Some Useful Distributions

Chapter 10 Some Useful Distributions Definition 101 The population median is any value MED(Y ) such that P (Y MED(Y )) 05 andp (Y MED(Y )) 05 (101) Definition 10 The population median absolute deviation is MAD(Y )=MED( Y MED(Y ) ) (10) Definition 103 The gamma function Γ(x) = t x 1 e t dt for x>0 0 Some properties of the gamma function follow i) Γ(k) =(k 1)! for integer k 1 ii) Γ(x +1)=x Γ(x) forx>0 iii) Γ(x) =(x 1) Γ(x 1) for x>1 iv) Γ(05) = π 1

101 The Beta Distribution If Y has a beta distribution, Y beta(δ, ν), then the probability density function (pdf) of Y is f(y) = where δ>0, ν>0and0 y 1 VAR(Y )= Γ(δ + ν) Γ(δ)Γ(ν) yδ 1 (1 y) ν 1 E(Y )= δ δ + ν δν (δ + ν) (δ + ν +1) Γ(δ + ν) f(y) = Γ(δ)Γ(ν) I [0,1](y)exp[(δ 1) log(y)+(ν 1) log(1 y)] is a P REF Hence Θ = (0, ) (0, ), η 1 = δ 1, η = ν 1and Ω=( 1, ) ( 1, ) If δ =1,thenW = log(1 Y ) EXP(1/ν) Hence T n = log(1 Y i ) G(n, 1/ν) andifr> n then Tn r is the UMVUE of E(Tn)= r 1 Γ(r + n) ν r Γ(n) If ν =1,thenW = log(y ) EXP(1/δ) Hence T n = log(y i ) G(n, 1/δ) andandifr> n then Tn r is the UMVUE of E(Tn)= r 1 Γ(r + n) δ r Γ(n) 10 The Bernoulli and Binomial Distributions If Y has a binomial distribution, Y BIN(k,ρ), then the probability mass function (pmf) of Y is ( ) k f(y) =P (Y = y) = ρ y (1 ρ) k y y 13

for y =0, 1,,k where 0 <ρ<1 If ρ =0,P(Y =0)=1=(1 ρ) k while if ρ =1,P(Y = k) =1=ρ k The moment generating function m(t) =[(1 ρ)+ρe t ] k, and the characteristic function c(t) =[(1 ρ)+ρe it ] k E(Y )=kρ VAR(Y )=kρ(1 ρ) The Bernoulli (ρ) distribution is the binomial (k = 1,ρ) distribution The following normal approximation is often used when kρ(1 ρ) > 9 Hence Y N(kρ, kρ(1 ρ)) P (Y y) Φ ( ) y +05 kρ kρ(1 ρ) Also ( 1 1 P (Y = y) exp 1 ) (y kρ) kρ(1 ρ) π kρ(1 ρ) See Johnson, Kotz and Kemp (199, p 115) This approximation suggests that MED(Y ) kρ, andmad(y ) 0674 kρ(1 ρ) Hamza (1995) states that E(Y ) MED(Y ) max(ρ, 1 ρ) and shows that E(Y ) MED(Y ) log() If k is large and kρ small, then Y Poisson(kρ) If Y 1,, Y n are independent BIN(k i,ρ)then Y i BIN( k i,ρ) ( ) [ ] k f(y) = (1 ρ) k ρ exp log( y 1 ρ )y is a 1P REF in ρ if k is known Thus Θ = (0, 1), ( ) ρ η =log 1 ρ 14

and Ω = (, ) Assume that Y 1,, Y n are iid BIN(k, ρ), then T n = Y i BIN(nk,ρ) If k is known, then the likelihood L(ρ) =cρ yi (1 ρ) nk y i, and the log likelihood Hence log(l(ρ)) = d +log(ρ) y i +(nk y i )log(1 ρ) d dρ log(l(ρ)) = yi + nk y i ( 1) set =0, 1 ρ ρ or (1 ρ) y i = ρ(nk y i ), or y i = ρnk or ˆρ = Y i /(nk) d dρ log(l(ρ)) = yi ρ nk yi < 0 (1 ρ) if 0 < y i <nkhence kˆρ = Y is the UMVUE, MLE and MME of kρ if k is known 103 The Burr Distribution If Y has a Burr distribution, Y Burr(φ, λ), then the pdf of Y is φy φ 1 f(y) = 1 λ (1 + y φ ) 1 λ +1 where y, φ, and λ are all positive The cdf of Y is [ ] log(1 + y φ ) F (y)=1 exp =1 (1 + y φ ) 1/λ for y > 0 λ MED(Y )=[e λ log() 1] 1/φ See Patel, Kapadia and Owen (1976, p 195) W =log(1+y φ )isexp(λ) 15

f(y) = 1 [ 1 λ φyφ 1 1+y exp 1 ] φ λ log(1 + yφ ) I(y >0) is a one parameter exponential family if φ is known If Y 1,, Y n are iid Burr(λ, φ), then T n = log(1 + Y φ i ) G(n, λ) If φ is known, then the likelihood L(λ) =c 1 [ λ exp 1 ] log(1 + y φ n i λ ), log(1 + y φ i ) Hence and the log likelihood log(l(λ)) = d n log(λ) 1 λ d n log(1 + y φ log(l(λ)) = dλ λ + i ) λ or log(1 + y φ i )=nλ or set =0, log(1 + Y φ i ) ˆλ = n d dλ log(l(λ)) = n λ log(1 + y φ λ i ) λ=ˆλ = ṋ λ nˆλ ˆλ 3 = n ˆλ < 0 Thus ˆλ is the UMVUE and MLE of λ if φ is known If φ is known and r> n, thent r n is the UMVUE of E(Tn)=λ r r Γ(r + n) Γ(n) 104 The Cauchy Distribution If Y has a Cauchy distribution, Y C(µ, σ), then the pdf of Y is f(y) = σ π 1 σ +(y µ) = 1 πσ[1 + ( y µ σ ) ] 16

where y and µ are real numbers and σ>0 The cumulative distribution function (cdf) of Y is F (y) = 1 π [arctan(y µ σ )+π/] See Ferguson (1967, p 10) This family is a location scale family that is symmetric about µ The moments of Y do not exist, but the chf of Y is c(t) =exp(itµ t σ) MED(Y )=µ, the upper quartile = µ + σ, and the lower quartile = µ σ MAD(Y )=F 1 (3/4) MED(Y )=σ If Y 1,, Y n are independent C(µ i,σ i ), then n n a i Y i C( a i µ i, i=1 i=1 n a i σ i ) In particular, if Y 1,, Y n are iid C(µ, σ), then Y C(µ, σ) 105 The Chi Distribution If Y has a chi distribution (also called a p dimensional Rayleigh distribution), Y chi(p,σ), then the pdf of Y is i=1 f(y) = yp 1 e 1 σ y σ p p 1 Γ(p/) where y 0andσ, p > 0 This is a scale family if p is known E(Y )=σ 1+p Γ( ) Γ(p/) +p VAR(Y )=σ Γ( ) ( Γ( 1+p Γ(p/) ) ), Γ(p/) and r+p E(Y r )= r/ r Γ( σ ) Γ(p/) 17

for r> p The mode is at σ p 1forp 1 See Cohen and Whitten (1988, ch 10) Note that W = Y G(p/, σ ) If p =1, then Y has a half normal distribution, Y HN(0,σ ) If p =, then Y has a Rayleigh distribution, Y R(0,σ) If p =3, then Y has a Maxwell Boltzmann distribution (also known as a Boltzmann distribution or a Maxwell distribution), Y MB (0,σ) If p is an integer and Y chi(p, 1), then Y χ p Since f(y) = 1 p 1 Γ(p/)σ p I(y >0) exp[(p 1) log(y) 1 σ y ], this family appears to be a P REF Θ = (0, ) (0, ), η 1 = p 1, η = 1/(σ ), and Ω = ( 1, ) (, 0) If p is known then y p 1 f(y) = p 1 Γ(p/) I(y >0) 1 [ ] 1 σ exp p σ y appears to be a 1P REF If Y 1,, Y n are iid chi(p, σ), then If p is known, then the likelihood and the log likelihood Hence or y i = npσ or T n = Y i G(np/, σ ) L(σ )=c 1 1 exp[ σnp y σ i ], log(l(σ )) = d np log(σ ) 1 σ y i d d(σ ) log(σ )= np σ + 1 (σ ) y i set =0, ˆσ = Y i np 18

d d(σ ) log(l(σ )) = np y (σ ) i (σ ) 3 = np σ =ˆσ (ˆσ ) npˆσ (ˆσ ) 3 = np (ˆσ ) < 0 Thus ˆσ is the UMVUE and MLE of σ when p is known If p is known and r> np/, then T r n is the UMVUE of E(Tn)= r r σ r Γ(r + np/) Γ(np/) 106 The Chi square Distribution If Y has a chi square distribution, Y χ p, then the pdf of Y is f(y) = y p 1 e y p Γ( p ) where y 0andp is a positive integer The mgf of Y is ( ) p/ 1 m(t) = =(1 t) p/ 1 t for t<1/ The chf ( ) p/ 1 c(t) = 1 it E(Y )=p VAR(Y )=p Since Y is gamma G(ν = p/,λ=), E(Y r )= r Γ(r + p/),r> p/ Γ(p/) MED(Y ) p /3 See Pratt (1968, p 1470) for more terms in the expansion of MED(Y ) Empirically, p MAD(Y ) 1483 (1 9p ) 09536 p 19

There are several normal approximations for this distribution The Wilson Hilferty approximation is ( ) 1 Y 3 N(1 p 9p, 9p ) See Bowman and Shenton (199, p 6) This approximation gives and P (Y x) Φ[(( x p )1/3 1+/9p) 9p/], χ p,α p(z α 9p +1 9p )3 where z α is the standard normal percentile, α =Φ(z α ) The last approximation is good if p> 14 log(α) See Kennedy and Gentle (1980, p 118) This family is a one parameter exponential family, but is not a REF since the set of integers does not contain an open interval 107 The Double Exponential Distribution If Y has a double exponential distribution (or Laplace distribution), Y DE(θ, λ), then the pdf of Y is f(y) = 1 ( ) λ exp y θ λ where y is real and λ>0 The cdf of Y is ( ) y θ F (y) =05exp if y θ, λ and ( ) (y θ) F (y) =1 05exp if y θ λ This family is a location scale family which is symmetric about θ The mgf m(t) =exp(θt)/(1 λ t ) for t < 1/λ, and the chf c(t) =exp(θit)/(1 + λ t ) 0

E(Y )=θ, and MED(Y )=θ VAR(Y )=λ,and MAD(Y )=log()λ 0693λ Hence λ =MAD(Y )/ log() 1443MAD(Y ) To see that MAD(Y )=λlog(), note that F (θ + λ log()) = 1 05 = 075 The maximum likelihood estimators are ˆθ MLE =MED(n) and ˆλ MLE = 1 n Y i MED(n) n A 100(1 α)% confidence interval (CI) for λ is ( n i=1 Y i MED(n), ) n i=1 Y i MED(n), χ n 1,1 χ α n 1, α and a 100(1 α)% CI for θ is MED(n) ± z n 1 α/ i=1 Y i MED(n) n n z1 α/ i=1 where χ p,α and z α are the α percentiles of the χ p and standard normal distributions, respectively See Patel, Kapadia and Owen (1976, p 194) W = Y θ EXP(λ) f(y) = 1 [ ] 1 λ exp y θ λ is a one parameter exponential family in λ if θ is known If Y 1,, Y n are iid DE(θ, λ) then If θ is known, then the likelihood T n = Y i θ G(n, λ) L(λ) =c 1 [ 1 λ exp n λ and the log likelihood ] yi θ, log(l(λ)) = d n log(λ) 1 λ yi θ 1

Hence d n log(l(λ)) = dλ λ + 1 yi θ set =0 λ or y i θ = nλ or Yi θ ˆλ = n d dλ log(l(λ)) = n λ yi θ = ṋ λ 3 λ nˆλ = n ˆλ 3 ˆλ < 0 λ=ˆλ Thus ˆλ is the UMVUE and MLE of λ if θ is known 108 The Exponential Distribution If Y has an exponential distribution, Y EXP(λ), then the pdf of Y is where λ>0 The cdf of Y is f(y) = 1 λ exp ( y ) I(y 0) λ F (y) =1 exp( y/λ), y 0 This distribution is a scale family The mgf m(t) =1/(1 λt) for t<1/λ, and the chf c(t) =1/(1 iλt) E(Y )=λ, and VAR(Y )=λ W =λy χ Since Y is gamma G(ν =1,λ), E(Y r )=λγ(r +1)forr> 1 MED(Y )=log()λ and MAD(Y ) λ/0781 since it can be shown that exp(mad(y )/λ) =1+exp( MAD(Y )/λ)

Hence 0781 MAD(Y ) λ The classical estimator is ˆλ = Y n and the 100(1 α)% CI for E(Y )=λ is ( n i=1 Y i, ) n i=1 Y i χ n,1 χ α n, α where P (Y χ n, )=α/ ify is χ n See Patel, Kapadia and Owen (1976, α p 188) f(y) = 1 [ ] 1 λ I(y >0) exp λ y is a 1P REF Hence Θ = (0, ), η = 1/λ and Ω = (, 0) Suppose that Y 1,, Y n are iid EXP(λ), then T n = Y i G(n, λ) The likelihood and the log likelihood L(λ) = 1 λ n exp [ 1 λ yi ], log(l(λ)) = n log(λ) 1 λ yi Hence or y i = nλ or d n log(l(λ)) = dλ λ + 1 set yi =0, λ ˆλ = Y Since d dλ log(l(λ)) = n λ yi λ 3 λ=ˆλ = ṋ λ nˆλ ˆλ 3 = n ˆλ < 0, the ˆλ is the UMVUE, MLE and MME of λ If r> n, thentn r is the UMVUE of E(Tn r Γ(r + n) )=λr Γ(n) 3

109 The Two Parameter Exponential Distribution If Y has a parameter exponential distribution, Y EXP(θ, λ) then the pdf of Y is f(y) = 1 ( ) λ exp (y θ) I(y θ) λ where λ>0 The cdf of Y is F (y)=1 exp[ (y θ)/λ)], y θ This family is an asymmetric location-scale family The mgf m(t) =exp(tθ)/(1 λt) for t<1/λ, and the chf c(t) =exp(itθ)/(1 iλt) E(Y )=θ + λ, and VAR(Y )=λ MED(Y )=θ + λ log() and MAD(Y ) λ/0781 Hence θ MED(Y ) 0781 log()mad(y ) See Rousseeuw and Croux (1993) for similar results Note that 0781 log() 144 To see that 0781MAD(Y ) λ, note that 05 = θ+λ log()+mad θ+λ log() MAD 1 exp( (y θ)/λ)dy λ =05[ e MAD/λ + e MAD/λ ] assuming λ log() > MAD Plug in MAD = λ/0781 to get the result If θ is known, then f(y) =I(y θ) 1 [ ] 1 λ exp (y θ) λ 4

is a 1P REF in λ Y θ EXP(λ) Let ˆλ = (Yi θ) n Then ˆλ is the UMVUE and MLE of λ if θ is known If Y 1,, Y n are iid EXP(θ, λ), then the likelihood L(θ, λ) = 1 [ 1 λ exp ] (yi θ) I(y n (1) θ), λ and the log likelihood log(l(λ)) = [ n log(λ) 1 λ (yi θ)]i(y (1) θ) For any fixed λ>0, the log likelihood is maximized by maximizing θ Hence ˆθ = Y (1), and the profile log likelihood is log(l(λ y (1) )) = n log(λ) 1 λ (yi y (1) ) is maximized by ˆλ = 1 n n i=1 (y i y (1) ) Hence the MLE ( (ˆθ, ˆλ) = Y (1), 1 n ) n (Y i Y (1) ) =(Y (1), Y Y (1) ) i=1 1010 The F Distribution If Y has an F distribution, Y F (ν 1,ν ), then the pdf of Y is f(y) = Γ( ν 1+ν ) Γ(ν 1 /)Γ(ν /) ( ) ν1 / ν1 where y>0andν 1 and ν are positive integers ν y (ν 1 )/ ) (ν1 +ν )/ ( 1+( ν 1 ν )y and E(Y )= ν ν, ν > 5

( ) ν (ν 1 + ν ) VAR(Y )= ν ν 1 (ν 4), ν > 4 E(Y r )= Γ( ν 1+r )Γ( ν ( ) r ) r ν, r < ν / Γ(ν 1 /)Γ(ν /) ν 1 Suppose that X 1 and X are independent where X 1 χ ν 1 and X χ ν Then W = (X 1/ν 1 ) (X /ν ) F (ν 1,ν ) If W t ν,theny = W F (1,ν) 1011 The Gamma Distribution If Y has a gamma distribution, Y G(ν, λ), then the pdf of Y is f(y) = yν 1 e y/λ λ ν Γ(ν) where ν, λ, and y are positive The mgf of Y is ( ) ν 1/λ m(t) = 1 t = λ for t<1/λ The chf c(t) = ( ) ν 1 1 iλt ( 1 ) ν 1 λt E(Y )=νλ VAR(Y )=νλ E(Y r )= λr Γ(r + ν) if r> ν (103) Γ(ν) Chen and Rubin (1986) show that λ(ν 1/3) < MED(Y ) <λν= E(Y ) Empirically, for ν>3/, MED(Y ) λ(ν 1/3), and MAD(Y ) λ ν 1483 6

This family is a scale family for fixed ν, soify is G(ν, λ) thency is G(ν, cλ) for c>0 If W is EXP(λ) thenw is G(1,λ) If W is χ p, then W is G(p/, ) Some classical estimates are given next Let [ w =log y n geometric mean(n) where geometric mean(n) =(y 1 y y n ) 1/n Then Thom s estimate (Johnson and Kotz 1970a, p 188) is ˆν 05(1 + 1+4w/3 ) w Also 05000876 + 0164885w 0054474w ˆν MLE w for 0 <w 0577, and ] ˆν MLE 8898919 + 9059950w +09775374w w(177978 + 11968477w + w ) for 0577 <w 17 If W>17then estimation is much more difficult, but a rough approximation is ˆν 1/w for w>17 See Bowman and Shenton (1988, p 46) and Greenwood and Durand (1960) Finally, ˆλ = Y n /ˆν Notice that ˆβ may not be very good if ˆν <1/17 Several normal approximations are available The Wilson Hilferty approximation says that for ν>05, Y 1/3 N Hence if Y is G(ν, λ) and ( (νλ) 1/3 (1 1 1 ), (νλ)/3 9ν 9ν α = P [Y G α ], ) then [ 1 G α νλ z α 9ν +1 1 9ν ] 3 where z α is the standard normal percentile, α =Φ(z α ) Bowman and Shenton (1988, p 101) include higher order terms 7

f(y) = [ ] 1 1 I(y >0) exp y +(ν 1) log(y) λ ν Γ(ν) λ is a P REF Hence Θ = (0, ) (0, ), η 1 = 1/λ, η = ν 1and Ω=(, 0) ( 1, ) If Y 1,, Y n are independent G(ν i,λ)then Y i G( ν i,λ) If Y 1,, Y n are iid G(ν, λ), then T n = Y i G(nν, λ) Since f(y) = 1 Γ(ν) exp[(ν 1) log(y)]i(y >0) 1 [ ] 1 λ exp ν λ y, Y is a 1P REF when ν is known If ν is known, then the likelihood [ 1 1 L(β) =c exp λnν λ The log likelihood yi ] log(l(λ)) = d nν log(λ) 1 λ yi Hence or y i = nνλ or d nν log(l(λ)) = dλ λ + yi λ ˆλ = Y/ν set =0, d nν log(l(λ)) = dλ λ yi λ 3 λ=ˆλ = nν ˆλ nνˆλ ˆλ 3 Thus Y is the UMVUE, MLE and MME of νλ if ν is known = nν ˆλ < 0 8

101 The Generalized Gamma Distribution If Y has a generalized gamma distribution, Y GG(ν, λ, φ), then the pdf of Y is f(y) = φyφν 1 λ φν Γ(ν) exp( yφ /λ φ ) where ν, λ, φ and y are positive This family is a scale family with scale parameter λ if φ and ν are known E(Y r )= λr Γ(ν)+ 1 φ Γ(ν) if r> φν (104) If φ and ν are known, then f(y) = φyφν 1 Γ(ν) [ ] 1 1 I(y >0) exp λφν λ φ yφ, which is a one parameter exponential family W = Y φ G(ν, λ φ ) If Y 1,, Y n are iid GG((ν, λ, φ) where φ and ν are known, then T n = n i=1 Y φ i G(nν, λ φ ), and Tn r is the UMVUE of E(Tn)=λ r φr Γ(r + nν) Γ(nν) for r> nν 1013 The Generalized Negative Binomial Distribution If Y has a generalized negative binomial distribution, Y GNB(µ, κ), then the pmf of Y is ( ) κ ( Γ(y + κ) κ f(y) =P (Y = y) = 1 κ ) y Γ(κ)Γ(y +1) µ + κ µ + κ for y =0, 1,, where µ>0andκ>0 This distribution is a generalization of the negative binomial (κ, ρ) distribution with ρ = κ/(µ + κ) andκ>0is an unknown real parameter rather than a known integer 9

The mgf is [ κ m(t) = κ + µ(1 e t ) for t< log(µ/(µ + κ)) E(Y )=µ and VAR(Y )=µ + µ /κ If Y 1,, Y n are iid GNB(µ, κ), then n i=1 Y i GNB(nµ, nκ) When κ is known, this distribution is a 1P REF IfY 1,, Y n are iid GNB(µ, κ) whereκ is known, then ˆµ = Y is the MLE, UMVUE and MME of µ 1014 The Geometric Distribution If Y has a geometric distribution, Y geom(ρ) then the pmf of Y is ] κ f(y) =P (Y = y) =ρ(1 ρ) y for y =0, 1,, and 0 <ρ<1 The cdf for Y is F (y) =1 (1 ρ) y+1 for y 0andF (y)=0fory<0 Here y is the greatest integer function, eg, 77 =7 E(Y )=(1 ρ)/ρ VAR(Y )=(1 ρ)/ρ Y NB(1,ρ) Hence the mgf of Y is for t< log(1 ρ) m(t) = ρ 1 (1 ρ)e t f(y) = ρ exp[log(1 ρ)y] is a 1P REF Hence Θ = (0, 1), η =log(1 ρ) andω=(, 0) If Y 1,, Y n are iid geom(ρ), then T n = Y i NB(n,ρ) The likelihood L(ρ) =ρ n exp[log(1 ρ) y i ], 30

and the log likelihood log(l(ρ)) = n log(ρ)+log(1 ρ) y i Hence d dρ log(l(ρ)) = n ρ 1 set yi =0 1 ρ or n(1 ρ)/ρ = y i or n nρ ρ y i =0or ˆρ = n n + Y i d n log(l(ρ)) = dρ ρ yi (1 ρ) < 0 Thus ˆρ is the MLE of ρ The UMVUE, MLE and MME of (1 ρ)/ρ is Y 1015 The Half Cauchy Distribution If Y has a half Cauchy distribution, Y HC(µ, σ), then the pdf of Y is f(y) = πσ[1 + ( y µ σ ) ] where y µ, µ is a real number and σ>0 The cdf of Y is F (y) = π arctan(y µ σ ) for y µ and is 0, otherwise This distribution is a right skewed locationscale family MED(Y )=µ + σ MAD(Y )=07305σ 1016 The Half Logistic Distribution If Y has a half logistic distribution, Y HL(µ, σ), then the pdf of y is f(y) = exp( (y µ)/σ) σ[1 + exp ( (y µ)/σ)] 31

where σ>0, y µ and µ are real The cdf of Y is F (y) = exp[(y µ)/σ] 1 1+exp[(y µ)/σ)] for y µ and 0 otherwise This family is a right skewed location scale family MED(Y )=µ +log(3)σ MAD(Y )=067346σ 1017 The Half Normal Distribution If Y has a half normal distribution, Y HN(µ, σ ), then the pdf of Y is f(y) = (y µ) exp ( ) πσ σ where σ>0andy µ and µ is real Let Φ(y) denote the standard normal cdf Then the cdf of Y is F (y) =Φ( y µ σ ) 1 for y > µ and F (y) = 0, otherwise E(Y )=µ + σ /π µ +0797885σ VAR(Y )= σ (π ) 0363380σ π This is an asymmetric location scale family that has the same distribution as µ + σ Z where Z N(0, 1) Note that Z χ 1 Hence the formula for the rth moment of the χ 1 random variable can be used to find the moments of Y MED(Y )=µ +06745σ MAD(Y )=03990916σ f(y) = [ I(y >µ)exp πσ ( 1 )(y µ) σ is a 1P REF if µ is known Hence Θ = (0, ), η= 1/(σ )andω= (, 0) 3 ]

W =(Y µ) G(1/, σ ) If Y 1,, Y n are iid HN(µ, σ ), then T n = (Y i µ) G(n/, σ ) If µ is known, then the likelihood L(σ )=c 1 [ σ exp ( 1 n σ ) ] (y i µ), and the log likelihood Hence log(l(σ )) = d n log(σ ) 1 σ (yi µ) d d(σ ) log(l(σ )) = or (y i µ) = nσ or n (σ ) + 1 (σ ) (yi µ) set =0, ˆσ = 1 n (Yi µ) d d(σ ) log(l(σ )) = n (σ ) (yi µ) ) (σ ) 3 = n σ =ˆσ (ˆσ ) nˆσ ( ˆσ ) 3 = n ˆσ < 0 Thus ˆσ is the UMVUE and MLE of σ if µ is known If r> n/ andifµ is known, then Tn r is the UMVUE of E(T r n)= r σ r Γ(r + n/)/γ(n/) 1018 The Hypergeometric Distribution If Y has a hypergeometric distribution, Y HG(C, N C, n), then the data set contains N objects of two types There are C objects of the first type (that you wish to count) and N C objects of the second type Suppose that n objects are selected at random without replacement from the N objects 33

Then Y counts the number of the n selected objects that were of the first type The pmf of Y is ( C N C ) y)( f(y) =P (Y = y) = n y ( N n) where the integer y satisfies max(0,n N + C) y min(n, C) The right inequality is true since if n objects are selected, then the number of objects of the first type must be less than or equal to both n and C The first inequality holds since n Y counts the number of objects of second type Hence n Y N C Let p = C/N Then E(Y )= nc N = np and VAR(Y )= nc(n C) N n N N 1 = np(1 p)n n N 1 If n is small compared to both C and N C then Y BIN(n, p) If n is large but n is small compared to both C and N C then Y N(np, np(1 p)) 1019 The Inverse Gaussian Distribution If Y has an inverse Gaussian distribution, Y IG(θ, λ), then the pdf of Y is [ ] λ λ(y θ) f(y) = πy exp 3 θ y where y, θ, λ > 0 The mgf is m(t) =exp [ ( )] λ 1 1 θ t θ λ for t < λ/(θ ) See Datta (005) and Schwarz and Samanta (1991) for additional properties The chf is [ ( )] λ φ(t) =exp 1 1 θ it θ λ 34

E(Y )=θ and f(y) = VAR(Y )= θ3 λ is a two parameter exponential family If Y 1,, Y n are iid IG(θ, λ), then [ λ 1 λ π eλ/θ I(y >0) exp y3 θ y λ ] 1 y n Y i IG(nθ, n λ) and Y IG(θ, nλ) i=1 If λ is known, then the likelihood and the log likelihood L(θ) =ce nλ/θ exp[ λ θ yi ], Hence or y i = nθ or log(l(θ)) = d + nλ θ λ θ yi d nλ log(l(θ)) = + λ set yi =0, dθ θ θ 3 ˆθ = Y d nλ log(l(θ)) = dθ θ 3 3λ yi θ 4 θ=ˆθ = nλ ˆθ 3 3nλˆθ ˆθ 4 = nλ ˆθ 3 < 0 Thus Y is the UMVUE, MLE and MME of θ if λ is known If θ is known, then the likelihood [ λ ] L(λ) =cλ n/ (yi θ) exp, θ y i 35

and the log likelihood Hence or log(l(λ)) = d + n log(λ) λ θ (yi θ) y i d dλ log(l(λ)) = n λ 1 θ (yi θ) y i ˆλ = nθ (yi θ) y i d n log(l(λ)) = dλ λ < 0 Thus ˆλ is the MLE of λ if θ is known set =0 Another parameterization of the inverse Gaussian distribution takes θ = λ/ψ so that f(y) = [ λ π e λψ 1 ψ I[y >0] exp y3 y λ ] 1, y where λ>0andψ 0 HereΘ=(0, ) [0, ), η 1 = ψ/, η = λ/ and Ω=(, 0] (, 0) Since Ω is not an open set, this is a parameter full exponential family that is not regular If ψ is known then Y is a 1P REF, but if λ is known the Y is a one parameter full exponential family When ψ = 0,Y has a one sided stable distribution with index 1/ See Barndorff Nielsen (1978, p 117) 100 The Inverted Gamma Distribution If Y has an inverted gamma distribution, Y INVG(ν, λ), then the pdf of Y is 1 f(y) = y ν+1 Γ(ν) I(y >0) 1 ( ) 1 λ exp 1 ν λ y where λ, ν and y are all positive It can be shown that W =1/Y G(ν, λ) This family is a scale family with scale parameter τ =1/λ if ν is known 36

If ν is known, this family is a 1 parameter exponential family If Y 1,, Y n are iid INVG(ν, λ) andν is known, then T n = n 1 i=1 Y i G(nν, λ) andtn r is the UMVUE of r Γ(r + nν) λ Γ(nν) for r> nν 101 The Largest Extreme Value Distribution If Y has a largest extreme value distribution (or Gumbel distribution), Y LEV (θ, σ), then the pdf of Y is f(y) = 1 σ exp( (y θ θ )) exp[ exp( (y σ σ ))] where y and θ are real and σ>0 The cdf of Y is F (y)=exp[ exp( ( y θ σ ))] This family is an asymmetric location scale family with a mode at θ The mgf m(t) =exp(tθ)γ(1 σt) for t < 1/σ E(Y ) θ +05771σ, and VAR(Y )=σ π /6 164493σ and MED(Y )=θ σ log(log()) θ +036651σ MAD(Y ) 0767049σ W =exp( (Y θ)/σ) EXP(1) f(y) = 1 σ eθ/σ e y/σ exp [ e θ/σ e y/σ] is a one parameter exponential family in θ if σ is known 37

If Y 1,, Y n are iid LEV(θ, σ)where σ is known, then the likelihood and the log likelihood L(σ) =ce nθ/σ exp[ e θ/σ e y i/σ ], Hence or or or log(l(θ)) = d + nθ σ eθ/σ e y i/σ d dθ log(l(θ)) = n σ 1 eθ/σ e y i /σ set =0, σ e θ/σ e y i/σ = n, e θ/σ = n e y i /σ, ( ) n ˆθ =log e Y i /σ Since d 1 log(l(θ)) = dθ σ eθ/σ e yi/σ < 0, ˆθ is the MLE of θ 10 The Logarithmic Distribution If Y has a logarithmic distribution, then the pmf of Y is f(y) =P (Y = y) = 1 θ y log(1 θ) y for y =1,, and 0 <θ<1 The mgf m(t) = log(1 θet ) log(1 θ) for t< log(θ) E(Y )= 1 θ log(1 θ) 1 θ 38

1 1 f(y) = log(1 θ) y exp(log(θ)y) is a 1P REF Hence Θ = (0, 1), η=log(θ) andω=(, 0) If Y 1,, Y n are iid logarithmic (θ), then Y is the UMVUE of E(Y ) 103 The Logistic Distribution If Y has a logistic distribution, Y L(µ, σ), then the pdf of y is f(y) = where σ>0andy and µ are real The cdf of Y is F (y) = exp ( (y µ)/σ) σ[1 + exp ( (y µ)/σ)] 1 1+exp( (y µ)/σ) = exp ((y µ)/σ) 1+exp((y µ)/σ) This family is a symmetric location scale family The mgf of Y is m(t) =πσte µt csc(πσt) for t < 1/σ, and the chf is c(t) =πiσte iµt csc(πiσt) wherecsc(t) is the cosecant of t E(Y )=µ, and MED(Y )=µ VAR(Y )=σ π /3, and MAD(Y )=log(3)σ 10986 σ Hence σ =MAD(Y )/ log(3) The estimators ˆµ = Y n and S = 1 n n 1 i=1 (Y i Y n ) are sometimes used Note that if q = F L(0,1) (c) = ec q then c =log( 1+e c 1 q ) Taking q = 9995 gives c = log(1999) 76 To see that MAD(Y )=log(3)σ, notethatf (µ +log(3)σ) =075, F (µ log(3)σ) =05, and 075 = exp (log(3))/(1 + exp(log(3))) 39

104 The Log-Cauchy Distribution If Y has a log Cauchy distribution, Y LC(µ, σ), then the pdf of Y is f(y) = 1 πσy[1 + ( log(y) µ σ ) ] where y>0, σ>0andµ is a real number This family is a scale family with scale parameter τ = e µ if σ is known It can be shown that W =log(y ) has a Cauchy(µ, σ) distribution 105 The Log-Logistic Distribution If Y has a log logistic distribution, Y LL(φ, τ), then the pdf of Y is f(y) = φτ(φy)τ 1 [1 + (φy) τ ] where y>0, φ>0andτ>0 The cdf of Y is F (y)=1 1 1+(φy) τ for y>0 This family is a scale family with scale parameter σ = φ 1 if τ is known MED(Y )=1/φ It can be shown that W =log(y ) has a logistic(µ = log(φ),σ =1/τ) distribution Hence φ = e µ and τ =1/σ Kalbfleisch and Prentice (1980, p 7-8) suggest that the log-logistic distribution is a competitor of the lognormal distribution 106 The Lognormal Distribution If Y has a lognormal distribution, Y LN(µ, σ ), then the pdf of Y is ( ) 1 (log(y) µ) f(y) = y πσ exp σ 40

where y>0andσ>0andµ is real The cdf of Y is ( ) log(y) µ F (y) =Φ for y>0 σ where Φ(y) is the standard normal N(0,1) cdf This family is a scale family with scale parameter τ = e µ if σ is known and E(Y )=exp(µ + σ /) VAR(Y )=exp(σ )(exp(σ ) 1) exp(µ) For any r, E(Y r )=exp(rµ + r σ /) MED(Y )=exp(µ) and exp(µ)[1 exp( 06744σ)] MAD(Y ) exp(µ)[1 + exp(06744σ)] f(y) = 1 [ 1 1 π σ exp( µ σ )1 I(y 0) exp y σ (log(y)) + µ ] σ log(y) is a P REF Hence Θ = (, ) (0, ), η 1 = 1/(σ ),η = µ/σ and Ω=(, 0) (, ) Note that W =log(y ) N(µ, σ ) f(y) = 1 π 1 σ [ ] 1 1 I(y 0) exp (log(y) µ) y σ is a 1P REF if µ is known, If Y 1,, Y n are iid LN(µ, σ )whereµ is known, then the likelihood L(σ )=c 1 [ 1 σ exp ] (log(yi ) µ), n σ and the log likelihood log(l(σ )) = d n log(σ ) 1 (log(yi ) µ) σ Hence d d(σ ) log(l(σ )) = n σ + 1 (log(yi ) µ) set =0, (σ ) 41

or (log(y i ) µ) = nσ or Since ˆσ = d (log(yi ) µ) n d(σ ) log(l(σ )) = n (σ ) (log(yi ) µ) (σ ) 3 = n σ =ˆσ (ˆσ ) nˆσ (ˆσ ) 3 = n (ˆσ ) < 0, ˆσ is the UMVUE and MLE of σ if µ is known Since T n = [log(y i ) µ] G(n/, σ ), if µ is known and r> n/ then Tn r is UMVUE of E(T r n )=r σ r Γ(r + n/) Γ(n/) If σ is known, f(y) = 1 1 1 1 [ I(y 0) exp( π σ y σ (log(y)) )exp( µ µ ] σ )exp σ log(y) is a 1P REF If Y 1,, Y n are iid LN(µ, σ ), where σ is known, then the likelihood [ L(µ) =c exp( nµ µ σ )exp ] log(yi ), σ and the log likelihood log(l(µ)) = d nµ σ + µ σ log(yi ) Hence d nµ log(yi ) set log(l(µ)) = + =0, dµ σ σ or log(y i )=nµ or log(yi ) ˆµ = n Note that d n log(l(µ)) = dµ σ < 0 4

Since T n = log(y i ) N(nµ, nσ ), ˆµ is the UMVUE and MLE of µ if σ is known When neither µ nor σ are known, the log likelihood log(l(σ )) = d n log(σ ) 1 (log(yi ) µ) σ Let w i =log(y i ) then the log likelihood is log(l(σ )) = d n log(σ ) 1 (wi µ), σ which is equivalent to the normal N(µ, σ ) log likelihood Hence the MLE (ˆµ, ˆσ) = 1 n W i, 1 n (W i W ) n n i=1 Hence inference for µ and σ is simple Use the fact that W i =log(y i ) N(µ, σ ) and then perform the corresponding normal based inference on the W i For example, a the classical (1 α)100% CI for µ when σ is unknown is S W S W (W n t n 1,1 α, W n + t n 1,1 α ) n n where S W = i=1 n n 1 ˆσ = 1 n 1 n (W i W ), and P (t t n 1,1 α )=1 α/ whent is from a t distribution with n 1 degrees of freedom Compare Meeker and Escobar (1998, p 175) 107 The Maxwell-Boltzmann Distribution If Y has a Maxwell Boltzmann distribution, Y MB(µ, σ), then the pdf of Y is (y µ) e 1 σ f(y) = (y µ) σ 3 π where µ is real, y µ and σ>0 This is a location scale family i=1 E(Y )=µ + σ 1 Γ(3/) 43

[ Γ( 5 VAR(Y )=σ ) ( ) ] 1 Γ(3/) Γ(3/) MED(Y )= µ + 153817σ and MAD(Y )= 046044σ This distribution a one parameter exponential family when µ is known Note that W =(Y µ) G(3/, σ ) If Z MB(0,σ), then Z chi(p = 3,σ), and for r> 3 The mode of Z is at σ r+3 E(Z r )= r/ r Γ( σ ) Γ(3/) 108 The Negative Binomial Distribution If Y has a negative binomial distribution (also called the Pascal distribution), Y NB(r,ρ), then the pmf of Y is ( ) r + y 1 f(y) =P (Y = y) = ρ r (1 ρ) y y for y =0, 1, where 0 <ρ<1 The moment generating function for t< log(1 ρ) E(Y )=r(1 ρ)/ρ, and [ m(t) = ρ 1 (1 ρ)e t ] r VAR(Y )= r(1 ρ) ρ ( ) r + y 1 f(y) =ρ r exp[log(1 ρ)y] y is a 1P REF in ρ for known r Ω=(, 0) Thus Θ = (0, 1), η= log(1 ρ) and 44

If Y 1,, Y n are independent NB(r i,ρ), then Yi NB( r i,ρ) If Y 1,, Y n are iid NB(r, ρ), then If r is known, then the likelihood and the log likelihood Hence or or nr ρnr ρ y i =0or T n = Y i NB(nr, ρ) L(p) =cρ nr exp[log(1 ρ) y i ], log(l(ρ)) = d + nr log(ρ)+log(1 ρ) y i d nr log(l(ρ)) = dρ ρ 1 set yi =0, 1 ρ 1 ρ ρ nr = y i, ˆρ = nr nr + Y i d nr 1 log(l(ρ)) = yi < 0 dρ ρ (1 ρ) Thus ˆρ is the MLE of p if r is known Y is the UMVUE, MLE and MME of r(1 ρ)/ρ if r is known 109 The Normal Distribution If Y has a normal distribution (or Gaussian distribution), Y N(µ, σ ), then the pdf of Y is ( ) 1 (y µ) f(y) = exp πσ σ 45

where σ>0andµ and y are real Let Φ(y) denote the standard normal cdf Recall that Φ(y) = 1 Φ( y) The cdf F (y) ofy does not have a closed form, but ( ) y µ F (y) =Φ, σ and Φ(y) 05(1 + 1 exp( y /π) ) for y 0 See Johnson and Kotz (1970a, p 57) The moment generating function is m(t) =exp(tµ + t σ /) The characteristic function is c(t) =exp(itµ t σ /) E(Y )=µ and VAR(Y )=σ E[ Y µ r ]=σ r r/ Γ((r +1)/) π for r> 1 If k is an integer, then E(Y k )=(k 1)σ E(Y k )+µe(y k 1 ) See Stein (1981) and Casella and Berger (1990, p 187) MED(Y )=µ and MAD(Y )=Φ 1 (075)σ 06745σ Hence σ =[Φ 1 (075)] 1 MAD(Y ) 1483MAD(Y ) This family is a location scale family which is symmetric about µ Suggested estimators are Y n =ˆµ = 1 n n i=1 Y i and S = S Y = 1 n 1 n (Y i Y n ) i=1 The classical (1 α)100% CI for µ when σ is unknown is S Y S Y (Y n t n 1,1 α, Y n + t n 1,1 α ) n n 46

where P (t t n 1,1 α )=1 α/ whent is from a t distribution with n 1 degrees of freedom If α =Φ(z α ), then z α m c o + c 1 m + c m 1+d 1 m + d m + d 3 m 3 where m =[ log(1 α)] 1/, c 0 =515517, c 1 =080853, c =001038, d 1 =143788, d =018969, d 3 =0001308, and 05 α For 0 <α<05, z α = z 1 α See Kennedy and Gentle (1980, p 95) To see that MAD(Y )=Φ 1 (075)σ, note that 3/4 =F (µ +MAD)sinceY is symmetric about µ However, ( ) y µ F (y) =Φ σ and ( ) 3 µ +Φ 1 4 =Φ (3/4)σ µ σ So µ +MAD=µ +Φ 1 (3/4)σ Cancel µ from both sides to get the result [ 1 1 f(y) = πσ exp( µ σ )exp σ y + µ ] σ y is a P REF Hence Θ = (0, ) (, ), η 1 = 1/(σ ),η = µ/σ and Ω=(, 0) (, ) If σ is known, [ ] 1 1 [ f(y) = exp πσ σ y exp( µ µ ] σ )exp σ y is a 1P REF Also the likelihood L(µ) =c exp( nµ σ )exp [ µ σ yi ] 47

and the log likelihood Hence or nµ = y i,or log(l(µ)) = d nµ σ + µ σ yi d nµ yi log(l(µ)) = + dµ σ σ ˆµ = Y set =0, Since d n log(l(µ)) = dµ σ < 0 and since T n = Y i N(nµ, nσ ), Y is the UMVUE, MLE and MME of µ if σ is known If µ is known, [ ] 1 1 f(y) = exp (y µ) πσ σ is a 1P REF Also the likelihood L(σ )=c 1 [ 1 σ exp ] (yi µ) n σ and the log likelihood log(l(σ )) = d n log(σ ) 1 σ (yi µ) Hence d dµ log(l(σ )) = n σ + 1 (yi µ) set =0, (σ ) or nσ = (y i µ),or ˆσ (Yi µ) = n Note that d dµ log(l(σ )) = n (σ ) (yi µ) (σ ) 3 = n σ =ˆσ (ˆσ ) nˆσ (ˆσ ) 3 48

= n (ˆσ ) < 0 Since T n = (Y i µ) G(n/, σ ), ˆσ is the UMVUE and MLE of σ if µ is known Note that if µ is known and r> n/, then Tn r is the UMVUE of E(T r n)= r σ r Γ(r + n/) Γ(n/) 1030 The One Sided Stable Distribution If Y has a one sided stable distribution (with index 1/), Y OSS(σ), then the pdf of Y is ( ) 1 σ 1 f(y) = σ exp πy 3 y for y>0andσ>0 This distribution is a scale family with scale parameter σ and a 1P REF Whenσ =1,Y INVG(ν =1/,λ =)where INVG stands for inverted gamma This family is a special case of the inverse Gaussian IG distribution It can be shown that W = 1/Y G(1/, /σ) This distribution is even more outlier prone than the Cauchy distribution See Feller (1971, p 5) and Lehmann (1999, p 76) For applications see Besbeas and Morgan (004) 1 Y i G(n/, /σ) Hence If Y 1,, Y n are iid OSS(σ) thent n = n i=1 T n /n is the UMVUE (and MLE) of 1/σ and Tn r is the UMVUE of for r> n/ 1 r Γ(r + n/) σ r Γ(n/) 1031 The Pareto Distribution If Y has a Pareto distribution, Y PAR(σ, λ), then the pdf of Y is f(y) = 1 λ σ1/λ y 1+1/λ 49

where y σ, σ > 0, and λ>0 ThemodeisatY = σ The cdf of Y is F (y) =1 (σ/y) 1/λ for y>µ This family is a scale family when λ is fixed for λ<1 E(Y )= σ 1 λ E(Y r )= σr for r<1/λ 1 rλ MED(Y )=σ λ X =log(y/σ)isexp(λ) andw =log(y )isexp(θ =log(σ),λ) f(y) = 1 λ σ1/λ I[y σ]exp [ (1 + 1λ ] )log(y) = 1 1 I[y σ]exp [ (1 + 1λ ] σ λ )log(y/σ) is a one parameter exponential family if σ is known If Y 1,, Y n are iid PAR(σ, λ) then T n = log(y i /σ) G(n, λ) If σ is known, then the likelihood L(λ) =c 1 [ (1 λ exp + 1 n λ ) ] log(y i /σ), and the log likelihood log(l(λ)) = d n log(λ) (1 + 1 λ ) log(y i /σ) Hence d n log(l(λ)) = dλ λ + 1 log(yi /σ) set =0, λ or log(y i /σ) =nλ or log(yi /σ) ˆλ = n 50

d dλ log(l(λ)) = n λ log(yi /σ) λ 3 = λ=ˆλ n ˆλ nˆλ ˆλ 3 = n ˆλ < 0 Hence ˆλ is the UMVUE and MLE of λ if σ is known If σ is known and r> n, thentn r is the UMVUE of E(Tn)=λ r r Γ(r + n) Γ(n) If neither σ nor λ are known, notice that f(y) = 1 [ ( )] 1 log(y) log(σ) y λ exp I(y σ) λ Hence the likelihood L(λ, σ) =c 1 λ n exp [ n ( ) ] log(yi ) log(σ) I(y (1) σ), i=1 λ and the log likelihood is [ log L(λ, σ) = d n log(λ) n ( ) ] log(yi ) log(σ) I(y (1) σ) i=1 λ Let w i =log(y i )andθ =log(σ), so σ = e θ Then the log likelihood is [ n ( ) ] wi θ log L(λ, θ) = d n log(λ) I(w (1) θ), λ which is equivalent to the log likelihood of the EXP(θ, λ) distribution Hence (ˆλ, ˆθ) =(W W (1),W (1) ), and by invariance, the MLE i=1 (ˆλ, ˆσ) =(W W (1),Y (1) ) 51

103 The Poisson Distribution If Y has a Poisson distribution, Y POIS(θ), then the pmf of Y is f(y) =P (Y = y) = e θ θ y for y =0, 1,, where θ>0 If θ =0,P(Y =0)=1=e θ The mgf of Y is m(t) =exp(θ(e t 1)), and the chf of Y is c(t) =exp(θ(e it 1)) E(Y )=θ, and Chen and Rubin (1986) and Adell and Jodrá (005) show that 1 < MED(Y ) E(Y ) < 1/3 VAR(Y )=θ The classical estimator of θ is ˆθ = Y n The approximations Y N(θ, θ) and Y N( θ, 1) are sometimes used y! f(y) =e θ 1 y! exp[log(θ)y] is a 1P REF ThusΘ=(0, ), η=log(θ) andω=(, ) If Y 1,, Y n are independent POIS(θ i )then Y i POIS( θ i ) If Y 1,, Y n are iid POIS(θ) then The likelihood and the log likelihood T n = Y i POIS(nθ) L(θ) =ce nθ exp[log(θ) y i ], log(l(θ)) = d nθ +log(θ) y i Hence or y i = nθ, or d dθ log(l(θ)) = n + 1 set yi =0, θ ˆθ = Y 5

d dθ log(l(θ)) = yi < 0 θ unless y i =0 Hence Y is the UMVUE and MLE of θ 1033 The Power Distribution If Y has a power distribution, Y POW(λ), then the pdf of Y is f(y) = 1 λ y 1 λ 1, where λ>0and0 y 1 The cdf of Y is F (y) =y 1/λ for 0 y 1 MED(Y )=(1/) λ W = log(y )isexp(λ) Y beta(δ =1/λ, ν =1) f(y) = 1 λ I [0,1](y)exp [( 1λ ] 1) log(y) = 1 λ I [0,1](y)exp [(1 1λ ] )( log(y)) is a 1P REF ThusΘ=(0, ), η=1 1/λ and Ω = (, 1) If Y 1,, Y n are iid POW(λ), then T n = log(y i ) G(n, λ) The likelihood L(λ) = 1 [( λ exp 1 n λ 1) ] log(y i ), and the log likelihood log(l(λ)) = n log(λ)+( 1 λ 1) log(y i ) Hence d n log(l(λ)) = dλ λ log(yi ) λ set =0, 53

or log(y i )=nλ, or ˆλ = log(y i ) n d dλ log(l(λ)) = n λ log(yi ) λ 3 = ṋ λ + nˆλ ˆλ 3 Hence ˆλ is the UMVUE and MLE of λ If r> n, thentn r is the UMVUE of = n ˆλ < 0 E(Tn)=λ r r Γ(r + n) Γ(n) λ=ˆλ 1034 The Rayleigh Distribution If Y has a Rayleigh distribution, Y R(µ, σ), then the pdf of Y is [ f(y) = y µ exp 1 ( ) ] y µ σ σ where σ>0, µ is real, and y µ See Cohen and Whitten (1988, Ch 10) This is an asymmetric location scale family The cdf of Y is [ F (y) =1 exp 1 ( ) ] y µ σ for y µ, andf (y) =0, otherwise E(Y )=µ + σ π/ µ +153314σ VAR(Y )=σ (4 π)/ 04904σ MED(Y )=µ + σ log(4) µ +117741σ Hence µ MED(Y ) 655MAD(Y )andσ 30MAD(Y ) Let σd =MAD(Y ) If µ =0, and σ =1, then 05 =exp[ 05( log(4) D) ] exp[ 05( log(4) + D) ] 54

Hence D 0448453 and MAD(Y ) 0448453σ It can be shown that W =(Y µ) EXP(σ ) Other parameterizations for the Rayleigh distribution are possible Note that f(y) = 1 [ σ (y µ)i(y µ)exp 1 ] (y µ) σ appears to be a 1P REF if µ is known If Y 1,, Y n are iid R(µ, σ), then T n = (Y i µ) G(n, σ ) If µ is known, then the likelihood [ L(σ 1 )=c exp 1 ] (yi µ), σn σ and the log likelihood log(l(σ )) = d n log(σ ) 1 (yi µ) σ Hence d d(σ ) log(l(σ )) = n σ + 1 (yi µ) set =0, σ or (y i µ) =nσ, or ˆσ (Yi µ) = n d d(σ ) log(l(σ )) = n (σ ) n (ˆσ ) nˆσ (ˆσ ) 3 = (yi µ) (σ ) 3 = σ =ˆσ n (ˆσ ) < 0 Hence ˆσ is the UMVUE and MLE of σ if µ is known If µ is known and r> n, thent r n is the UMVUE of E(T r n )=r σ 55 r Γ(r + n) Γ(n)

1035 The Smallest Extreme Value Distribution If Y has a smallest extreme value distribution (or log-weibull distribution), Y SEV (θ, σ), then the pdf of Y is f(y) = 1 σ exp(y θ θ )exp[ exp(y σ σ )] where y and θ are real and σ>0 The cdf of Y is F (y) =1 exp[ exp( y θ σ )] This family is an asymmetric location-scale family with a longer left tail than right E(Y ) θ 05771σ, and VAR(Y )=σ π /6 164493σ MED(Y )=θ σlog(log()) MAD(Y ) 0767049σ Y is a one parameter exponential family in θ if σ is known If Y has a SEV(θ, σ) distribution, then W = Y has an LEV( θ, σ) distribution 1036 The Student s t Distribution If Y has a Student s t distribution, Y t p, then the pdf of Y is f(y) = Γ( p+1 ) y p+1 (1 + (pπ) 1/ Γ(p/) p ) ( ) where p is a positive integer and y is real This family is symmetric about 0 The t 1 distribution is the Cauchy(0, 1) distribution If Z is N(0, 1) and is independent of W χ p, then Z ( W p )1/ is t p E(Y )=0forp MED(Y )=0 56

VAR(Y )=p/(p ) for p 3, and MAD(Y )=t p,075 where P (t p t p,075 )=075 If α = P (t p t p,α ), then Cooke, Craven, and Clarke (198, p 84) suggest the approximation t p,α p[exp( w α p ) 1)] where w α = z α(8p +3) 8p +1, z α is the standard normal cutoff: α =Φ(z α ), and 05 α If 0 <α<05, then t p,α = t p,1 α This approximation seems to get better as the degrees of freedom increase 1037 The Truncated Extreme Value Distribution If Y has a truncated extreme value distribution, Y TEV(λ), then the pdf of Y is f(y) = 1 ( ) λ exp y ey 1 λ where y>0andλ>0 The cdf of Y is [ ] (e y 1) F (y)=1 exp λ for y>0 MED(Y )=log(1+λlog()) W = e Y 1isEXP(λ) f(y) = 1 [ ] 1 λ ey I(y 0) exp λ (ey 1) is a 1P REF Hence Θ = (0, ), η= 1/λ and Ω = (, 0) If Y 1,, Y n are iid TEV(λ), then T n = (e Y i 1) G(n, λ) 57

The likelihood L(λ) =c 1 [ 1 λ exp n λ and the log likelihood ] log(e y i 1), log(l(λ)) = d n log(λ) 1 λ log(e y i 1) Hence d n log(e y log(l(λ)) = dλ λ + i 1) λ or log(e y i 1) = nλ, or ˆλ = log(e Y i 1) n d dλ log(l(λ)) = n λ log(e y i 1) λ 3 = ṋ λ nˆλ ˆλ 3 = n ˆλ < 0 Hence ˆλ is the UMVUE and MLE of λ If r> n, thentn r is the UMVUE of E(Tn)=λ r r Γ(r + n) Γ(n) 1038 The Uniform Distribution set =0, λ=ˆλ If Y has a uniform distribution, Y U(θ 1,θ ), then the pdf of Y is 1 f(y) = I(θ 1 y θ ) θ θ 1 The cdf of Y is F (y) =(y θ 1 )/(θ θ 1 )forθ 1 y θ This family is a location-scale family which is symmetric about (θ 1 + θ )/ By definition, m(0) = c(0) = 1 For t 0, the mgf of Y is m(t) = etθ e tθ 1 (θ θ 1 )t, 58

and the chf of Y is c(t) = eitθ e itθ 1 (θ θ 1 )it E(Y )=(θ 1 + θ )/, and MED(Y )=(θ 1 + θ )/ VAR(Y )=(θ θ 1 ) /1, and MAD(Y )=(θ θ 1 )/4 Note that θ 1 =MED(Y ) MAD(Y )andθ =MED(Y )+MAD(Y ) Some classical estimates are ˆθ 1 = y (1) and ˆθ = y (n) 1039 The Weibull Distribution If Y has a Weibull distribution, Y W (φ, λ), then the pdf of Y is f(y) = φ λ yφ 1 e yφ λ where λ, y, and φ are all positive For fixed φ, this is a scale family in σ = λ 1/φ The cdf of Y is F (y) =1 exp( y φ /λ) fory>0 E(Y )=λ 1/φ Γ(1 + 1/φ) VAR(Y )=λ /φ Γ(1 + /φ) (E(Y )) E(Y r )=λ r/φ Γ(1 + r φ ) for r> φ MED(Y )=(λ log()) 1/φ Note that λ = (MED(Y ))φ log() W = Y φ is EXP(λ) W =log(y ) has a smallest extreme value SEV(θ =log(λ 1/φ ),σ =1/φ) distribution f(y) = φ [ ] 1 λ yφ 1 I(y 0) exp λ yφ is a one parameter exponential family in λ if φ is known 59

If Y 1,, Y n are iid W (φ, λ), then T n = Y φ i If φ is known, then the likelihood and the log likelihood L(λ) =c 1 λ n exp [ 1 λ G(n, λ) log(l(λ)) = d n log(λ) 1 λ ] y φ i, y φ i Hence or y φ i = nλ, or d n log(l(λ)) = dλ λ + y φ i λ Y φ i ˆλ = n d dλ log(l(λ)) = n λ y φ i λ 3 set =0, λ=ˆλ = ṋ λ nˆλ ˆλ 3 Hence ˆλ is the UMVUE and MLE of λ If r> n, thentn r is the UMVUE of = n ˆλ < 0 E(Tn)=λ r r Γ(r + n) Γ(n) 1040 The Zeta Distribution If Y has a Zeta distribution, Y Zeta(ν), then the pmf of Y is f(y) =P (Y = y) = 1 y ν ζ(ν) 60

where ν>1andy =1,, 3, Here the zeta function ζ(ν) = y=1 1 y ν for ν>1 This distribution is a one parameter exponential family E(y) = ζ(ν 1) ζ(ν) for ν>, and VAR(Y )= ζ(ν ) ζ(ν) [ ζ(ν 1) ζ(ν) for ν>3 E(Y r ζ(ν r) )= ζ(ν) for ν>r+1 This distribution is sometimes used for count data, especially by linguistics for word frequency See Lindsey (004, p 154) 1041 Complements Many of the distribution results used in this chapter came from Johnson and Kotz (1970a,b) and Patel, Kapadia and Owen (1976) Cohen and Whitten (1988), Ferguson (1967), Castillo (1988), Cramér (1946), Kennedy and Gentle (1980), Lehmann (1983), Meeker and Escobar (1998), Bickel and Doksum (1977), DeGroot and Schervish (001), Hastings and Peacock (1975) and Leemis (1986) also have useful results on distributions Also see articles in Kotz and Johnson (198ab, 1983ab, 1985ab, 1986, 1988ab) Often an entire book is devoted to a single distribution, see for example, Bowman and Shenton (1988) ] 61