3.4 Let X,, X be a random sample of size n from the U(Q,62) distribution, 6,...
3.3 Let X, ., X, be a random sample of size n from the U(0, e) distribution, Be Ω (0, o), and let Yz be the largest order statistic of the X,'s. Then (i) Employ formula (29) in Chapter 6 in order to obtain the p.d.f. of Y,. (ii) Use part (i) in order to construct an unbiased estimate of θ depending only on (iii) By Example 6 here (with α-0 and A-0) in conjunction with Theorem 3, show that...
1. Let X1, ..., Xn be a random sample of size n from a normal distribution, X; ~ N(M, 02), and define U = 21-1 X; and W = 2-1 X?. (a) Find a statistic that is a function of U and W and unbiased for the parameter 0 = 2u – 502. (b) Find a statistic that is unbiased for o? + up. (c) Let c be a constant, and define Yi = 1 if Xi < c and...
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)
7.46. Let Yi < Y2 Y, be the order statistics of a random sample of size 3 from the distribution with p.d.f. zero elsewhere. Find the Joint p.d.f. of Z.-,, Z,-, and Z,- YYY,. The corresponding transformation maps the space 12 Show that z, and z, are joint sufficient statistics for θ1 and θ2. 7.46. Let Yi
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a) Show that θ = y is an unbiased estimator for θ (b) Show that θ = 1Y is a minimum-variance estimator for θ.
Let X, denote the mean of a random sample of size n from a distribution that has pdf (9xe-3x, x>0 f(x) = 0, otherwise Let Yn = mn (Ăn – ). Find the limit distribution of O N(0, 1) O N(0, 0) O N(o, ž) O N(0, 3) other
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)? 8. Let X1,...,Xn denote a random...
Let Y, Y2, ..., Yn be n i.i.d random variables drawn from the population distribution of Y-(My, oy). Suppose we want to estimate My and we are asked to choose between two possible estimators of Wy: (1)Y, and (2) Y = (x + 3) (a) Show both estimators are unbiased (2 points) (b) Derive the variance of both estimators and discuss which estimator is more efficient (3 points)