Let X1,…,Xn∼ Uniform(a,b) where a and b are unknown paramaters and a&lt;b. (a) Find the method of moments estimators for a and b. (b) Find the MLE a^ and b^.

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Let X1,…,Xn∼ Uniform(a,b)  where a and b are unknown paramaters and a&amp;lt;b.  (a) Find the method of moments estimators for a and b.  (b) Find the MLE a^ and b^.

James Etheridge · Accepted Answer

The first moment is∫abxf(x)dx=∫abxdxb−a=12⋅b2−a2b−a=b+a2.The second moment is∫abx2f(x)dx=∫abx2dxb−a=13⋅b3−a3b−a=b2+ba+a23.So equate the sample moments with the population moments found above:x1+…+xnn=x―=b+a2...(1)x12+…+xn2n=b2+ba+a23...(2)It&#039;s routine to solve (1) for b. Plug that expression into (2) wherever you see b. You get a quadratic equation in a. Solving a quadratic equation can be done by a known algorithm. You get two solutions. The estimate of a will be the smaller of the two (Exercise: Figure out why it&#039;s the smaller one).A bit of algebra that may be useful in simplifying the answer is this:x12+…+xn2n−(x1+…+xn{n)2=(x1−x―)2+…+(xn−x―)2n with x― as above. An alternative approach is to let m be the midpoint of the interval [a,b] and let c be the half-length of the interval, so that the interval is [m-c,m+c]. Then you&#039;d havex1+…+xnn=m,x12+…+xn2n=m2+c23It&#039;s easy to solve that for m and c, and above you&#039;re given a and b as functions of m and c.

Let X_{1}, ..., X_{n}\sim Uniform(a,b) where a and b are unknow

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