F71SM1 STATISTICAL METHODS TUTORIAL ON 7 ESTIMATION SOLUTIONS

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1 F7SM STATISTICAL METHODS TUTORIAL ON 7 ESTIMATION SOLUTIONS RJG. (a) E[X] = 0 xf(x)dx = θ 0 y e y dy = θγ(3) = θ [or note X gamma(, /θ) wth mean θ (from Yellow Book)] Settng X = θ MME θ = X/ (b) L(θ) = kθ n exp( x /θ) l(θ) = n log θ x /θ + const. U(θ) = dl dθ = n + θ θ. Note U(θ) = n ( ) θ n θ () I(θ) = E[ d l dθ ] = E[ θ 3 n θ ] = n.n.θ θ3 θ = n θ (c) U(θ) = 0 ˆθ = X/ E[ˆθ] = E[ X/] = E[X/] = θ/ = θ, so unbased So MSE[ˆθ] = V [ˆθ] = V [ X]/4 = V [X]/4n = θ /4n = θ /n CRLBound = /I(θ) = θ /n Asymptotc dstrbuton: ˆθ N(θ, θ /n) [ ] snce V [X] = θ (from Yellow Book) = V [ˆθ] so CRLBound s attaned Note: we can deduce drectly from the representaton of U(θ) above n () that ˆθ = X/, that ˆθ s unbased, and that t attans the CRLBound, wth V [ˆθ] = θ /n. (d) Bas of a X s gven by B[a X] = E[a X] θ = aθ θ = (a )θ V [a X] = a V [ X] = a θ /n MSE[a X] = {(a )θ} + a θ /n = {a + n(a ) }θ /n Mnmsng M = MSE[a X] dm : da = {4a + 4n(a )}θ /n = 0 a = [You can check that ths s a mnmum f you care to.] n n + So, on a mnmse MSE crteron, the optmum estmator whch s a multple of X s θ* = n n + X, wth MSE[θ*] = θ (check) n + (e) Comparson: n MSE[θ*] MSE[ˆθ] MSE[θ*]/MSE[ˆθ] n θ /(n + ) θ /n n/(n + ) θ /3 θ / θ / θ / θ /0 θ / On the mnmse MSE crteron ˆθ s effectvely as good as θ* for large samples.. P(bulb fals by tme t 0 ) = e λt 0, so # number of bulbs whch fal by t 0, X b(n, e λt 0 ). MME: we have observaton (x) of X, whch has mean n( e λt 0 ), so we set x = n( e λt 0 ) whch λ = t 0 log( X/n)

2 MLE: the MLE of a bnomal probablty s the observed sample proporton, so we set x/n = e λt 0 whch ˆλ = t 0 log( X/n) [as above MLE and MME are the same] [OR start from the lkelhood functon L(λ) = k ( e λt 0) x (e λt 0 ) n x ] 3. P ( or more clams) = P (no clam) = e λ λ 0 /0! = e λ # of polcyholders who make clams, X b(n, e λ ) MLE: the MLE of a bnomal probablty s the observed sample proporton, so we set X/n = e λ whch ˆλ = log( X/n) Second part: there are n x polces wth no clams, m polces wth clam, and x m wth clams, so L(λ) = ( e λ) n x ( λe λ ) m ( λ e λ / ) x m = ke nλ λ x m l(λ) = nλ + (x m) log λ + const. dl = n + (x m)/λ dλ Settng dl dλ = 0 gves ˆλ = (X M)/n 4. (a) E[Y ] = V [Y ] = λx We mnmse S = (Y E[Y ]) = (Y λx ) ds dλ = x (Y λx ) = 0 λ* = Y x E[λ*] = x E[ x Y ] = x E[Y ] = x λx = λ so unbased V [λ*] = ( x ) V [ x Y ] = ( x x 3 x V [Y ] = λ ) ( x ) (b) f(y ; λ) = exp( λx )(λx ) y /y!, =,..., n L(λ) = k exp( λ x )λ P y l(λ) = λ x + y log λ + const. dl dλ = x + y = 0 λ ˆλ Y = E[ˆλ] = E[Y ] = λ = λ so unbased V [ˆλ] = Notes: ( x ) V [Y ] = ( λ x ) = λ. We can begn from Y P osson(λ x ) from whch the MLE of λ follows mmedately, snce the MLE of the Posson mean λ x s the sngle observaton Y.. We can wrte U(λ) = dl dλ = λ ( y λ) gvng the MLE, and extendng the result about the form of the score functon to ths new stuaton (the rvs are not dentcally dstrbuted here), we get the varance of the MLE as well.

3 5. (a) Let X be # throws requred to get a 6. E[X] =.θ +.θ( θ) + 3.θ( θ) +... = θ[ ( θ)] = /θ Note: X geometrc(θ) as Yellow Book page 8 wth k =, p = θ MME: settng X = /θ θ = / X. Here θ = /4. = 0.44 MLE: f(x) = θ( θ) x, x =,,... L(θ) = θ n ( θ) P x n l(θ) = n log θ + ( x n) log( θ) + const dl dθ = n θ n = 0 θ ˆθ = / X. Here ˆθ = /4. = 0.44 (same as MME) (b) MME = MLE = X. Here = 4.30cm (c) MME: θ = X. Here =.67 = 3.34* ; MLE: ˆθ = max(x). Here = 3.4 *Note ths estmate s nadmssble snce we have an observaton greater than ˆβ = Y / x, Y N(βx, σ ), ˆβ N(β, σ / x ) E[Y ] = σ + β x, E[ ˆβ ] = σ x + β (n )ˆσ = Y ˆβ x Y + ˆβ x = Y ˆβ x E[(n )ˆσ ] = E[ Y ˆβ x ] = {σ + β x } x = nσ + β x σ β x = (n )σ hence result 7. We know h(x) = (n )as, E[S ] = σ, (n )S σ χ n Bas: B(h) = E[(n )as ] σ = {(n )a }σ ( ) σ χ n has varance n, so V [S ] = (n ) = σ4 n n V [h] = (n ) a σ4 n = (n )a σ 4 MSE[h] = (n )a σ 4 + [(n )a ] σ = f(a)σ 4 where f(a) = (n )a + [(n )a ] { σ x + β } f (a) = 4(n )a + (n )[(n )a ] = 0 a = /(n + ) clearly a mnmum So mnmum MSE estmator of the gven form s h(x) = (X n + X) = n n + S Usual estmator s S, whch has MSE σ4 n ; ths estmator s n n + S, whch has MSE σ4 n + 8. (a) E[X] = θx θ dx =... = θ θ ; MME: settng X = θ θ θ = X X (b) L(θ) = kθ n x θ l(θ) = n log θ θ log x + const. dl dθ = n θ log x = 0 ˆθ n = log X [ d ] l I(θ) = E dθ = n θ ese(ˆθ) = ˆθ θ n. Asymptotc dstrbuton: ˆθ N(θ, n ) (c) MME: θ = 5.7 MLE: ˆθ = 5.

4 [ 9. (a) R ( ) ] + R N(r, σ ) E R + R π = π( σ + r ) = A + π σ σ. Bas = π [ π ] (b) E (R + R) = π..(σ + r ) = A + πσ. Bas = πσ (c) E[πR R ] = π r r = πr = A. Bas = 0 0. X N(µ, σ /n), X N(µ, σ /n), X, X ndependent (a) E[µ*] = E[w X + ( w) X ] = we[ X ] + ( w)e[ X ] = wµ + ( w)µ = µ unbased (b) V [µ*] = V [w X + ( w) X ] = w V [ X ] + ( w) V [ X ] = w σ /n + ( w) σ /n d dw V [µ*] = wσ /n ( w)σ /n = 0 w = σ σ + clearly a mnmum σ σ σ σ σ (c) V [µ*] = n (σ + σ ) (σ + σ) = n (σ + σ ) Let µ** = ( X + X ) ; V [µ** ] = 4 n (σ + σ) σ σ n (σ Relatve effcency of µ* and µ** = + σ ) = 4σ σ 4n (σ + σ) (σ + σ ). (a) L(θ) = kθ n A{θ( θ)} n B{θ( θ) } n C {( θ) 3 } n D = kθ n A+n B +n C ( θ) n B+n C +3n D l(θ) = (n A + n B + n C ) log θ + (n B + n C + 3n D ) log( θ) + const. U(θ) = dl dθ = (n A + n B + n C )/θ (n B + n C + 3n D )/( θ) = 0 ˆθ n A + n B + n C = n A + n B + 3n C + 4n D (b) I(θ) = E[ d l dθ ] = E[(n A + n B + n C )/θ + (n B + n C + 3n D )/( θ) ] = θ.n{θ + θ( θ) + θ( θ) } + ( θ).n{θ( θ) + θ( θ) + 3( θ) 3 } = n(3 3θ + θ ) θ( θ) (c) () ˆθ = 78/35 = , I(ˆθ) = 338 ese(ˆθ) = 338 / = 0.07(34) () Expected frequences are 00ˆθ etc..e. A : 09.5 B : 49.5 C :.4 D : 8.5 The sample contans many more A and fewer B nvestors that the model suggests. Note : A formal test of goodness-of-ft of the model shows that s some evdence to justfy rejectng the model as t does not provde a good enought ft to the data. The man problem s wth category B nvestors we observed 50% more than suggested by the model. so

5 θ. (a) E[X x ] = θ θ dx = θ /3. Settng x /n = θ /3 MME: θ 3 = X n MLE: ˆθ = max { X, =,,..., n} (.e. the mamum absolute value of an observaton) (b) P (X > k) = e kθ MLE of e 4000θ s 4/00 = 0.4 Notng that e 8000θ = {e 4000θ } t follows that the MLE of e 8000θ s 0.4 = (c) Let N be the total number of spongos n the area. Estmate of the proporton tagged, n/n, s k/m, so estmate of N s mn/k. (d) The dancers are numbered,,..., k. A sensble model for the number of the dancer who comes out, X, s a unform dstrbuton on {,,..., k}, snce all dancers are equally lkely to be the one who comes out, so P (X = x) = /k, x =,,..., k. We have a sngle observaton x = 7. MME: E[X] = (k + )/ so settng x = E[X] gves 7 = (k + )/ k = 33 MLE: L(k) = /k, k x (and = 0 otherwse). L(k) s mamsed when k s mnmsed, and the smallest value of k consstent wth the data s k = x. So ˆk = x. Here ˆk = 7 (e) Y = X θ exp() E[X θ] = E[X] = θ + E[ X] = E[X] = θ + so X s based MME: settng X = θ + θ = X MLE: All the observatons exceed θ. L(θ) s non-zero for θ < x, =,,..., n. In ths range L(θ) ncreases to a mamum as θ ncreases to ts mamum allowable value, whch s the smallest observaton. So ˆθ = mn {X, =,,..., n} So answers are No, No, No. (f) MLE of µ s X, so MLE of θ = µ 3 s ˆθ = X 3 E[θ*] = E[X X X 3 ] = E[X ]E[X ]E[X 3 ] (by ndependence), so E[θ*] = µ 3 = θ, so θ* s unbased E[(θ*) ] = E[X ]E[X ]E[X 3 ] = ( + µ ) 3 So MSE[θ*] = V [θ*] = ( + µ ) 3 (µ 3 ) = + 3µ + 3µ 4 (g) Sample sze n. Let X be the number of members of the sample whch possess the property concerned. X b(n, π) N(nπ, nπ( π)) As π ncreases from 0 to 0.5, π( π) ncreases, so the most varable/uncertan case for π n the range 0 π 0. occurs when π = 0.. So, takng π = 0. n V [X] gves X N(nπ, 0.09n). Our estmate of the populaton proporton s X/n and we requre P (nπ 0.0n X nπ + 0.0n) n n P ( < Z < ) n 0.09n 0.0n n > n n > 0 n <.059 or > 3.65 n 558