8. Let X1, , Xn be an ID random sample from the normal density with parameters...
Let XI, X2, , Xn İs a random sample from the probability density function Use factorization theorem to show that X(1) = min(X1 , . . . , Xn) is sufficient for θ Is X(1) minimal sufficient for θ? a. b.
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?
a) Show that Σ.1X, and Σηι x? are jointly sufficient statistics for two un known parameters of the normal distribution N(01,02) (based on the data sample X1,..., Xn) in two ways: by factorization theorem and using the property of the exponential family. b) Show that X and s2 are jointly sufficient statistics for the same distribution. c) Give yet another example of a couple of jointly sufficient statistics. Hint: Example 6.7-5 in Hogg et al. Anyway, make sure to include...
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.
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 X = (X1, . . . , Xn) be a random sample of size n with mean μ and variance σ2. Consider Tm i=1 (a) Find the bias of μη(X) for μ. Also find the bias of S2 and ỡXX) for σ2. (b) Show that Hm(X) is consistent. (c) Suppose EIXI < oo. Show that S2 and ỡXX) are consistent. Let X = (X1, . . . , Xn) be a random sample of size n with mean μ...
3. Let X1, X2, ,Xn be a random sample from N(μ, σ2), and k be a positive integer. Find E(S2). In particular, find E(S2) and var(s2).
Suppose that X1, ..., Xn is a random sample from a normal distribution with mean μ and variance σ2. Two unbiased estimators of σ2 are 1?n 1 i=1 σˆ12 =S2 = n−1 Find the efficiency of σˆ12 relative to σˆ2. (Xi −X̄)2, and σˆ2= 2(X1 −X2)2
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.
Problem 3. Consider a random sample X1, X2,..., Xn from a distribution with log-normal pdf (density function): for t 0 and 0 otherwise. Both μ and σ 0 are unknown parameters. Find the method of moments estinates μ and σ. Hint: computing moments, change of variable y = Int might be useful.