Concept explainers
Let Y1, Y2, …, Yn denote a random sample of size n from a Pareto distribution (see Exercise 6.18). Then the methods of Section 6.7 imply that Y(1) = min(Y1, Y2, … , Yn) has the distribution
Use the method described in Exercise 9.26 to show that Y(1) is a consistent estimator of β.
Want to see the full answer?
Check out a sample textbook solutionChapter 9 Solutions
Mathematical Statistics with Applications
- Let X₁, X2, ..., Xn be a random sample from a normal population with mean μ and variance o² taken independently from another random sample Y₁, Y2, ..., Ym from a normal population with mean Ⓒ and variance 0². Let Z₁ = Xi- Derive the distribution of T = z² o (Y₁-P) 772-1arrow_forwardSuppose that X1, X2, X3 are independent and identically distributed random variables with distribution function: Fx (x) = 1 – 2 for x >0 and Fx (x) = 0 for x 1).arrow_forwardSuppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).arrow_forward
- Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known. Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.arrow_forwardB) Let the random variable X have the moment generating function e3t M(t) for -1arrow_forwardSuppose Y1, Y2, Y3, and Y4 are mutually independent and identically distributed exponential random variables with mean 1. a) Find the moment generating function of the sum T = Y+Y2+Y3+Y4. What is the distribution of T? b) Find the pdf of the maximum order statistic Y(4).arrow_forward5. Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.arrow_forwardse 3 Let Y1, Y2,..., Yn denotes a random sample from the uniform distribution on the interval (0,0 + 1). Let Ô = Y -2 and 02 =Y(n)- n+1 %3D Show that both 6, and 02 are consistent estimators for 0.arrow_forwardB) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.arrow_forward5. Let Y,, Y2, ., Yn be independent, exponentially distributed random variables with mean 0/2. Show that the variance of the minimum, Y1) = min(Y,, Y2, ...,n), are E(Y1)) Var(Ya)) = and 2n 02 4n²°arrow_forwardExercise 2 Let Y1, Y2, ... , Yn denotes a random sample from the uniform distribution on the interval (0,0 + 1). Let 1 and 2 =Y(n) 2 %3D n+1 Show that both ô, and ô2 are unbiased estimators of 0. Find the efficiency of ô1 relative to ê1.arrow_forwardLet X1 ,X be a random sample from N(u, o?). The sample variance S2 is defined by (X- %3D n-1 im1 (n-1)S2 (a) Using the fact that two constants a and b satisfying: has a xn-1) distribution, explain how we can get (n-1)S2 S6 =1-a, (1) where a xo/2(n-1) and b xa/2(n-1) are critical values that can be obtained from a x-table. %3Darrow_forwardarrow_back_iosarrow_forward_ios
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning