a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the interval (0-1,0+1). . Find the method of moment estimator of θ. Is your estimator an unbiased estimator of θ? . Given the following n 5 observations of X, give a point estimate of θ: 6.61 7.70 6.98 8.36 7.26
0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n (b) [1 point] If K is a biased estimator for o?, state the function of K that would make it an unbiased...
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
5. Suppose that X1, X2, , Xn s a random sample from a uniform distribution on the interval (9,8 + 1). (a) Determine the bias of the estimator X, the sample mean. (b) Determine the mean-square error of X as an estimator of θ. (c) Find a function, a, of that is an unbiased estimator of θ. Determine the mean-square error of θ.
Consider a random sample (X1, Y1),(X2, Y2), . . . ,(Xn, Yn) where Y | X = x is modeled by a N(β0 + βx, σ2 ) distribution, where β0, β1 and σ 2 are unknown. (a) Prove that the mle of β1 is an unbiased estimator of β1. (b) Prove that the mle of β0 is an unbiased estimator of β0.
Suppose X1, X2, . . . , Xn are a random sample from a Uniform(0, θ) distribution, where θ > 0. Consider two different estimators of θ: R1 = 2X¯ R2 =(n + 1)/n max(X1, . . . , Xn) (a) For each of the estimators R1 and R2, assess whether it is an unbiased estimator of θ. (b) Compute the variances of R1 and R2. Under what conditions will R2 have a smaller variance than R1?
PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution - 5) +1 if 0 <r <1 fx(2) = 10 0. 0.w.. where @ € (-2, 2) is an unknown parameter. We define the estimate ēn as: ô, = 12X – 6 to estimate . (a) Is ên an unbiased estimator of e? (b) Is Ôn a consistent estimator of e?
4. Let X1, X2, ...,Xn be a random sample from a normal distribution with mean 0 and unknown variance o2. (a) Show that U = <!-, X} is a sufficient statistic for o?. [4] (c) Show that the MLE of o2 is Ô = 2-1 X?. [4] (c) Calculate the mean and variance of Ô from (b). Explain why ő is also the MVUE of o2. [6]
1. Let X1, X2,... .Xn be a random sample of size n from a Bernoulli distribution for which p is the probability of success. We know the maximum likelihood estimator for p is p = 1 Σ_i Xi. ·Show that p is an unbiased estimator of p.
Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0 unbiased estimator of e. estimator eCYis an (c) Get the lower bound for the variance of the unbiased estimator found in (b) Let X1, X2,..., Xn be a random sample from Poisson(0), 0 > 0. X. Determine the value of a constant c such that the (b) Let Y =1 -0...