Suppose that (X1, · · · , Xn) is a sample from the normal distribution N(θ,...

Suppose that (X1, · · · , Xn) is a sample from the normal distribution N(θ, θ^2 ) with θ ∈ R. Find an MLE of θ.

Expert Solution

orchestra answered 1 year ago

Suppose X1, . . . , Xn is a random sample from the Normal(μ, σ2) distribution,...

Suppose X1, . . . , Xn is a random sample from the Normal(μ, σ2) distribution, where μ is unknown but σ2 is known, and it is of interest to test H0: μ = μ0 versus H1: μ ̸= μ0 for some value μ0. The R code below plots the power curve of the test Reject H0 iff |√n(X ̄n − μ0)/σ| > zα/2 for user-selected values of μ0, n, σ, and α. For a sequence of values of μ,...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are...

Suppose X1, ..., Xn are i.i.d. from an exponential distribution with mean θ. If we are testing H0 : θ = θ0 vs Ha : θ > θ0. Suppose we reject H0 when ( X¯n/ θ0) > 1 + (1.645/ √n) (a) (10 points) Calculate the power function G(ζ). You may leave your answer in terms of the standard normal cdf Φ(x). (b) (5 points) Is this test consistent?

Suppose that X1,. . . , Xn is an m.a. of a distribution U (0, θ]....

Suppose that X1,. . . , Xn is an m.a. of a distribution U (0, θ]. (a) Find the most powerful test of size α to test H0: θ = θ0 vs Ha: θ = θa, where θa <θ0. (b) Is the test obtained in part (a) the UMP (α) to test H0: θ = θ0 vs Ha: θ <θ0 ?. (c) Find the most powerful test of size α to test H0: θ = θ0 vs Ha: θ =...

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the...

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the interval [0, θ]. Show that T = max(X1, ..., Xn) is a sufficient statistic for θ.

X1, . . . , Xn are random sample from Uniforma(0, θ), θ > 0. Construct...

X1, . . . , Xn are random sample from Uniforma(0, θ), θ > 0. Construct a consistent estimator for θ based on X(n) .

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ)...

Suppose X1; : : : ; Xn is i.i.d Exponential distribution with density f(xjθ) = (1/θ) * e(-x/θ); 0 ≤ x < 1; θ > 0: (a) Find the UMVUE (the best unbiased estimator) of θ. (b) What is the Cramer-Rao lower bound of all unbiased estimator of all unbiased estimator of θ. Does the estimator from (a) attain the lower bound? Justify your answer. (c) What is the Cramer-Rao lower bound of all unbiased estimator of θ^2? 3 (d)...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine...

Let X1,…, Xn be a sample of iid N(0, ?) random variables with Θ=(0, ∞). Determine a) the MLE ? of ?. b) E(? ̂). c) the asymptotic variance of the MLE of ?. d) the MLE of SD(Xi ) = √ ?.

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter µ. a. What is the mean and variance of this distribution? b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator? Why or why not? c. Find the moment estimator and MLE of µ. d. Show the estimators in (c) are unbiased. e. Find the MSE of the estimators in (c). Given the frequency table below: X 0...

: Let X1, X2, . . . , Xn be a random sample from the normal...

: Let X1, X2, . . . , Xn be a random sample from the normal distribution N(µ, 25). To test the hypothesis H0 : µ = 40 against H1 : µne40, let us define the three critical regions: C1 = {x¯ : ¯x ≥ c1}, C2 = {x¯ : ¯x ≤ c2}, and C3 = {x¯ : |x¯ − 40| ≥ c3}. (a) If n = 12, find the values of c1, c2, c3 such that the size of...

. Let X1, X2, . . . , Xn be a random sample from a normal...

. Let X1, X2, . . . , Xn be a random sample from a normal population with mean zero but unknown variance σ 2 . (a) Find a minimum-variance unbiased estimator (MVUE) of σ 2 . Explain why this is a MVUE. (b) Find the distribution and the variance of the MVUE of σ 2 and prove the consistency of this estimator. (c) Give a formula of a 100(1 − α)% confidence interval for σ 2 constructed using the...

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