Let X₁, X₂,..., Xn be a random sample from an exponential distribution with the pdfƒ(x; ß) = ¹⁄e¯ª/³, for 0 < x <∞, and zero otherwise. Which statement is correct?X(1) 8 = is an efficient estimator of ß, whose variance achieves the lower bound of theCramer-Rao Inequality which is n(II) 3 = X is an efficient estimator whose variance achieves the lower bound of theCramer-Rao Inequality which is(III) = X is an efficient estimator whose variance achieves the lower bound of theCramer-Rao Inequality which is n(IV)O (IV)Cramer-Rao Inequality which isO (III)1XO (II)O (1)is an efficient estimator whose variance achieves the lower bound of theB

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Answered: Let X₁, X₂,..., Xn be a random sample…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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2. A random sample X₁, X2, X3 of size 3 is drawn from a population with mean and variance o2. Let 1 3 Đi (Xi+X+X3) - (X₁ + X2 + X3), (3X₁ — X₂ +2X3). - 4 (a) Show that ₁ and ₂ are both unbiased estimators of the population 2 = mean . (b) Compute the variance of ₁ and 2. (c) Which estimator would you use? Explain your answer.
2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. 1 f(x; 0) = B 0, e (x-4) B for x > 4 otherwise f) Is the maximum likelihood estimator of the parameter ß unbiased or not? g) What is the maximum likelihood estimator of g (B) = 2(B + 1). h) Is the maximum likelihood estimator of g (B) = 2(ß + 1) unbiased or not?
Suppose that Y₁, Y₂, ..., Ym is a random sample of size m from Gamma (a = 3, B = 0), where 0 is not known. Check whether or not the maximum likelihood estimator Ô is a minimum variance unbiased estimator of the parameter 8.
2) Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function Where, ß > 0. If fram e ¹- {** f(x;0)=B for x > 4 otherwise "1 g) What is the maximum likelihood estimator of g (B) = 2(ß + 1). h) Is the maximum likelihood estimator of g (B) = 2(B + 1) unbiased or not?
Suppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.
2 If X₁ and X₂ are the means of independent ran- dom samples of sizes n₁ and n₂ from a normal population with the mean u and the variance o2, show that the vari- ance of the unbiased estimator 2 w X₁ + (1-w). X₂ is a minimum when @ = n1 n₁ + n₂ idnu
18. Show that for type lII population - x/0 dp O
Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.
Consider a set of data x1, x2, n n i=1 ..., n taken from a population with mean µ. - Show that (x-μ)² = Σ(x₂ − x)² + n(x − µ)². i=1
7. For simple linear regression, we assume that Y = Bo + BIX +e, where e - N(0,0?) and X is fixed (not random). We collect n i.i.d, training sample (x),y1)....(XYn)). Prove that the (Bo.B1) estimated through minimizing RSS equals to the one through maximizing likelihood.
If the roots of the quadratic equation x – ax + b = 0 are real and b is positive but otherwise unknown, what are the expected values of the roots of the equation. Assume that b has a uniform distribution in the permissible range. -
Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |

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