![2 Suppose that we observe the continuous random variable X (X1,.., Xn) with state space S, whose distribution we do not know but we are assuming that its p.d.f. belongs to a known family of distributions {fe;Be Θ). We construct an estimator for the unknown parameter θ(X) (a) Explain why it is wrong to write E(ex) and correct it. 12 marks (b) Explain the difference between pdf and likelihood function. [1 mark] (c) Explain the different between estimate and estimator. [1 mark d) In point estimation, we are aiming to find the estimator that minimises a given cost function. What parameters does the cost function depend on? State the exact optimisation problem, describing the range of these parameters. [2 marks](http://img.homeworklib.com/questions/9ecd2e90-5d1e-11ea-ba2f-ad738d333f68.png?x-oss-process=image/resize,w_560)
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2 Suppose that we observe the continuous random variable X (X1,.., Xn) with state space S,...
I'm not sure if there any differences in writing E(^θ(X)) and E(^θ)......If not, why is the...
I'm not sure if there any differences in writing E(^θ(X)) and
E(^θ)......If not, why is the above format wrong?
I'm confused what the question is asking.....I have no idea what
kind of parameters the cost function depends on.....Also, refer to
the exact optimisation problem, is it saying the situation is
unbiased? But then, what is the range of these parameters?
A lot of thanks if you could answer my questions!
Suppose that we observe the continuous random variable X -...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood fun...
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
2. (18 marks) Suppose that X1, ..., Xn constitute a random sample of size n from...
2. (18 marks) Suppose that X1, ..., Xn constitute a random sample of size n from a population XN (4,0). Define n n 1 S2 (Xx - X)2 and S. (X-X) n-1 k=1 (2a) Compare S2 and Sas two estimators of o2; (25) Show that S2 satisfies the inequality: var(S2) > 1/nI(o), where I (a) is the Fisher's information function given by I(oP) = -E 1 (2 - )2 a(02)2 ex 202 02 (20) We have known the fact that...
Problem 5 (15pts). Suppose that we observe a random sample X. from the density Xn 1 0 2 0, else, where m is a known constant which is greater than zero, and 0>0. (a) Find the most powerful test...
Problem 5 (15pts). Suppose that we observe a random sample X. from the density Xn 1 0 2 0, else, where m is a known constant which is greater than zero, and 0>0. (a) Find the most powerful test for testing Ho : θ Bo against b) Indicate how you would find the power of the most powerful test when θ-e-Do not perform (c) Is the resulting test uniformly most powerful for testing Ho :0-00 against Ha :e> et Explain...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
I'm not sure if there any differences in writing E(^θ(X)) and
E(^θ)......If not, why is the above format wrong?
I'm confused what the question is asking.....I have no idea what
kind of parameters the cost function depends on.....Also, refer to
the exact optimisation problem, is it saying the situation is
unbiased? But then, what is the range of these parameters?
A lot of thanks if you could answer my questions!
Suppose that we observe the continuous random variable X -...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
2. (18 marks) Suppose that X1, ..., Xn constitute a random sample of size n from a population XN (4,0). Define n n 1 S2 (Xx - X)2 and S. (X-X) n-1 k=1 (2a) Compare S2 and Sas two estimators of o2; (25) Show that S2 satisfies the inequality: var(S2) > 1/nI(o), where I (a) is the Fisher's information function given by I(oP) = -E 1 (2 - )2 a(02)2 ex 202 02 (20) We have known the fact that...
Problem 5 (15pts). Suppose that we observe a random sample X. from the density Xn 1 0 2 0, else, where m is a known constant which is greater than zero, and 0>0. (a) Find the most powerful test for testing Ho : θ Bo against b) Indicate how you would find the power of the most powerful test when θ-e-Do not perform (c) Is the resulting test uniformly most powerful for testing Ho :0-00 against Ha :e> et Explain...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
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