MLE in a Gaussian sequence modelConsider the Gaussian sequence model y i = θ i + w i n , for 1 ≤ i ≤ nand w i ∼ N ( 0 , 1 ) are i.i.d. My goal is to estimate { θ i } i = 1 n based on y 1 , . . . , y n .Assume that there exists an S ⊂ { 1 , 2 , . . . , n } such that | S | = s and θ i ∈ { − 1 , + 1 } for i ∈ S and θ i = 0 for i ∈ S c . What should be the MLE of θ = ( θ 1 , . . . , θ n )?The likelihood in this case becomes: L ( θ ) = ( n 2 π ) n exp ⁡ { − n 2 ( ∑ i ∈ S c y i 2 + ∑ i ∈ S ( y i − θ i ) 2 ) } .We need to minimize the exponent and thus we need θ i = y i for i ∈ S, but in order to respect the fact that θ i = ± 1, we need y i = ± 1 for i ∈ S, but it is not always the case. So, how should I go about this?

Question

MLE in a Gaussian sequence modelConsider the Gaussian sequence model      y    i    =      θ    i    +            w      i              n        ,     for     1  ≤  i  ≤  nand       w    i    ∼  N  (  0  ,  1  ) are i.i.d. My goal is to estimate   {      θ    i        }          i      =      1              n       based on       y    1    ,  .  .  .  ,      y    n  .Assume that there exists an   S  ⊂  {  1  ,  2  ,  .  .  .  ,  n  } such that       |    S      |    =  s and       θ    i    ∈  {  −  1  ,  +  1  } for   i  ∈  S and       θ    i    =  0 for   i  ∈      S          c      . What should be the MLE of   θ  =  (      θ    1    ,  .  .  .  ,      θ    n    )?The likelihood in this case becomes:  L  (  θ  )  =            (                        n                          2          π                    )              n        exp  ⁡      {    −          n      2              (              ∑                  i          ∈                      S                          c                                                  y        i        2            +              ∑                  i          ∈          S                    (              y        i            −              θ        i                    )        2            )        }    .We need to minimize the exponent and thus we need       θ    i    =      y    i   for   i  ∈  S, but in order to respect the fact that       θ    i    =  ±  1, we need       y    i    =  ±  1 for   i  ∈  S, but it is not always the case. So, how should I go about this?

canhaulatlt · Accepted Answer

Step 1      θ    i    =  sgn  (      y    i    ) Step 2and use the fact that             (      x      −      sgn      (      x      )      )        2    ≤            (      x      +      sgn      (      x      )      )        2

Consider the Gaussian sequence model y_i=theta_i+\frac{w_i}{sqrt{n}}, for 1 <= i <= n

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