Partially missing at random and ignorable inferences for parameter subsets with missing data
Parameter Families Random - UCSD Mathematics
Transcript of Parameter Families Random - UCSD Mathematics
1- Parameter Families of Random Variables
we will often have occasion to considert - parameter families of random variables
{ Xp : te la,b) ) - es sa - b Sos
we will study many stochastic processesof this form ,
but they also come up more
innocuously .
Eg .
Let X be a random variable in Crisp ) .
Then etx 30 for all te IR,so
14×111 #letx ) = fetxdp exists in ↳ a ]
Note : if Hk e then txsltxlselxl
If Fe > o sk ed" e Lt i. fetxdlpsfe""DIP -
- Ile" "Yes .
i. CHEL' ft E. C.-GE) .
Prop .16.30 Let crit
,g) be a measure space ,and f : Ca
,b) xr → IR St
.
t.
w↳ fit,
w) is measurable for each tha,b )
2. fl to
,
. ) E Lt Crisp) for some to e- Ca,
b )
3.
2 flat Usw ) exists for m - a.e.
w and for every becqb)4. There is gehtCrim) sit . 1¥, lt.ws/sguo)for µ - a
-e - w and for every te Ca
,b) -
Then fit; ) c- L' for all to Ca
,b)
,
ti-ffct.co)Mdw) is differentiable en la,b),
and dq.ffct.wsMdw) = fHgtgIµ Idw) -
Pf . tgfqllsw) = LIE n - ( fifth , w) - fugw)) between thet
Mean Value Thm : lflt.wf.f.fct.in/=lffqCt9wJ/sgcw)i. Hans it!.ws .ci#lgupj.iflts:Y ,
"
" ".
Define GH) - fflt.wsMdw) .
Then
GHI - GH = f fairy-fit;D
t - to t.fm/dw)
t.m.am#I=fisi.f ""
IH ftp.n.ll-pvdw)by DCT . H
Eg .
Let fit,w ) = etxlw)
.
measurable ft V
If RM t Lt,then so is ft; ) for Gay ) te -- f
¥.
. dateTX = xetx exists ft t
Let attest ) Then 1×1 sea""
i. IX et Xt s PEK ! e
" - 04×1=61×1
-:# Is e""
ft msn.ae
"
ft!!" -ne
tht
i. at , Itt se.
-
'
- by deal day My = dat # let'T = # Hetty = #Hetty
Repeat : ¥ynµ×µ ) = #(xn et 'T fo ltke .
i. Mx'"
co) = LENI .⇐ moments of X .
Actually,we can do this all at once . Note : etx.no?tnqXh .
Prep : If e""t Lt then Mx is analytic on te, e ) and
Mxlt) - Ia tf.
TECH la se.
Pf. Let fnlt.ws .- Eat
.
Hugh
-
'
- I falls as I s Intl Nwsl"f Einen
.
I NP = e'" 'as et
.
fo Lt.
. ) → etx one
i. by DCT ⑤ Half .) f → El et'll aMx H)
"
El! in = E. In.
#oxy .
i . as > Ef et'T = ftp./" " ) - E. tf #KY .
CautionIt's really important that the conditions of the
differentiate- under - an - integral propositionheld as
. independently of t .
Eg . Lebesgue measure ( cell,Basil? X)
fltw) = Dust = Degli Iwl .
1-.
For fixed t, Gee
,× ,V measurable .
2. If It,
-H Sl and Hegel) ekes so fllg.de L'
3- Film '- Itf 's . 138 }
-
- { §' fgt!:
.
.
- o as .
4. gzo C- Lt . V
daftest , di = ddtffctwjlldw) = 1¥, DX = fodteo .
= days ↳ H -
- deft -1.
*.