In each of the following questions, you are given an i.i.d. sample and two hypotheses. For any ?∈(0,1), define a test with asymptotic level ?, then give a formula for the asymptotic ?-value of your test.
a)
b)
c)
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In each of the following questions, you are given an i.i.d.
sample and two hypotheses. For...
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so...
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so that the test has asymptotic level a. choosing a test so that Can does not depend on the estimate of A Note that here we are (If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa. Recall the convention in this course that P (Z...
Using the plug-in method determine A >0 such p (1-p) (1- A),p (1-p)(1+ A)] is a confidence interval for p (1 -p) wit...
Using the plug-in method determine A >0 such p (1-p) (1- A),p (1-p)(1+ A)] is a confidence interval for p (1 -p) with asymptotic level 95%. Note that A should only depend on n and p (Enter hatp for p. If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa for any numerical value a Recall the convention in this course that P (Z< qa) =1-a for Z~ N (0,1).)
Using the...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) =...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) = 0x0-11(0<x< 1) where 0 is some positive number. Using the MLE Ô, find the shortest confidence interval for 6 with asymptotic level 85% using the plug-in method. To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V (Ô), and/or I for the Fisher information I (Ô) evaluated at ê, or you may enter your answer without using...
2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using...
2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using the standard CDF Φ(z) = P(Z 2) where Z ~ N(0,1), instead of real numbers. (a) For X Binomial(n, 1/4), P(X/Vn 0.5). (b) For X1, , , , Xn iid Uniform(0, 1), PK, < 2/(3 ) (c) For Y ~ χ2(n), P(Y < n).
Consider a sample of i.i.d. random variables X1,..., X, and assume their common density is given...
Consider a sample of i.i.d. random variables X1,..., X, and assume their common density is given by fo(a) = exp (3) 1(220), where is an unknown parameter Consider the following set of hypotheses: H:0=1 and H:0 1. You will perform the likelihood ratio test at significance level 7% for these hypotheses. Assume that the sample size is n=500. You observe that • The sample mean is 1 , X; = 0.86: • The (biased) sample variance is 15. (X- Xn)...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
only a-i T or F lit khd where it came from 4. You do not need to simplify results, unless otherwise stated. 1. (20pts.) Indicate whether each of the following questions is True or False by writin...
only a-i T or F
lit khd where it came from 4. You do not need to simplify results, unless otherwise stated. 1. (20pts.) Indicate whether each of the following questions is True or False by writing the words "True" or "False" No explanation is needed. (a) If S is a set of linearly independent vectors in R" then the set S is an orthogonal set (b) If the vector x is orthogonal to every vector in a subspace W...
Suppose you are given the following production function Q= 21x + 9x2-x3 Compose a table calculating...
Suppose you are given the following production function
Q= 21x + 9x2-x3
Compose a table calculating the Total Product (Q), Marginal
Product (MP) and Average Product (AP), for x ranging from 0 to 9
units.
Using the information from the table, graph these individual
curves and identify the rate of which diminishing marginal returns
are evident. Delineate the stages of production and
explain why Stages I and III are considered as
irrational whereas Stage II is termed as rational. What...
You roll a six-sided die. Find the probability of each of the following scenarios. (a) Rolling...
You roll a six-sided die. Find the probability of each of the following scenarios. (a) Rolling a 6 or a number greater than 3 (b) Rolling a number less than 4 or an even number (c) Rolling a 4 or an odd number (a) P(6 or number> 3)- (Round to three decimal places as needed) (b) P/1 or 2 or 3 or 4 or 6)-( Round to three decimal places as needed.) (c) P(4 or 1 or 3 or 5)...
You test the hypotheses Ho A 1 vs H1 : > # 1 by using the test =1 (|X, 5> Can) Find the smallest threshold Can so that the test has asymptotic level a. choosing a test so that Can does not depend on the estimate of A Note that here we are (If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa. Recall the convention in this course that P (Z...
Using the plug-in method determine A >0 such p (1-p) (1- A),p (1-p)(1+ A)] is a confidence interval for p (1 -p) with asymptotic level 95%. Note that A should only depend on n and p (Enter hatp for p. If applicable, enter Phi(z) for the cdf (z) of a normal variable Z, q(alpha) for the quantile qa for any numerical value a Recall the convention in this course that P (Z< qa) =1-a for Z~ N (0,1).)
Using the...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
Let X1, ..., X, bei.i.d. random variable with pdf fe defined as follows: fo (2) = 0x0-11(0<x< 1) where 0 is some positive number. Using the MLE Ô, find the shortest confidence interval for 6 with asymptotic level 85% using the plug-in method. To avoid double jeopardy, you may use V for the appropriate estimator of the asymptotic variance V (Ô), and/or I for the Fisher information I (Ô) evaluated at ê, or you may enter your answer without using...
2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using the standard CDF Φ(z) = P(Z 2) where Z ~ N(0,1), instead of real numbers. (a) For X Binomial(n, 1/4), P(X/Vn 0.5). (b) For X1, , , , Xn iid Uniform(0, 1), PK, < 2/(3 ) (c) For Y ~ χ2(n), P(Y < n).
Consider a sample of i.i.d. random variables X1,..., X, and assume their common density is given by fo(a) = exp (3) 1(220), where is an unknown parameter Consider the following set of hypotheses: H:0=1 and H:0 1. You will perform the likelihood ratio test at significance level 7% for these hypotheses. Assume that the sample size is n=500. You observe that • The sample mean is 1 , X; = 0.86: • The (biased) sample variance is 15. (X- Xn)...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
only a-i T or F
lit khd where it came from 4. You do not need to simplify results, unless otherwise stated. 1. (20pts.) Indicate whether each of the following questions is True or False by writing the words "True" or "False" No explanation is needed. (a) If S is a set of linearly independent vectors in R" then the set S is an orthogonal set (b) If the vector x is orthogonal to every vector in a subspace W...
Suppose you are given the following production function
Q= 21x + 9x2-x3
Compose a table calculating the Total Product (Q), Marginal
Product (MP) and Average Product (AP), for x ranging from 0 to 9
units.
Using the information from the table, graph these individual
curves and identify the rate of which diminishing marginal returns
are evident. Delineate the stages of production and
explain why Stages I and III are considered as
irrational whereas Stage II is termed as rational. What...
You roll a six-sided die. Find the probability of each of the following scenarios. (a) Rolling a 6 or a number greater than 3 (b) Rolling a number less than 4 or an even number (c) Rolling a 4 or an odd number (a) P(6 or number> 3)- (Round to three decimal places as needed) (b) P/1 or 2 or 3 or 4 or 6)-( Round to three decimal places as needed.) (c) P(4 or 1 or 3 or 5)...
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