chapter 1

Chapter 6: Point Estimation38.a. The likelihood is( ) ( )( ) 2 ( ) 2µµ⎜⎛µµ ⎟⎞x −y −122 122 1⎝ Σ x i − i +Σ y i − in⎠− i i− i i−σσ2σ2Π e ⋅ e == 1 222ei2 n2πσ2πσπσ( )22 Σ( x µ i) (i µ i)likelihood is thus ( ) i − +Σ y −zero giveslikelihood gives2( ) d− nln2πσ−2. Taking2σdµ ixi+ yiµ ˆi= . Substituting these estimates of theis2'⎛2( )⎜⎜ ⎛ x + ⎛ +1iyi⎞xiyi⎞− nln 2πσ− ∑⎜− ⎟ + ∑⎜− ⎟⎟ ⎟2x2iyσi⎝ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎠22d1 1= −nln ( 2πσ) −2( Σ( x i− y ) ). Now takingsolving for2σ22σ gives the desired result.i2dσ2. The logand equating toˆµ into the log2⎞, equating to zero, and1b. E( σ ˆ) = E Σ( X − Y )E4n−Y( ) 2 = ⋅ ΣE( X − Y )412iinXi, but22 22( X ) = V ( X −Y) + [ E( −Y)] = 2σ + 0 = 2σiii22 σ1 2( ˆ ) Σ( 2σ)1 2E σ = = 2nσ=4n4n2. Thus, so the mle is definitely not unbiased; theexpected value of the estimator is only half the value of what is being estimated!218