**QUESTION**

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Please finish the question without R or SAS. question: LetX, Y be two continuous random variables with joint density function 8 xy / q 4if0 < y < x< q f(x,y½q)=

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LetX, Y be two continuous random variables with joint density function

8 xy / q ^{4}if0 < y < x< q

f(x,y½q)=

We want to estimate q relative to Mean Square Error (MSE).From lecture we already know that X is a sufficient statistic forq and has density function

f_{X }(x½q)=4 x^{3} / q^{4} if0 < x < q

0 otherwise

Consider the estimatord (x,y) =3 x y. Without computing any MSE find an estimator that will have smaller MSE than the estimatord (x,y)=3 x y and explain why that should be true.

**ANSWER**

To find an estimator that will have a smaller Mean Square Error (MSE) than the estimator (d(x,y) = 3xy), we need to consider a new estimator that is unbiased and has a smaller variance.

Let’s denote the new estimator as d'(x, y). Since X is a sufficient statistic for q, we can use the properties of sufficiency to construct a new estimator. The factorization theorem states that if we can factorize the joint density function as the product of a function of the data and a function of the parameter, then the statistic representing the function of the data is sufficient for the parameter.

In this case, the joint density function is given by:

f(x, y|q) = 8xy/q^4 if 0 < y < x < q, otherwise 0.

To find a sufficient statistic, we can write the joint density function as follows:

f(x, y|q) = g(x, y) * h(q),

where g(x, y) = 8xy/q^4 if 0 < y < x < q, and h(q) is a function of q only.

Since X is a sufficient statistic for q, we can express the joint density function as:

f(x, y|q) = g(x, y) * h(q) = g(x, y) * 1, (since h(q) = 1, as q is known)

Now, using the factorization theorem, we can conclude that d'(x, y) = g(x, y) is a sufficient estimator for q.

To construct an unbiased estimator, we need to calculate the expected value of d'(x, y) and set it equal to q:

E[d'(x, y)] = E[g(x, y)] = ∫∫g(x, y) * f(x, y|q) dx dy,

where f(x, y|q) is the joint density function given earlier.

Given the joint density function, we can calculate the expected value of g(x, y) using the limits of integration corresponding to the region of non-zero density:

E[d'(x, y)] = ∫∫g(x, y) * f(x, y|q) dx dy

= ∫∫(8xy/q^4) * (8xy/q^4) dx dy

= ∫∫(64x^2y^2/q^8) dx dy.

Now, to find an estimator with a smaller MSE, we need to minimize the variance. The variance of d'(x, y) can be calculated as:

Var[d'(x, y)] = E[d'(x, y)^2] – E[d'(x, y)]^2.

To minimize the variance, we need to minimize E[d'(x, y)^2], which can be calculated as:

E[d'(x, y)^2] = ∫∫(g(x, y))^2 * f(x, y|q) dx dy

= ∫∫((8xy/q^4))^2 * ((8xy/q^4)) dx dy

= ∫∫(64x^2y^2/q^8) dx dy.

Comparing the expression for E[d'(x, y)^2] with the previous expression for E[d'(x, y)], we can see that they are equal. Therefore, the estimator d'(x, y) = g(x, y) = 8xy/q^4 is unbiased.

Since d'(x, y) = g(x, y) = 8xy/q^4 is an unbiased estimator, the estimator (d(x, y) = 3xy

) cannot have a smaller MSE. Therefore, we cannot find an estimator with a smaller MSE than (d(x, y) = 3xy) based on the given information.