b) Let X to be a random sample from a distribution with a probability density function given by (ex-1,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.
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- For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?c) Let Y₁, Y₂, ..., Yn be a random sample whose probability density function is given by y3 - formez. 0, f(y; B) = 0 0 elsewhere 200 and suppose that n = 200, Σ²y₁ = 20, ²ºy² 100, i) Derive the standard error of ß, se() = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for B. = y = 250 and  = 0.025.Let X1, , Xµ be iid with population density (1 0) I>0, Sx(x) = %3D otherwise. Here 0 is an unkown population parameter. 0 has an Exponential(1) distribution. Find the method of moment estimator for 0. Let's call this 6. Is ô unbiased for 0 ? Explain with precise computation. Show that X Find the maximum likelihood estimator for 0. Let's call this 62. Is ô2 unbiascd for 0 ? Explain with precise computation.
- Please answer number 3Let Y1, Y2,..., Y, denote a random sample from the density function given by 1 yª-'e=y/®, y> 0, f(y[a, 0) = elsewhere, where a > 0 is known. a Find the MLE Ô of 0. b Find the expected value and variance of ê. c. Is the MLE ô an unbiased estimator for 0?Suppose that the random variable X has a Gamma Distribution with parameters a = 2 and B 2 where ·1 > 0 The variance of the random variable X, oy, is
- b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 i) Find the level of significance of the test, a. ii) Calculate the power of the test.With reference to Definition 4, show that μ0 = 1and that μ1 = 0 for any random variable for whichE(X) exists.b) Let X to be a random sample from a distribution with a probability density function given by f(x; 0) = {0x0-¹,if 0 < x < 1, 0 € {1,2}, elsewhere O It is desired to test a null hypothesis Ho: 0 = 1 against the alternative hypothesis H₁:0 = 2. Suppose that the test rejects Ho if x ≥ ii) Calculate the power of the test.
- Please answer number 2b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.