Let X1, · · · , Xn ∼ Uni[0, θ], here θ is an unknown quantity. Find a 90% confidence interval of θ.
Let X1, . . . , Xn ∼ iid Unif(θ − 1/2 , θ + 1/2 ) for θ unknown. Find an asymptotic confidence interval for θ.
Let X1, . . . , Xn ∼ iid Unif(θ − 1/2 , θ + 1/2 ) for θ unknown. Find an asymptotic confidence interval for θ.
Let X1, ..., Xn be IID observations from Uniform(0, θ). T(X) = max(X1, . . . Xn) is a sufficient statistic (additionally, T is the MLE for θ). Find a (1 − α)-level confidence interval for θ. [Note: The support of this distribution changes depending on the value of θ, so we cannot use Fisher’s approximation for the MLE because not all of the regularity assumptions hold.]
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0 >0 is unknown (a) Find a 1-a confidence interval for 0 by pivoting the cdf of X(n) = max{X1, ... , Xn}. (b) Show that the confidence interval in (a) can also be obtained using a pivotal quantity Let X1,... , Xn be a random sample from a population with pdf 3x2/03,E(0, 0), f(x|0) = otherwise 0, where 0...
Let X1 , . . . , xn be n iid. random variables with distribution N (θ, θ) for some unknown θ > 0. In the last homework, you have computed the maximum likelihood estimator θ for θ in terms of the sample averages of the linear and quadratic means, i.e. Xn and X,and applied the CLT and delta method to find its asymptotic variance. In this problem, you will compute the asymptotic variance of θ via the Fisher Information....
Let X1,... Xn i.i.d. random variable with the following riemann density: with the unknown parameter θ E Θ : (0.00) (a) Calculate the distribution function Fo of Xi (b) Let x1, .., xn be a realization of X1, Xn. What is the log-likelihood- function for the parameter θ? (c) Calculate the maximum-likelihood-estimator θ(x1, , xn) for the unknown parameter θ
Let X1,... , Xn be a random sample from the Pareto distribution with pdf { f (r0)= 0, where 0>0 is unknown (a) Find a uniformly most powerful (UMP) test of size a for testing Ho 0< 0 versus where 0o>0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for 0 with confidence coefficient 1-a by pivoting a ran- dom variable based on T = log Xi. (Use quantiles...
Let X1, · · · ,Xn be iid from Uniform(−θ,θ), where θ > 0. Let X(1) < X(2) < ... < X(n) denotes the order statistics. (a) Find a minimal sufficient statistics for θ (d) Find the UMVUE for θ. (e) Find the UMVUE for τ(θ) = P(X1 > k).
et (X1,··· ,Xn) be a sample from U[0,θ], where θ ∈ (0,1) is unknown, and θ has a prior distribution U[0,1].