Probability and Statistics

Fisher Information and Asymptotic Normality of the MLE.Consider the statistical model (R,{Pθ}θ∈R) associated to the statistical experiment X1,…,Xn∼iidPθ∗, where θ∗ is the true parameter. Assume that the conditions of the theorem for the convergence of the MLE hold. Which of the following statements about the Fisher information I(θ) is true?

Fisher Information and Asymptotic Normality of the MLE 1 point possible (graded) Consider the statistical model (R,{Pθ}θ∈R) associated to the statistical experiment X1,…,Xn∼iidPθ∗, where θ∗ is the true parameter. Assume that the conditions of the theorem for the convergence of the MLE hold. Which of the following statements about the Fisher information I(θ) is true? …

Fisher Information and Asymptotic Normality of the MLE.Consider the statistical model (R,{Pθ}θ∈R) associated to the statistical experiment X1,…,Xn∼iidPθ∗, where θ∗ is the true parameter. Assume that the conditions of the theorem for the convergence of the MLE hold. Which of the following statements about the Fisher information I(θ) is true? Read More »

As learned from the course and slides, we know that under some regularity conditions, an unbiased estimator Tx) is efficient for the parameter off and only if 1(0)T()-01 alog Ple:) 80 where @) is the Fisher information. A) (10 pts. Use the definition of Fisher information to prove that the Fisher information (0) is non-negative. b) (10 pts. Suppose that Fisher information is positive and the maximum likelihood estimator can be found by 0, then prove that if an unbiased efficient estimator 7(a) exists, then it must be the MLE ala

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14. (a) Let Y be an exponential random variable with mean land X = 01 +02Y, 02 > 0. Find the pdf of X and remember to state the support of X. X is said to follow a shifted exponential distribution with location parameter 01 and scale parameter 02. (b) Let X1, X2, … , Xn be a random sample which Xį are identically distributed as X. Find the method-of-moments estimator for 61 and 02. (c) When 02 is fixed, show that the likelihood function is strictly increasing in 0, when 01 X(1), where x(1) = min{X1, X2, … , Xn} is the sample minimum. Hence find the maximum likelihood estimator of 01 and 02.

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2.16 Two students have an argument over who can turn an angle better. To resolve the argument, they agree to each measure a single angle 10 times. The results of the observations are: Student A Student B 108°26’10”, 108*26’10”, 108°26’08”, 108°26’12”, 108°26’11”, 108*26’09”, 108°26’10”, 108*26’10”, 108°26’05”, 108°26’13”, 108°26’12”, 108°26’01”, 108°26’04”, 108° 26’10”, 108°26’11”, 108° 26’01”, 108°26’11”, 108° 26’14”, 108° 26’05” 108°26’03” (a) What are the means and variances of both data sets? (b) Construct a histogram of each data set using a 3″ class width. (e) Which student performed the best in this situation?

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4.6.4 The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. (Use normal approximation. Round your answers to 4 decimal places.) Statistical Tables. and Cha Part 1 XIncorrect Approximate the probability that more than 25 are defective. 0.1294 the absolute tolerance is +/-0.0001

EXPERT ANSWER X ~ Binomial (n,p) Where n = 1000, p = 0.02 For binomial distribution, Mean = n * p = 1000 * 0.02 = 20 Standard deviation = Sqrt( np(1-p)) = sqrt( 1000 * 0.02 * 0.98) = 4.4272 Using normal approximation to binomial distribution and using continuity correction, P( X > 25) …

4.6.4 The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. (Use normal approximation. Round your answers to 4 decimal places.) Statistical Tables. and Cha Part 1 XIncorrect Approximate the probability that more than 25 are defective. 0.1294 the absolute tolerance is +/-0.0001 Read More »

4.6.3 [ Your answer is partially correct. Try again. There were 49.7 million people with some type of long-lasting condition or disability living in the United States in 2000. This represented 19.3 percent of the majority of civilians aged five and over (http://factfinder.census.gov). A sample of 1000 persons is selected at random Use normal approximation. Round the answers to four decimal places (e.g. 98.7654). (a) Approximate the probability that more than 210 persons in the sample have a disability. 0.0804 (b) Approximate the probability that between 180 and 300 people in the sample have a disability. 0.8603 Click if you would like to Show Work for this question: Open Show Work LINK TO TEXT

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