Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0

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Question

Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0

Define f(x) as follows

θ² -1<=x<0

1-θ² 0<=x>1

0 otherwise

Let X1, … Xn be iid random variables with density f for some unknown θ (0,1),

Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b.

Find he Maximum likelihood estimator of θ?
Is it asymptotically normal in this sample?
Find the asymptotic variance
Consider the following hypotheses:
H0: X is Unif (-1,1)
H1: X is not distributed as Unif (-1,1)
Write down the test statistics (Wald)Tn. Use the value of θ thst defines H0 as the argument of the asymptotic variance V(θ): Hint: write the hypotheses in terms of θ.

math Statistics-And-Probability

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Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0

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Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0

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