Let us define an estimator of location as the value of µ that minimizes                         f (µ) = n i=1 |Xi − µ| 3/2 , giving an estimator part way between the median (exponent 1) and mean...

Let us define an estimator of location as the value of µ that minimizes

                        f (µ) = n i=1 |Xi − µ| 3/2 ,

giving an estimator part way between the median (exponent 1) and mean (exponent 2).

(a) Show that the derivative f (µ) is monotone increasing.

(b) Where does the second derivative(µ) not exist?

(c) Find an interval in which the minimum of f must be attained.

(d) Show that neither Newton’s method nor the secant method are guaranteed to work.

(e) Can regula falsi also fail?

(f ) Of the many methods, what is the best method for computing this estimate?