12. Suppose XIX, iid X, P(θ, l), where P(0,1) is the one-parameter Pareto distribution with density...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
Suppose that {X}}=1 are iid random variables uniformly distributed random variables with density fr A f(x; 0) = S (0 – 10)- € (10,0) 0 otherwise (i) Derive the MLE of e. (ii) Obtain the asymptotic sampling properties of 0. Is the distribution of the MLE asymptotically normal?
If X follows a two-parameter Pareto distribution with a = 3 and θ = 100, find the density function of Y, where Y- 1.5X
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ), derive the randomized UMP level α test for l, P-Po p reassigned oE versus H p Po where p, is a number between 0 and 1 8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ),...
Suppose that Xi, X2, , xn is an iid sample from a U(0,0) distribution, where θ 0. În turn, the parameter 0 is best regarded as a random variable with a Pareto(a, b) distribution, that is, bab 0, otherwise, where a 〉 0 and b 〉 0 are known. (a) Turn the "Bayesian crank" to find the posterior distribution of θ. I would probably start by working with a sufficient statistic (b) Find the posterior mean and use this as...
7.2.10 Suppose that X, .., X, are iid with the Rayleigh distribution, that is the common pdf is where θ(> 0) is the unknown parameter. Find the MLE for θ, is the MLE sufficient for θ?
Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X...
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Exercise 3.16: A sample of n independent observations is taken on a rv. X having a logarithmic series distribution, x=1, 2, EWT-0), , x In . Show that the MLE θ of θ where θ is an unknown parameter in the range (0,1) satisfies the equation e+ ž(1-0) ln(1-9-0, Fuercio ti tample mean. Find the asymptotie distribution oftå. Exercise 3.16: A sample of n independent observations is taken on a rv. X having a logarithmic series distribution, x=1, 2, EWT-0),...