Mathematics, 10.01.2020 06:31 jayme2407
Let $$x_1, x_2, $$ be uniformly distributed on the interval 0 to a. recall that the maximum likelihood estimator of a is $$a = max(x_i)$$. argue intuitively why aˆ cannot be an unbiased estimator for a. b. suppose that e(a) = na/(n + 1). is it reasonable that aˆ consistently underestimates a? show that the bias in the estimator approaches zero as n gets large. c. propose an unbiased estimator for a. d. let $$y = max(x_i)$$. use the fact that y ≤ y if and only if each $$x_i ≤ y$$ to derive the cumulative distribution function of y . then show that the probability density function of y is. $$f(y) = [ny^n - ^1/a^n 0$$, 0 ≤ y ≤ a otherwise, use this result to show that the maximum likelihood estimator for a is biased. e. we have two unbiased estimators for a: the moment estimator $$a_1=2\overline{\mbox{x}}$$ and $$a_2 = [(n + 1)/n] max(x_i)$$, where max $$(x_i)$$ is the largest observation in a random sample of size n. it can be shown that $$v(a_1) = a^2/(3n)$$ and that $$v(a_2) = a^2/[n(n + 2)]$$. show that if n > 1, aˆ2 is a better estimator than aˆ. in what sense is it a better estimator of a?
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Let $$x_1, x_2, $$ be uniformly distributed on the interval 0 to a. recall that the maximum likeliho...
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