Question

Q1 and Q2 (please also show the steps):

Q1 Prove that MSE) = Var(ë) + Bias(@?, i.e., El(Ô – 9)2) = E[(O - ECO)?] + [ECO) – 6)2. Q2 Suppose X1, X2, ..., X, are i.i.d. Bernoulli random variables with probability of success p. It is known that = is an unbiased estimator for p. n 1. Find E(2) and show that p2 is a biased estimator for p? (Hint: make use of the distribution of x. and the fact that Var(Y) = E(Y?) - E(Y)?) 2. Suggest an unbiased estimator for p. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= is a biased estimator for p. 4. For what values of p, MSE() is smaller than MSE(P)? MacBook Ai

Answer 1

FE MSE[6) (6-0) --LE() + +(01-01 * 16-263] + 2 6–6C6) [+16) ] + [e 197-07] e/lo-eco] + 2 [6-42e)] [e com- 0) + E [CO) J? E(-E66)] + 2 [e[e [6-F(6) J*+ 2x + [cco-ol? Var (0) + Bias (?) [2(6):6)]E (6)-0) + (606)-07) (2) X, X... X, ind Be moulli (p) E(X)= p Vcxi ) - P(1-p) We know that xi N Binomial (n. p) and - EXi is an unbrated estimator of p. 14 p 1) E () Fle] h I & E(X,?) v (xi) ip lip tx [.p (1-p-pl] E(X)= plip) = p (1-p-p p[l-p- 8) = p-p² - p = p. 2p h 2 Elp) + p² 2 n2 not ube

T= {Xi NB (nip). 2. We know that E (T) = E(EX) amp Var (1) - Var (Exi) Eng Cl-p). E(+²) - up-up²+ mph [ Var (7) = E(+²) {E (7)] E(T_T)= e(t²) - E(T) - np-up + sp² - up mpe (m-1 2 E [7²_T 7²_1 i an ube of p² n(n-1) EX: [Ex:-) ube of p //

is a beased estimatoo of P. nt 4 n74 3) Ex: +2 (p) I Elexi +2 [E(Exi) + 2 - 1 (upt 2) npt2 Elp) & n+4 int 4 4). Now Var Var (5) = (n+7 ye Var (Exit2) Var (Exi) mpClope (n+4) (+4) I Var (Ex1) = xp (1-2) apcep Var hz Var (8) Var (8) < Var (f) when pup 2