1. (20) Suppose X1,..., X, are i.i.d. with common probability density function: fo(x) = *€ (-1,1), Ger 2sinh(0)' where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter. In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1 if Xi > 0 and Yi 0 otherwise. (a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn- (Give the equation only, no need to solve ÔMLE. X). (b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn. (Give the equation only, no need to solve мOM, X). (c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi- mize the likelihood function of Y₁...., Yn). (d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.

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