/* 8-2-2023     exact mean MISE    v known    mu-hat = Xb   calculation Mathematica  */

p[x_,m_,v_] := Exp[-(x-m)^2/2/v] / Sqrt[ 2 Pi v ]

/*  fhat = N(x,xb,v)   f = n(x,m,v)  Xb = f(xb,m,v/n)   */

ise = Integrate[ ( p[x,xb,v]-p[x,m,v] )^2 , {x,-Infinity,Infinity}, Assumptions -> {v>0,n>1} ]

mise = Integrate[ ise p[xb,m,v/n] , {xb,-Infinity,Infinity}, Assumptions -> {v>0,n>1} ]

Series[ mise, {n,Infinity,2} ]





/*   exact   MISE   S2      */

del = p[x, 0, s2] - p[x, 0, v]
ise$s2 = Integrate[del^2, {x, -Infinity, Infinity}, Assumptions -> {s2 > 0, v > 0}]

  /*  error in class   */
fs2 = (v s2 /(n-1))^((n-3)/2) Exp[ -v s2 / (n-1)/2] / 2^((n-1)/2) / Gamma[ (n-1)/2 ] v/(n-1)

fs2v = ((n-1) s2/v )^((n-3)/2) Exp[ -(n-1) s2 / ( 2 * v ) ] / ( 2^((n-1)/2) * Gamma[ (n-1)/2 ] ) * (n-1)/v

     /* corrected 9-24-2023  *(/
     
fs2v = (n-1)^((n-1)/2) / ( 2^((n-1)/2) * Gamma[ (n-1)/2 ] * v^((n-1)/2) ) * s2^((n-3)/2) * Exp[-(n-1)*s2/(2*v) ]

fs2 = (n-1)^((n-1)/2) * s2^((n-3)/2) * Exp[ -(n-1)*s2/2 ] / ( 2^((n-1)/2) * Gamma[ (n-1)/2 ] )  /* corrected  v=1 case */

,

Integrate[ fs2v,{ s2,0,Infinity}, Assumptions -> { v>0,n>2} ]
mise$s2 = Integrate[ ise$s2 fs2v,{ s2,0,Infinity}, Assumptions -> { v>0,n>2} ]

Series[ mise$s2, {n,Infinity,1} ]   /* still a mess */

LogLogPlot[ mise$2, {n,25,1000} ]  /* ugh  */

     /*  for some reason only works for n odd   9-25-2023   */

ttt = Table[{Log[k], ( mise$s2 /. {v -> 1, n -> k})}, {k, 3, 41, 2}];

ListLogPlot[ttt]






/*   Taylor Series     same   Xb   v known                */

dfhat = (x-xb) Exp[ -(x-xb)^2/2/v ] / Sqrt[ 2 Pi ] / v^(3/2)

ise_ts = Integrate[ dfhat^2 , {x,-Infinity,Infinity}, Assumptions -> v>0 ]





/*   Taylor Series     S^2=S2 unbiased   mu known                */

fhatS2 = Exp[-(x-m)^2/2/S2] / Sqrt[2 Pi S2]

dfhatS2 = D[ fhatS2, S2 ]

wk1 = ( Integrate[dfhatS2^2, {x, -Infinity, Infinity}, Assumptions -> S2 > 0] ) /. { S2 -> s^2 }

var$S2 = 2 s^4 / (n-1)

mise$S2 = var$S2 wk1

mise$S2 = Simplify[ var$S2 wk1 , Assumptions -> {s>0}]  /*  PowerExpand does not req Assumptions  */

mise = Normal[ Series[ mise$S2, {n,Infinity,2}]]