Let Y1, Y2, ..., Yn denote a random sample from an exponential distribution with mean θ. Find the rejection region for the likelihood ratio test of H0 : θ = 2 versus Ha : θ ≠ 2 with α = 0.09 and n = 14.
Rejection region =
Let Y1, Y2, ..., Yn denote a random sample from an exponential distribution with mean θ
Let Y1 , Y2 , . . . , Yn denote a random sample from the uniform distribution on the interval (θ, θ+1). Let a. Show that both ? ̂1 and ? ̂2 are unbiased estimators of θ.
Let Y1, Y2, . . . , Yn be independent random variables with Exponential distribution with mean β. Let Y(n) = max(Y1,Y2,...,Yn) and Y(1) = min(Y1,Y2,...,Yn). Find the probability P(Y(1) > y1,Y(n) < yn).
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a population with Rayleigh distribution (Weibull distribution with parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ > 0, y > 0 Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn}, and ˆθ2 = 1 n Xn i=1 Y 2 i . ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased estimators, and in...
Let Y1,Y2, …… Yn be a random sample from the distribution f(y) = θxθ-1 where 0 < x < 1 and 0 < θ < ∞. Show that the maximum likelihood estimator (MLE) for θ is
QUESTION 3 Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by (Parcto distribution). Consider the estimator β-Yu)-min(n, Y, where β is unknown (a) Derive the bias of the estimator β. (b) Derive the mean square error of B. , Yn).
QUESTION8 Let Y,,Y2, ..., Yn denote a random sample of size n from a population whose density is given by (a) Find the maximum likelihood estimator of θ given α is known. (b) Is the maximum likelihood estimator unbiased? (c) is a consistent estimator of θ? (d) Compute the Cramer-Rao lower bound for V(). Interpret the result. (e) Find the maximum likelihood estimator of α given θ is known.
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following questions. (a) decide whether Z1 = Y2 and Z2=Y4-Y2 are stochastically independent or not. (hint. first find the joint p.d.f. of Y2 and Y4) (b) show that Z1 = nY1, Z2= (n-1)(Y2-Y1), Z3=(n-2)(Y3-Y2), ...., Zn=Yn-Yn-1 are stocahstically independent and that each Zi has the exponential distribution.(hint use change of variable technique)
. Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from an exponential distribution with parameter θ = 1. (a) Find the pdf of Yr. (b) Find the pdf of U = e −Yr .
QUESTION 7 Let Y, Y2, ....Yn denote a random sample of size n from a population whose density is given by (a) Find an estimator for θ by the maximum likelihood method. (b) Find the maximum likelihood estimator for E( Y4).
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a normal population with mean μ and variance 2 . Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....