Module 7. Lecture 7: Statistical parameter estimation

Transcription

1 Lecture 7: Statstcal parameter estmato

2 Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato of parameter:a estmate of a parameter s sad to be ubased estmate, f E( ) =

3 1) Method of Matchg Pots I a data set, 75 % values are less tha, It s assumed to follow the dstrbuto, e f() = ; > 0, estmate the parameter. P[X 3] = 0.75 = F()= f ( ) d = 0 0 e d e = = 1-e ( 1 ) e = 0.75 or = or =.164

4 ) Method of Momets Gve a fucto f(,...,,) ad values,..., we eed to fd,,... j j 1 Geerate umber of equatos by takg momets of the dstrbuto ( ) Take, ay dstrbuto, lke f() = ( ) e - α < < + α 1 1 Take the 1st momet Mea about the org α ( 1 ) 1 ( f) d = ( ) e d α E(X) = µ = = 1 ( ) α α e ( ) 1 d

5 ) Method of Momets Cotd... ( 1) substtutg, y = ( ) = y + 1 E(X) = d = dy α y 1 ( + 1) α y e d E(X) = 1 α α y α y e ydy + 1 e dy α 1 E(X) = + α y 0 1 e dy α [As odd multpler, h(-y) = -h(y) 1 = 0 + = 1 = E( ) = µ 1

6 ) Method of Momets Cotd Secod momet about the mea, E (X- µ ) = σ = ( µ ) f ( ) d α α = α α ( 1 ) 1 µ e d ( ) ( ) Substtutg, = µ 1 ad y = µ ad we wll get, σ =

7 3) Mamum Lkelhood method Sample, 1 1 = We have the followg, f( ; ) f( ; ) f( ; ) 1 3 f ( ; ) pdf evaluated at = Product of f( ; ) f( ; ) s "lkelhood " L 1 3 If L( ; ) > f( ; ), the s the estmate preferred, whch mamzes the lkelhood fucto.

8 3) Mamum Lkelhood method Cotd Because l(l) s a creasg fucto of L, t reaches mamum value l( L) at the same pt., as l(l) does, = 0 (Whe there s o other method feasble, ths method s best oe) β f( ) = βe ; > 0 β s a parameter { },,, Sample avalable; 1 L = f(, β) f(, β) f(, β) f(, β) 1 3 = β β β 1 e e e β β β β β ( ) = 1 = β e = β e (formulato of lke lhood fucto)

9 3) Mamum Lkelhood method Cotd Now set L mamum = 1 l( L) = l β βe l( L) = β 0 as we wat to set the value of β l( L) = = β β = β = = = 1 = Arthmetc average Ma. lkelhood estmate

10 3) Mamum Lkelhood method Cotd 1 µ = 1 f( ) ep σ σ [Take, σ as parameter ot S.D. ad µ also] or µ µσ µσ µσ 1 1 L = f( 1,, ) f(,, ) f(,, ) = ep σ ( σ ) = 1 µ = = or or µ = σ 1 l( L) l( ) = σ 1 l( L) l( L) = 0 = set µ & σ,now µ σ l( L) µ = 0 = µ = µ ( X ) 0 = 1 σ

11 3) Mamum Lkelhood method Cotd 1 1 µ ( ) ep f = σ σ [Take, σ as parameter ot S.D. ad µ also] 1 ( 1 µ 1, µσ, ) (, µσ, ) (, µσ, ) ep L = f f f = ( σ σ ) or L 1 µ σ = σ = 1 l( ) l( ) or l( L) l( L) = 0 = µ σ set µ & σ,now l( L) µ = 0 = µ = 1 σ ( X µ ) = 0

12 3) Mamum Lkelhood method Cotd or µ = = 1 = ad l( L) 4 σ 1 ( µ ) ( ) = 0 = µ 3 σ = 1 σ or ( µ ) - + = 3 σ σ = 1 0 or σ = = 1 ( µ ) [But t s ot the best method. It depeds upo stuato]

13 Hghlghts the Module I hydrology, most of the pheomea are radom ature. E.g. rafall-ruoff model Radom varables volved a hydrologcal process may be depedet or depedet. The radom varables X & Y are stochastcally depedet f ad oly f ther jot desty s equal to the product of margal desty fuctos. Jot desty fucto : Smultaeous occurrece Margal desty fucto : Dstrbuto of oe varable rrespectve of the value of the other varables Codtoed dstrbuto: Dstrbuto of oe varable codtoed o the other varable.

14 Hghlghts the Module Cotd Measures of Cetral Tedecy: Mea Arthmetc average (for sample) Mode Meda Measures of Spread or Dsperso: Rage [(ma-m)] Relatve Rage [=(rage/mea)] Varace Stadard devato, Coeffcet of varato

15 Hghlghts the Module Cotd Measures of Symmetry: Coeffcet of skewess, Kurtoss Correlato coeffcet shows the degree of lear assocato betwee two radom varables Commoly used dstrbutos hydrology : Gamma dstrbuto Normal dstrbuto Log-ormal dstrbuto Uform dstrbuto Etreme value dstrbuto Epoetal dstrbuto

16 Hghlghts the Module Cotd Methods of statstcal parameter estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method