, be two independent sets of regression points. It is assumed that

where xji are constants and eji are i.i.d. N(0, σ2). Let

where j and j are the respective sample means.

Section 5.5

5.5.1 Consider the following raw data (Draper and Smith, 1966, p. 178).

Section 5.6

5.6.1 Let X1, …, Xn be i.i.d. random variables having a binomial distribution B(1, θ), 0 < θ < 1. Find the MLE of

5.6.2 Let X1, …, Xn be i.i.d. P(λ), 0 < λ < ∞. What is the MLE of p(j;λ) = e–λ · λj/j!, j = 0, 1, …?

5.6.3 Let X1, …, Xn be i.i.d. N(μ, σ2), -∞ < μ < ∞, 0 < σ < ∞. Determine the MLEs of

5.6.4 Using the delta method (see Section 1.13.4), determine the large sample approximation to the expectations and variances of the MLEs of Problems 1, 2, and 3.

5.6.5 Consider the normal regression model

where x1, …, xn are constants such that , and e1, …, en are i.i.d. N(0, σ2).

5.6.6 Let (xi, Ti), i = 1, …, n be specified in the following manner. x1, …, xn are constants such that are independent random variables and Ti ∼ , i = 1 = 1, …, n.

5.6.7 Consider the MLE of the parameters of the normal, logistic, and extreme–value tolerance distributions (Section 5.6.6). Let x1 < … < xk be controlled experimental levels, n1, …, nk the sample sizes and J1, …, Jk the number of response cases from those samples. Let pi = (Ji + 1/2)/(ni + 1). The following transformations:

5.6.8 Consider a trinomial according to which (J1, J2) has the trinomial distribution M(n, P(θ)), where

This is the Hardy – Weinberg model.

5.6.9 A minimum chi–squared estimator (MCE) of θ in a multinomial model M(n, P(θ)) is an estimator n minimizing

For the model of Problem 8, show that the MCE of θ is the real root of the equation

Section 5.7

5.7.1 Let X1, …, Xn be i.i.d. random variables having a common rectangular distribution R(0, θ), 0 < θ < ∞.

5.7.2 Let X1, …, Xn be i.i.d. random variables having a common location–parameter Cauchy distribution, i.e., f(x;μ) = , -∞ < x < ∞; -∞ < μ < ∞. Show that the Pitman estimator of μ is

where Y(i) = X(i) – X(1), i = 2, …, n. Or, by making the transformation ω = (1 + u2)−1 one obtains the expression

This estimator can be evaluated by numerical integration.

5.7.3 Let X1, …, Xn be i.i.d. random variables having a N(μ, σ2) distribution. Determine the Pitman estimators of μ and σ, respectively.

5.7.4 Let X1, …, Xn be i.i.d. random variables having a location and scale parameter p.d.f. f(x;μ, σ) = , where -∞ < μ < ∞, 0 < σ < ∞ and (z) is of the form

Determine the Pitman estimators of μ and σ for (i) and (ii).

Section 5.8

5.8.1 Let X1, …, Xn be i.i.d. random variables. What are the MEEs of the parameters of

5.8.2 It is a common practice to express the degree and skewness and kurtosis (peakness) of a p.d.f. by the coefficients

and

Provide MEEs of and β2 based on samples of n i.i.d. random variables X1, …, Xn.

5.8.3 Let X1, …, Xn be i.i.d. random variables having a common distribution which is a mixture α G(λ, ν1) + (1 – α) G(λ, ν2), 0 < α < 1, 0 < λ, ν1, ν2 < ∞. Construct the MEEs of α, λ, ν1, and ν2.

5.8.4 Let X1, …, Xn be i.i.d. random variables having a common truncated normal distribution with p.d.f.

where n(x| μ, σ2) = . Determine the MEEs of (μ, σ, ξ).

Section 5.9

5.9.1 Consider the fixed–effects two–way ANOVA (Section 4.6.2). Accordingly, Xijk, i = 1, …, r1; j = 1, … r2, k = 1, …, n, are independent normal random variables, N(μ ij, σ2), where

Construct PTEs of the interaction parameters and the main–effects (i = 1, …, r1; j = 1, …, r2). [The estimation is preceded by a test of significance. If the test indicates nonsignificant effects, the estimates are zero; otherwise they are given by the value of the contrasts.]

5.9.2 Consider the linear model Y = Aβ + e, where Y is an N× 1 vector, A is an N × p matrix (p < N) and β a p× 1 vector. Suppose that rank (A) = p. Let β′ = (β}(1), β′(2)), where β(1) is a k× 1 vector, 1 ≤ k < p. Construct the LSE PTE of β (1). What is the expectation and the covariance of this estimator?

5.9.3 Let be the sample variance of n1 i.i.d. random variables having a N(μ1, ) distribution and the sample variance of n2 i.i.d. random variables having a N(μ2, ) distribution. Furthermore, and are independent. Construct PTEs of and . What are the expectations and variances of these estimators? For which level of significance, α, these PTEs have a smaller MSE than and separately.

Section 5.10

5.10.1 What is the asymptotic distribution of the sample median Me when the i.i.d. random variables have a distribution which is the mixture

L(μ, σ) designates the Laplace distribution with location parameter μ and scale parameter σ.

5.10.2 Suppose that X(1) ≤ … ≤ X(9) is the order statistic of a random sample of size n = 9 from a rectangular distribution R(μ – σ, μ + σ). What is the expectation and variance of

5.10.3 Simulate N = 1000 random samples of size n = 20 from the distribution of X∼ 10 + 5t[10]. Estimate in each sample the location parameter μ = 10 by , Me, GL, .10 and compare the means and MSEs of these estimators over the 1000 samples.

PART IV: SOLUTIONS OF SELECTED PROBLEMS

5.2.1

5.2.3 The m.s.s. is nX(1) and U = ;

Thus is UMVU of λ, provided n> 2. X(1) ∼ μ + G(nλ, 1). Thus, E, since E{U} = . Thus, = X(1) – is a UMVU.

Thus,

provided n> 3. Since X(1) and U are independent,

for n> 1.

5.2.4

5.2.6 X1, …, Xn ∼ i.i.d. NB(, ν), 0 < < ∞ ν is known. The m.s.s. is T = .

5.2.8 . Hence, , where . Therefore,

Set t = we get

Thus, or .

5.2.9 The probability of success is p = e-Δ/θ. Accordingly, θ = -Δ/log (p). Let . Thus, . The estimator of θ is n = -Δ/log (n). The bias of n is B(n) = -Δ . Let . Taylor expansion around p yields

where |p -p| < |n – p|. Moreover,

Furthermore,

Hence,

5.2.17 X1, …, Xn ∼ G(λ, ν), ν ≥ 3 fixed.

5.3.3 This is continuation of Example 5.8. We have a sample of size n of vectors (X, Y), where . We have seen that . We derive now the variance of 2 = (QX + QY)/2n, where QX = and QY = . Note that QY| QX ∼ σ2(1 – ρ2)χ2. Hence,

and

It follows that

Finally, since (QX + QY, PXY) is a complete sufficient statistic, and since is invariant with respect to translations and change of scale, Basu’s Theorem implies that 2 and are independent. Hence, the variance – covariance matrix of (2, ) is

Thus, the efficiency of (2, ), according to (5.3.13), is

5.4.3 X1, …, X5 are i.i.d. having a Laplace distribution with p.d.f. , where -∞ < μ < ∞, 0 < σ < ∞, and (x) = , -∞ < x < ∞. The standard c.d.f. (μ = 0, σ = 1) is

Let X(1) < X(2) < X(3) < X(4) < X(5) be the order statistic. Note that Me = X(3). Also X(i) = μ + σ U(i), i = 1, …, 5, where U(i) are the order statistic from a standard distribution.

The densities of U(1), U(3), U(5) are

Since (u) is symmetric around u = 0, F(-u) = 1-F(u). Thus, p(3)(u) is symmetric around u = 0, and U(1) ∼ -U(5). It follows that α1 = E{U(1)} = – α5 and α3 = 0. Moreover,

Accordingly, α′ = (-1.58854, 0, 1.58854), V{U(1)} = V{U(5)} = 1.470256, V{U(3)} = 0.35118.

Thus,

Let

The matrix V is singular, since U(5) = -U(1). We take the generalized inverse

We then compute

According to this, the estimator of μ is = X(3), and that of σ is = 0.63X(3)-0.63X(1). These estimators are not BLUE. Take the ordinary LSE given by

then the variance covariance matrix of these estimators is

Thus, V{} = 0.0391 < V{} = 0.3512, and V{} = 0.5827 > V{} = 0.5125. Due to the fact that V is singular, is a better estimator than .

5.6.6 (xi, Ti), i = 1, …, n. Ti ∼ (α + β xi)G(1, 1), i = 1, …, n.

The MLEs of α and β are the roots of the equations:

(I)

(II),

The Newton – Raphson method for approximating numerically (α, β) is

The matrix

where wi = .

We assume that |D(α, β)| > 0 in each iteration. Starting with (α1, β1), we get the following after the lth iteration:

The LSE initial solution is

5.6.9 . For the Hardy – Weinberg model,

where

Note that J3 = n – J1 – J2. Thus, the MCE of θ is the root of N(θ) ≡ 0.

5.7.1 X1, …, Xn are i.i.d. R(θ, θ), 0 < θ < ∞.

5.7.3