2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient...
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independent and identically distributed sample from 2. 1 f(ru,02) x < 00, -00 < u < 00, o20 - 00 27TO2 (a) Derive the sample variance, S2, for this random sample (b) Derive the maximum likelihood estimator (MLE) of u and o2, denoted fi and o2, respectively (c) Find the MLE of 2 (d) Derive the method of moment...
Let X1, ..., Xn be a random sample (i.i.d.) from a normal distribution with parameters µ, σ2 . (a) Find the maximum likelihood estimation of µ and σ 2 . (b) Compare your mle of µ and σ 2 with sample mean and sample variance. Are they the same?
Let X1, . . . , Xn be i.i.d. from N(µ1, σ2 ), and Y1, . . . , Ym be i.i.d. from N(µ2, σ2 ). If the two samples are independent, find the maximum likelihood estimates for µ1, µ2, and the common variance σ 2 .
Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show that X(1)+X(n) )/2- 1/2 is also an unbiased estimator of θ, whereX(1) is the minimum order statistic and X(n) is the maximum order statistic. If X - 1/2 is also an unbiased estimator of θ which of the two estimators would you prefer to use.
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the density f(x) = 1 µ e −x/µ for x > 0. (a) Let ˆµ1 = X. Show ˆµ1 is a minimum variance unbiased estimator. (b) Let ˆµ2 = (X1 +X2)/2. Show ˆµ2 is unbiased. Calculate V ar(ˆµ2). Confirm V ar(ˆµ1) < V ar(ˆµ2). Calculate the efficiency of ˆµ2 relative to ˆµ1. (c) Show X is consistent and sufficient. (d) Show ˆµ2 is not consistent...
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and o2 is the unknown parameter. Let y() = 02. (i) Find the CRLB for yo?). (ii) Recall that S2 is an unbiased estimator for o2. Compare the Var(S2) to that of the CRLB for
Let X1....,Xn be a sample of size n from a distribution with expectation u and variance sigma^2 and let u = (2X1+X2+...+Xn-1+2Xn)/(n+1) be an estimator for u. u is consistent,asymptotically unbiased ,unbiased?
1. Suppose that {X1, ... , Xn} is a random sample from a normal distribution with mean p and and variance o2. Let sa be the sample variance. We showed in lectures that S2 is an unbiased estimator of o2. (a) Show that S is not an unbiased estimator of o. (b) Find the constant k such that kS is an unbiased estimator of o. Hint. Use a result from Student's Theorem that (n − 1)52 ~ x?(n − 1)...
Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.