. Let Xl' ... ' Xn rv Uniform(0,θ) and let θˆ = max{Xl , ... ,...

. Let Xl' ... ' Xn rv Uniform(0,θ) and let θˆ = max{Xl , ... , Xn}. Find the bias, se, and MSE of this estimator. + If we assume that θˆ is asymptotically normal, what is the 90% percentile confidence interval? Suppose my data was 1.0 2.0 3.0 4.0 5.0 and report some numbers.

Expert Solution

orchestra answered 2 years ago

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.

Let X1, . . . , Xn i.i.d. Uniform(θ, θ + 1). Show that: ˆθ1 =...

Let X1, . . . , Xn i.i.d. Uniform(θ, θ + 1). Show that: ˆθ1 = X¯ − 1 2 and ˆθ2 = X(n) − n n + 1 are both consistent estimators for θ.

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...

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 ) = √ ?.

Let Y1,...,Yn be a sample from the Uniform density on [0,2θ]. Show that θ = max(Y1,...

Let Y1,...,Yn be a sample from the Uniform density on [0,2θ]. Show that θ = max(Y1, . . . , Yn) is a sufficient statistic for θ. Find a MVUE (Minimal Variance Unbiased Estimator) for θ.

For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta (1,θ) density. i) Find...

For a fixed θ>0, let X1,X2,…,Xn be i.i.d., each with the beta (1,θ) density. i) Find θ^ that is the maximum likelihood estimate of θ. ii) Let X have the beta (1,θ) density. Find the density of −log⁡(1−X). Recognize this as one of the famous ones and provide its name and parameters. iii) Find f that is the density of the MLE θ^ in part (i).

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) .

Let X1. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x =...

Let X1. . . . Xn be i.i.d f(x; θ) = θ(1 − θ)^x x = 0.. Is there a function of θ for which there exists an unbiased estimator of θ whose variance achieves the CRLB? If so, find it

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 =...

6. Let X1; : : : ;Xn be i.i.d. samples from Uniform(0;theta ). (a) Find cn...

6. Let X1; : : : ;Xn be i.i.d. samples from Uniform(0;theta ). (a) Find cn such that Theta1 = cn min(X1; : : : ;Xn) is an unbiased estimator of theta (b) It is easy to show that theta2 = 2(X-bar) is also an unbiased estimator of theta(you do not need to show this). Compare theta1 and theta2. Which is a better estimator of theta? Specify your criteria.

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