![(c) Hence give a function of Y that is an unbiased estimator of 0 and find its variance (d) Consider H0 : θ-: θο against H1 : θ < θο. Show that the likelihood ratio test L41 of these hypotheses is to reject Ho if Z Sc for some c. (e) To test the null hypothesis Ho : θ 2 against H1 : θ < 2 a statistician takes a random sample of size 11 and proposes to reject Ho if Z s 1.5. i. Show that for given θ > 0 the density of Z is 11210 i. What is the probability of committing a Type I error? iii. If θ < 1.5 what is the probability of committing a Type II error? iv. I 0 1.6 what is the probability of committing a Type II error? v. For θ > 1.5 give an expression for the power of the test. [2] [2) [2](http://img.homeworklib.com/questions/e4eecf90-4023-11ea-9641-dd8b80a46f99.png?x-oss-process=image/resize,w_560)
Homework Answers
HERE WE HAVE APPLIED SOME
BASIC RESULTS OF ORDER STATISTICS AND BASIC KNOWLEDGE ON TEST OF
HYPOTHESIS IS USED TO CALCULATE THE TYPE 1 AND TYPE 2 ERROR
PROBABILITIES AND POWER OF THE TEST IS CALCULATED BY
DEFINITION.
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2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+...
1. Let Y1, . . . ,Y,, be a random sample from a population with density...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x >...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
5. A bowl contains ten marbles of which 0 are white while the others are black....
5. A bowl contains ten marbles of which 0 are white while the others are black. In order to test the null hypothesis H0 : θ-1 against H1: θ > 1, five of the marbles are randomly drawn with replacement and the null hypothesis is rejected if at least two of the five drawn are white. (a) What is the probability of committing a Type I error? (21 (b) What is the probability of committing a Type II error when...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0....
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (c) Show that there is an appropriate statistic T - T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for versusS You must give an explicit expression...
Can anyone help me with this problem? Thank you! 7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1: θ θο. (a) For a sample of size n, find a uniformly most powerful (UM...
Can anyone help me with this problem? Thank you!
7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1: θ θο. (a) For a sample of size n, find a uniformly most powerful (UMP) size-a test if such exists. (b) Take n-?, θ0-1, and α-.05, and sketch the power function of the UMP test.
7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1:...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
2. Let Xi,... Xn be a random sample from the density f(x:0) 1o otherwise Suppose n = 2m+1 for some integer m. Let Y be the sample median and Z = (a) Apply the usual formula for the density of an order statistic to show the density max(X1) be the sample maximum. of Y is 0) 6 3) (b) Note that a beta random variable X has density re+ β22 a-1 (1-2)8-1 with mean μ α/G + β) and variance...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
5. A bowl contains ten marbles of which 0 are white while the others are black. In order to test the null hypothesis H0 : θ-1 against H1: θ > 1, five of the marbles are randomly drawn with replacement and the null hypothesis is rejected if at least two of the five drawn are white. (a) What is the probability of committing a Type I error? (21 (b) What is the probability of committing a Type II error when...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (c) Show that there is an appropriate statistic T - T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for versusS You must give an explicit expression...
Can anyone help me with this problem? Thank you!
7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1: θ θο. (a) For a sample of size n, find a uniformly most powerful (UMP) size-a test if such exists. (b) Take n-?, θ0-1, and α-.05, and sketch the power function of the UMP test.
7. Let X1,.. , Xn denote a random sample from (1-9)/0 x; Test Ho: θ Bo versus H1:...
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