The observations are Y1, . . . , Yn. The model is Yi = βxi +...

The observations are Y1, . . . , Yn. The model is Yi = βxi + i , i = 1, . . . , n, where (i) x1, . . . , xn are known constants, and (ii) 1, . . . , n are iid N(0, σ2 ). Find the MLEs of β and σ^ 2 . Are they jointly sufficient for β and σ ^2 ?

Expert Solution

milcah answered 8 months ago

Let Y1,Y2,...,Yn be a Bernoulli distributed random sample with P(Yi = 1) = p and P(Yi...

Let Y1,Y2,...,Yn be a Bernoulli distributed random sample with P(Yi = 1) = p and P(Yi = 0) = 1−p for all i. (a) Prove that E(¯ Y ) = p and V (¯ Y ) = p(1−p)/n2, for the sample mean ¯ Y of Y1,Y2,...,Yn, and find a suﬃcient statistic U for p and show it is suﬃcient for p. (b) Find MVUE for p and show it is unbiased for p.

Suppose that the random variable Y1,...,Yn satisfy Yi = ?xi + ?i i=1,...,n. where the set...

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1. . Let X1, . . . , Xn, Y1, . . . , Yn be...

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If you conduct and experiment 1500 times independently, i=1,2,3,...1500. Let y1, y2.... yN be i.i.d observations...

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

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True or False (a) For any distribution, the sample data, Y1, . . . Yn, is...

True or False (a) For any distribution, the sample data, Y1, . . . Yn, is always a sufficient statistic. (b) Biased estimators are always preferred to unbiased estimators. (c) Maximum likelihood estimators are always unbiased.

Let Y1, Y2, . . ., Yn be a random sample from a uniform distribution on...

Let Y1, Y2, . . ., Yn be a random sample from a uniform distribution on the interval (θ - 2, θ). a) Show that Ȳ is a biased estimator of θ. Calculate the bias. b) Calculate MSE( Ȳ). c) Find an unbiased estimator of θ. d) What is the mean square error of your unbiased estimator? e) Is your unbiased estimator a consistent estimator of θ?

Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent random samples, with...

Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent random samples, with the variables Xi normally distributed with mean μ1 and variance σ12 and the variables Yi normally distributed with mean μ2 and variance σ22. The difference between the sample means, X − Y, is then a linear combination of m + n normally distributed random variables and, by this theorem, is itself normally distributed. (a) Find E(X − Y). (b) Find V(X − Y). (c)...

Suppose Y1, . . . , Yn are independent random variables with common density fY(y) =...

Suppose Y1, . . . , Yn are independent random variables with common density fY(y) = eμ−y y > μ. Derive a 95% confidence interval for μ. Find the MLE for μ to use as the estimator

Assume Y1, Y2, . . . , Yn are i.i.d. Normal (μ, σ2) where σ2 is...

Assume Y1, Y2, . . . , Yn are i.i.d. Normal (μ, σ2) where σ2 is known and fixed and the unknown mean μ has a Normal (0,σ2/m) prior, where m is a given constant. Give a 95% credible interval for μ.

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