# Question: The information about in a random sample of size

The information about θ in a random sample of size n is also given by

Where f (x) is the value of the population density at x, provided that the extremes of the region for which f(x) ≠ 0 do not depend on θ. The derivation of this formula takes the following steps:

(a) Differentiating the expressions on both sides of

With respect to θ, show that

By interchanging the order of integration and differentiation.

(b) Differentiating again with respect to θ, show that

Where f (x) is the value of the population density at x, provided that the extremes of the region for which f(x) ≠ 0 do not depend on θ. The derivation of this formula takes the following steps:

(a) Differentiating the expressions on both sides of

With respect to θ, show that

By interchanging the order of integration and differentiation.

(b) Differentiating again with respect to θ, show that

## Relevant Questions

Rework Example 10.5 using the alternative formula for the information given in Exercise 10.19. Example 10.5 Show that is a minimum variance unbiased estimator of the mean µ of a normal population. If X1, X2, and X3 constitute a random sample of size n = 3 from a normal population with the mean µ and the variance σ2, find the efficiency of X1 + 2X2 + X3 / 4 relative to X1 + X2 + X3 / 3 as estimates of µ. Show that if Θ is a biased estimator of θ, then To show that an estimator can be consistent with-out being unbiased or even asymptotically unbiased, consider the following estimation procedure: To estimate the mean of a population with the finite variance σ2, we first ...Given a random sample of size n from a population that has the known mean µ and the finite variance σ2, show thatPost your question