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) .

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orchestra answered 1 year ago

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 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.

Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show that for...

Let X1, . . . , Xn be i.i.d. samples from Uniform(0, θ). Show that for any α ∈ (0, 1), there is a cn,α, such that [max(X1,...,Xn),cn,α max(X1,...,Xn)] is a 1−α confidence interval of θ.

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 =...

Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that...

Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that the maximum likelihood estimator (MLE) of a is ˆ a = max(Xi). a) Let Y = max(Xi). Use the fact that Y ≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y. b) Find the probability density function of Y from cdf. c) Use the obtained pdf to show that MLE for a (ˆ a =...

Let X1, ..., Xn be a random sample from U(0, 3). Recall that this means fXi...

Let X1, ..., Xn be a random sample from U(0, 3). Recall that this means fXi (xi) = 1 3 , 0 < xi < 3, i = 1, ..., n, and that all Xi are mutually independent. Let X(1) ≤ X(2) ≤ ... ≤ X(n) be the order statistics of the random sample. Denote Y1 = X(1). • Derive FXi (xi). • Find FY1 (y) = P(Y1 ≤ y). Hint: Use the complement rule of probability. That is, P(Y1...

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR...

Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Is this family MLR in Y = X(n)? (b) Find the UMP size-α test for H0 : θ ≤ θ0 vs H1 : θ > θ0. (c) Find the UMP size-α test for H0 : θ ≥ θ0 vs H1 : θ < θ0. (d) Letting R1 be the rejection region for the test in part (b) and R2 be the rejection region for the test in part...

2. Let X1, . . . , Xn be a random sample from the distribution with...

2. Let X1, . . . , Xn be a random sample from the distribution with pdf given by fX(x;β) = β 1(x ≥ 1). xβ+1 (a) Show that T = ni=1 log Xi is a sufficient statistic for β. Hint: Use n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1 (b) Find the pdf of Y = logX, where X ∼ fX(x;β). (c) Find the distribution of T . Hint: Identify the distribution of Y and use mgfs. (d) Find...

: 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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