"Frontmatter". In: Analysis of Financial Time Series

274 VALUEATRISK7.5.2.1 The Parametric ApproachTwo parametric approaches are available. They are the maximum likelihood andregression methods.Maximum likelihood methodAssuming that the subperiod minima {r n,i } follow a generalized extreme value distributionsuch that the pdf of x i = (r n,i −β n )/α n is given in Eq. (7.17), we can obtainthe pdf of r n,i by a simple transformation asf (r n,i ) =⎧ [1⎪⎨ 1 + k n(r n,i − β n )α n α n{1 rn,i − β n⎪⎩ expα nα n] {1/kn −1 [exp − 1 + k ] }n(r n,i − β n ) 1/knα n[ ]}rn,i − β n− expα nif k n ̸= 0if k n = 0,where it is understood that 1 + k n (r n,i − β n )/α n > 0ifk n ̸= 0. The subscript nis added to the shape parameter k to signify that its estimate depends on the choiceof n. Under the independence assumption, the likelihood function of the subperiodminima isl(r n,1 ,...,r n,g | k n ,α n ,β n ) =g∏f (r n,i ).Nonlinear estimation procedures can then be used to obtain maximum likelihoodestimates of k n , β n ,andα n . These estimates are unbiased, asymptotically normal,and of minimum variance under proper assumptions. We apply this approach to somestock return series later.Regression methodThis method assumes that {r n,i } g i=1is a random sample from the generalized extremevalue distribution in Eq. (7.14) and make uses of properties of order statistics; seeGumbel (1958). Denote the order statistics of the subperiod minima {r n,i } g i=1 asi=1r n(1) ≤ r n(2) ≤···≤r n(g) .Using properties of order statistics (e.g., Cox and Hinkley, 1974, p. 467), we haveE{F ∗ [r n(i) ]} =i ,g + 1i = 1,...,g. (7.19)For simplicity, we separate the discussions into two cases depending on the valueof k. First, consider the case of k ̸= 0. From Eq. (7.14), we have