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.

Expert Solution

orchestra answered 1 year ago

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?

Let Y1, Y2, ..., Yn be a random sample from an exponential distribution with mean theta....

Let Y1, Y2, ..., Yn be a random sample from an exponential distribution with mean theta. We would like to test H0: theta = 3 against Ha: theta = 5 based on this random sample. (a) Find the form of the most powerful rejection region. (b) Suppose n = 12. Find the MP rejection region of level 0.1. (c) Is the rejection region in (b) the uniformly most powerful rejection region of level 0.1 for testing H0: theta = 3...

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.

Let Y1, ... , Yn be a random sample that follows normal distribution N(μ,2σ^2) i)get the...

Let Y1, ... , Yn be a random sample that follows normal distribution N(μ,2σ^2) i)get the mle for σ^2 ii)prove using i) that it is an efficient estimator

Suppose that Y1 ,Y2 ,...,Yn is a random sample from distribution Uniform[0,2]. Let Y(n) and Y(1)...

Suppose that Y1 ,Y2 ,...,Yn is a random sample from distribution Uniform[0,2]. Let Y(n) and Y(1) be the order statistics. (a) Find E(Y(1)) (b) Find the density of (Y(n) − 1)2 (c) Find the density of Y(n) − Y (1)

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y...

Consider a random sample (X1, Y1), (X2, Y2), . . . , (Xn, Yn) where Y | X = x is modeled by Y=β0+β1x+ε, ε∼N(0,σ^2), where β0,β1and σ^2 are unknown. Let β1 denote the mle of β1. Derive V(βhat1).

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

1. . Let X1, . . . , Xn, Y1, . . . , Yn be mutually independent random variables, and Z = 1 n Pn i=1 XiYi . Suppose for each i ∈ {1, . . . , n}, Xi ∼ Bernoulli(p), Yi ∼ Binomial(n, p). What is Var[Z]? 2. There is a fair coin and a biased coin that flips heads with probability 1/4. You randomly pick one of the coins and flip it until you get a...

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 ?

Decide if the statement is True or False. A sampling distribution of sample means has a...

Decide if the statement is True or False. A sampling distribution of sample means has a mean equal to the population mean, μ, divided by the sample size.

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.

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