Some Useful Distributions

Chapter 0 Some Useful Distributios Defiitio 0 The populatio media is ay value MED(Y ) such that P(Y MED(Y )) 05 ad P(Y MED(Y )) 05 (0) Defiitio 0 The populatio media absolute deviatio is MAD(Y ) = MED( Y MED(Y ) ) (0) Fidig MED(Y ) ad MAD(Y ) for symmetric distributios ad locatio scale families is made easier by the followig lemma Let F(y α ) = P(Y y α ) = α for 0 < α < where the cdf F(y) = P(Y y) Let D = MAD(Y ), M = MED(Y ) = y 05 ad U = y 075 Lemma 0 a) If W = a + by, the MED(W) = a + bmed(y ) ad MAD(W) = b MAD(Y ) b) If Y has a pdf that is cotiuous ad positive o its support ad symmetric about µ, the MED(Y ) = µ ad MAD(Y ) = y 075 MED(Y ) Fid M = MED(Y ) by solvig the equatio F(M) = 05 for M, ad fid U by solvig F(U) = 075 for U The D = MAD(Y ) = U M c) Suppose that W is from a locatio scale family with stadard pdf f Y (y) that is cotiuous ad positive o its support The W = µ + σy where σ > 0 First fid M by solvig F Y (M) = 05 After fidig M, fid D by solvig F Y (M + D) F Y (M D) = 05 The MED(W) = µ + σm ad MAD(W) = σd 74

Defiitio 03 The gamma fuctio Γ(x) = 0 t x e t dt for x > 0 Some properties of the gamma fuctio follow i) Γ(k) = (k )! for iteger k ii) Γ(x + ) = x Γ(x) for x > 0 iii) Γ(x) = (x ) Γ(x ) for x > iv) Γ(05) = π Some lower case Greek letters are alpha: α, beta: β, gamma: γ, delta: δ, epsilo: ɛ, zeta: ζ, eta: η, theta: θ, iota: ι, kappa: κ, lambda: λ, mu: µ, u: ν, xi: ξ, omicro: o, pi: π, rho: ρ, sigma: σ, upsilo: υ, phi: φ, chi: χ, psi: ψ ad omega: ω Some capital Greek letters are gamma: Γ, theta: Θ, sigma: Σ ad phi: Φ For the discrete uiform ad geometric distributios, the followig facts o series are useful Lemma 0 Let, ad be itegers with, ad let a be a costat Notice that i= a i = + if a = a) a i = a a+, a a i= b) a i =, a < a c) i=0 a i = a, a < a d) a i = a, a < a i= f) e) i = i = ( + ) ( + )( + ) 6 75

See Gabel ad Roberts (980, p 473-476) for the proof of a) d) For the special case of 0, otice that i=0 a i = a + a, a To see this, multiply both sides by ( a) The ( a) a i = ( a)( + a + a + + a + a ) = i=0 + a + a + + a + a a a a a + = a + ad the result follows Hece for a, a i = a i i= i=0 i=0 a i = a + a a a = a + a a The biomial theorem below is sometimes useful Theorem 03, The Biomial Theorem For ay real umbers x ad y ad for ay iteger 0, (x + y) = i=0 ( ) x i y i = (y + x) = i 0 The Beta Distributio i=0 ( ) y i x i i If Y has a beta distributio, Y beta(δ, ν), the the probability desity fuctio (pdf) of Y is f(y) = where δ > 0, ν > 0 ad 0 y Γ(δ + ν) Γ(δ)Γ(ν) yδ ( y) ν E(Y ) = 76 δ δ + ν

Notice that f(y) = VAR(Y ) = δν (δ + ν) (δ + ν + ) Γ(δ + ν) Γ(δ)Γ(ν) I [0,](y)exp[(δ )log(y) + (ν )log( y)] is a P REF Hece Θ = (0, ) (0, ), η = δ, η = ν ad Ω = (, ) (, ) If δ =, the W = log( Y ) EXP(/ν) Hece T = log( Y i ) G(, /ν) ad if r > the T r is the UMVUE of E(T) r = Γ(r + ) ν r Γ() If ν =, the W = log(y ) EXP(/δ) Hece T = log(y i ) G(, /δ) ad ad if r > the T r is the UMVUE of E(T r ) = Γ(r + ) δ r Γ() 0 The Beta Biomial Distributio If Y has a beta biomial distributio, Y BB(m, ρ, θ), the the probability mass fuctio of Y is ( ) m B(δ + y, ν + m y) P(Y = y) = y B(δ, ν) for y = 0,,,, m where 0 < ρ < ad θ > 0 Here δ = ρ/θ ad ν = ( ρ)/θ, so ρ = δ/(δ + ν) ad θ = /(δ + ν) Also B(δ, ν) = Γ(δ)Γ(ν) Γ(δ + ν) Hece δ > 0 ad ν > 0 The E(Y ) = mδ/(δ + ν) = mρ ad V(Y ) = mρ( ρ)[+(m )θ/(+θ)] If Y π biomial(m, π) ad π beta(δ, ν), the Y BB(m, ρ, θ) 77

03 The Beroulli ad Biomial Distributios If Y has a biomial distributio, Y BIN(k, ρ), the the probability mass fuctio (pmf) of Y is ( ) k f(y) = P(Y = y) = ρ y ( ρ) k y y for y = 0,,, k where 0 < ρ < If ρ = 0, P(Y = 0) = = ( ρ) k while if ρ =, P(Y = k) = = ρ k The momet geeratig fuctio m(t) = [( ρ) + ρe t ] k, ad the characteristic fuctio c(t) = [( ρ) + ρe it ] k E(Y ) = kρ VAR(Y ) = kρ( ρ) The Beroulli (ρ) distributio is the biomial (k =, ρ) distributio Pourahmadi (995) showed that the momets of a biomial (k, ρ) radom variable ca be foud recursively If r is a iteger, ( 0 0) = ad the last term below is 0 for r =, the r ( ) r r ( ) r E(Y r ) = kρ E(Y i ) ρ E(Y i+ ) i i i=0 i=0 The followig ormal approximatio is ofte used whe kρ( ρ) > 9 Hece Y N(kρ, kρ( ρ)) P(Y y) Φ ( ) y + 05 kρ kρ( ρ) Also P(Y = y) ( exp ) (y kρ) kρ( ρ) π kρ( ρ) 78

See Johso, Kotz ad Kemp (99, p 5) This approximatio suggests that MED(Y ) kρ, ad MAD(Y ) 0674 kρ( ρ) Hamza (995) states that E(Y ) MED(Y ) max(ρ, ρ) ad shows that E(Y ) MED(Y ) log() If k is large ad kρ small, the Y Poisso(kρ) If Y,, Y are idepedet BIN(k i, ρ) the Y i BIN( k i, ρ) Notice that ( ) [ ] k f(y) = ( ρ) k ρ exp log( y ρ )y is a P REF i ρ if k is kow Thus Θ = (0, ), ( ) ρ η = log ρ ad Ω = (, ) Assume that Y,, Y are iid BIN(k, ρ), the T = Y i BIN(k, ρ) If k is kow, the the likelihood ad the log likelihood L(ρ) = c ρ P y i ( ρ) k P y i, log(l(ρ)) = d + log(ρ) y i + (k y i )log( ρ) Hece d dρ log(l(ρ)) = y i + k ρ ρ y i ( ) set = 0, or ( ρ) y i = ρ(k y i), or y i = ρk or ˆρ = y i /(k) 79

This solutio is uique ad d dρ log(l(ρ)) = y i ρ k y i < 0 ( ρ) if 0 < y i < k Hece kˆρ = Y is the UMVUE, MLE ad MME of kρ if k is kow Let ˆρ = umber of successes / ad let P(Z z α/ ) = α/ if Z N(0, ) Let ñ = + z α/ ad ρ = ˆρ + 05z α/ + z α/ The the large sample 00 ( α)% Agresti Coull CI for ρ is ρ( ρ) p ± z α/ ñ Let W = Y i bi( k i, ρ) ad let w = k i Ofte k i ad the w = Let P(F d,d F d,d (α)) = α where F d,d has a F distributio with d ad d degrees of freedom The the Clopper Pearso exact 00 ( α)% CI for ρ is ( ) 0, for W = 0, + w F w,(α) ( ) w w + F,w ( α), for W = w, ad (ρ L, ρ U ) for 0 < W < w with ρ L = W W + ( w W + )F (w W+),W( α/) ad ρ U = W + W + + ( w W)F (w W),(W+)(α/) 80

04 The Burr Distributio If Y has a Burr distributio, Y Burr(φ, λ), the the pdf of Y is φy φ f(y) = λ ( + y φ ) λ + where y, φ, ad λ are all positive The cdf of Y is [ ] log( + y φ ) F(y) = exp = ( + y φ ) /λ for y > 0 λ MED(Y ) = [e λlog() ] /φ See Patel, Kapadia ad Owe (976, p 95) W = log( + Y φ ) is EXP(λ) Notice that f(y) = λ φyφ + y φ exp [ ] λ log( + yφ ) I(y > 0) is a oe parameter expoetial family if φ is kow If Y,, Y are iid Burr(λ, φ), the T = log( + Y φ i ) G(, λ) If φ is kow, the the likelihood L(λ) = c λ exp [ λ ] log( + y φ i ), ad the log likelihood log(l(λ)) = d log(λ) λ log( + yφ i ) Hece or log( + yφ i ) = λ or d log(l(λ)) = dλ λ + log( + yφ i ) λ set = 0, ˆλ = log( + yφ i ) 8

This solutio is uique ad d dλ log(l(λ)) = λ log( + yφ i ) λ λ=ˆλ = ṋ λ ˆλ ˆλ 3 = ˆλ < 0 Thus ˆλ = log( + Y φ i ) is the UMVUE ad MLE of λ if φ is kow If φ is kow ad r >, the T r is the UMVUE of E(T) r = λ rγ(r + ) Γ() 05 The Cauchy Distributio If Y has a Cauchy distributio, Y C(µ, σ), the the pdf of Y is f(y) = σ π σ + (y µ) = πσ[ + ( y µ σ ) ] where y ad µ are real umbers ad σ > 0 The cumulative distributio fuctio (cdf) of Y is F(y) = π [arcta(y µ σ ) + π/] See Ferguso (967, p 0) This family is a locatio scale family that is symmetric about µ The momets of Y do ot exist, but the characteristic fuctio of Y is c(t) = exp(itµ t σ) MED(Y ) = µ, the upper quartile = µ + σ, ad the lower quartile = µ σ MAD(Y ) = F (3/4) MED(Y ) = σ If Y,, Y are idepedet C(µ i, σ i ), the a i Y i C( a i µ i, a i σ i ) I particular, if Y,, Y are iid C(µ, σ), the Y C(µ, σ) If W U( π/, π/), the Y = ta(w) C(0, ) 8

06 The Chi Distributio If Y has a chi distributio (also called a p dimesioal Rayleigh distributio), Y chi(p, σ), the the pdf of Y is f(y) = yp e σ y σ p p Γ(p/) where y 0 ad σ, p > 0 This is a scale family if p is kow ad VAR(Y ) = σ E(Y ) = σ Γ(+p ) Γ(p/) Γ(+p ) Γ(p/) ( Γ( +p ) Γ(p/) E(Y r ) = r/ σ rγ(r+p) Γ(p/) ) for r > p The mode is at σ p for p See Cohe ad Whitte (988, ch 0) Note that W = Y G(p/, σ ) Y geeralized gamma (ν = p/, λ = σ, φ = ) If p =, the Y has a half ormal distributio, Y HN(0, σ ) If p =, the Y has a Rayleigh distributio, Y R(0, σ) If p = 3, the Y has a Maxwell Boltzma distributio (also kow as a Boltzma distributio or a Maxwell distributio), Y MB (0, σ) If p is a iteger ad Y chi(p, ), the Y χ p Sice f(y) = p Γ(p/)σ, pi(y > 0)exp[(p )log(y) σ y ], this family appears to be a P REF Notice that Θ = (0, ) (0, ), η = p, η = /(σ ), ad Ω = (, ) (, 0) If p is kow the y p f(y) = p Γ(p/) I(y > 0) [ ] σ exp p σ y 83

appears to be a P REF If Y,, Y are iid chi(p, σ), the T = Y i G(p/, σ ) If p is kow, the the likelihood ad the log likelihood Hece or y i = pσ or L(σ ) = c exp[ σp σ yi ], log(l(σ )) = d p log(σ ) σ d d(σ ) log(σ ) = p σ + (σ ) This solutio is uique ad d d(σ ) log(l(σ )) = p (σ ) ˆσ = y i (σ ) 3 y i p = σ =ˆσ y i yi set = 0, p (ˆσ ) pˆσ (ˆσ ) 3 = p (ˆσ ) < 0 Thus ˆσ ˆσ = Y i p is the UMVUE ad MLE of σ whe p is kow If p is kow ad r > p/, the T r is the UMVUE of E(T) r = r σ r Γ(r + p/) Γ(p/) 84

07 The Chi square Distributio If Y has a chi square distributio, Y χ p, the the pdf of Y is f(y) = y p e y p Γ( p ) where y 0 ad p is a positive iteger The mgf of Y is ( ) p/ m(t) = = ( t) p/ t for t < / The characteristic fuctio c(t) = E(Y ) = p VAR(Y ) = p Sice Y is gamma G(ν = p/, λ = ), ( ) p/ it E(Y r ) = r Γ(r + p/), r > p/ Γ(p/) MED(Y ) p /3 See Pratt (968, p 470) for more terms i the expasio of MED(Y ) Empirically, p MAD(Y ) 483 ( 9p ) 09536 p There are several ormal approximatios for this distributio The Wilso Hilferty approximatio is ( ) Y 3 N( p 9p, 9p ) See Bowma ad Sheto (99, p 6) This approximatio gives P(Y x) Φ[(( x p )/3 + /9p) 9p/], 85

ad χ p,α p(z α 9p + 9p )3 where z α is the stadard ormal percetile, α = Φ(z α ) The last approximatio is good if p > 4 log(α) See Keedy ad Getle (980, p 8) This family is a oe parameter expoetial family, but is ot a REF sice the set of itegers does ot cotai a ope iterval 08 The Discrete Uiform Distributio If Y has a discrete uiform distributio, Y DU(θ, θ ), the the pmf of Y is f(y) = P(Y = y) = θ θ + for θ y θ where y ad the θ i are itegers Let θ = θ + τ where τ = θ θ + The cdf for Y is F(y) = y θ + θ θ + for θ y θ Here y is the greatest iteger fuctio, eg, 77 = 7 This result holds sice for θ y θ, F(y) = y θ θ + i=θ E(Y ) = (θ + θ )/ = θ + (τ )/ while V (Y ) = (τ )/ The result for E(Y ) follows by symmetry, or because E(Y ) = θ y=θ y θ θ + = θ (θ θ + ) + [0 + + + + (θ θ )] θ θ + where last equality follows by addig ad subtractig θ to y for each of the θ θ + terms i the middle sum Thus E(Y ) = θ + (θ θ )(θ θ + ) (θ θ + ) = θ + θ θ sice i = ( + )/ by Lemma 0e with = θ θ 86 = θ + θ

To see the result for V (Y ), let W = Y θ + The V (Y ) = V (W) ad f(w) = /τ for w =,, τ Hece W DU(, τ), E(W) = τ τ w = τ(τ + ) τ = + τ, ad E(W) = τ by Lemma 0 So τ w = τ(τ + )(τ + ) 6τ = (τ + )(τ + ) 6 V (Y ) = V (W) = E(W ) (E(W)) = (τ + )(τ + ) 6 ( ) + τ = (τ + )(τ + ) 3(τ + ) = 4(τ + ) (τ + ) 3(τ + ) τ + τ + τ (τ + )[(τ + ) ] 3(τ + ) = (τ + ) τ = τ Let Z be the set of itegers ad let Y,, Y be iid DU(θ, θ ) The the likelihood fuctio L(θ, θ ) = (θ θ + ) I(θ Y () )I(θ Y () )I(θ θ )I(θ Z)I(θ Z) is maximized by makig θ θ as small as possible where itegers θ θ So eed θ as small as possible ad θ as large as possible, ad the MLE of (θ, θ ) is (Y (), Y () ) 09 The Double Expoetial Distributio If Y has a double expoetial distributio (or Laplace distributio), Y DE(θ, λ), the the pdf of Y is f(y) = ( ) y θ λ exp λ 87 = =

where y is real ad λ > 0 The cdf of Y is ( ) y θ F(y) = 05exp λ ad ( ) (y θ) F(y) = 05exp λ if y θ, if y θ This family is a locatio scale family which is symmetric about θ The mgf m(t) = exp(θt)/( λ t ) for t < /λ, ad the characteristic fuctio c(t) = exp(θit)/( + λ t ) E(Y ) = θ, ad MED(Y ) = θ VAR(Y ) = λ, ad MAD(Y ) = log()λ 0693λ Hece λ = MAD(Y )/log() 443MAD(Y ) To see that MAD(Y ) = λlog(), ote that F(θ +λlog()) = 05 = 075 The maximum likelihood estimators are ˆθ MLE = MED() ad ˆλ MLE = Y i MED() A 00( α)% cofidece iterval (CI) for λ is ( ) Y i MED(), Y i MED(), χ, χ α, α ad a 00( α)% CI for θ is MED() ± z α/ Y i MED() z α/ where χ p,α ad z α are the α percetiles of the χ p ad stadard ormal distributios, respectively See Patel, Kapadia ad Owe (976, p 94) W = Y θ EXP(λ) 88

Notice that f(y) = [ ] λ exp y θ λ is a oe parameter expoetial family i λ if θ is kow If Y,, Y are iid DE(θ, λ) the T = Y i θ G(, λ) If θ is kow, the the likelihood ad the log likelihood L(λ) = c λ exp [ λ ] y i θ, log(l(λ)) = d log(λ) λ y i θ Hece d log(l(λ)) = dλ λ + λ or y i θ = λ or ˆλ = y i θ This solutio is uique ad d dλ log(l(λ)) = λ y i θ λ 3 λ=ˆλ y i θ set = 0 = ṋ λ ˆλ ˆλ 3 = ˆλ < 0 Thus ˆλ = Y i θ is the UMVUE ad MLE of λ if θ is kow 89

00 The Expoetial Distributio If Y has a expoetial distributio, Y EXP(λ), the the pdf of Y is where λ > 0 The cdf of Y is f(y) = λ exp( y ) I(y 0) λ F(y) = exp( y/λ), y 0 This distributio is a scale family with scale parameter λ The mgf m(t) = /( λt) for t < /λ, ad the characteristic fuctio c(t) = /( iλt) E(Y ) = λ, ad VAR(Y ) = λ W = Y/λ χ Sice Y is gamma G(ν =, λ), E(Y r ) = λγ(r + ) for r > MED(Y ) = log()λ ad MAD(Y ) λ/078 sice it ca be show that exp(mad(y )/λ) = + exp( MAD(Y )/λ) Hece 078 MAD(Y ) λ The classical estimator is ˆλ = Y ad the 00( α)% CI for E(Y ) = λ is ( ) Y i, Y i χ, χ α, α where P(Y χ, ) = α/ if Y is χ See Patel, Kapadia ad Owe (976, α p 88) Notice that f(y) = [ ] λ I(y 0)exp λ y is a P REF Hece Θ = (0, ), η = /λ ad Ω = (, 0) Suppose that Y,, Y are iid EXP(λ), the T = Y i G(, λ) 90

The likelihood ad the log likelihood L(λ) = λ exp [ λ ] y i, log(l(λ)) = log(λ) λ y i Hece or y i = λ or d log(l(λ)) = dλ λ + λ ˆλ = y y i set = 0, Sice this solutio is uique ad d dλ log(l(λ)) = λ y i λ 3 λ=ˆλ = ṋ λ ˆλ ˆλ 3 = ˆλ < 0, the ˆλ = Y is the UMVUE, MLE ad MME of λ If r >, the T r is the UMVUE of E(T) r = λr Γ(r + ) Γ() 0 The Two Parameter Expoetial Distributio If Y has a parameter expoetial distributio, Y EXP(θ, λ) the the pdf of Y is f(y) = ( ) (y θ) λ exp I(y θ) λ where λ > 0 ad θ is real The cdf of Y is F(y) = exp[ (y θ)/λ)], y θ 9

This family is a asymmetric locatio-scale family The mgf m(t) = exp(tθ)/( λt) for t < /λ, ad the characteristic fuctio c(t) = exp(itθ)/( iλt) E(Y ) = θ + λ, ad VAR(Y ) = λ ad MED(Y ) = θ + λlog() MAD(Y ) λ/078 Hece θ MED(Y ) 078 log()mad(y ) See Rousseeuw ad Croux (993) for similar results Note that 078 log() 44 To see that 078MAD(Y ) λ, ote that 05 = θ+λ log()+mad θ+λ log() MAD exp( (y θ)/λ)dy λ = 05[ e MAD/λ + e MAD/λ ] assumig λlog() > MAD Plug i MAD = λ/078 to get the result If θ is kow, the f(y) = I(y θ) [ ] λ exp (y θ) λ is a P REF i λ Notice that Y θ EXP(λ) Let ˆλ = (Y i θ) The ˆλ is the UMVUE ad MLE of λ if θ is kow If Y,, Y are iid EXP(θ, λ), the the likelihood [ ] L(θ, λ) = λ exp (y i θ) I(y () θ), λ 9

ad the log likelihood log(l(θ, λ)) = [ log(λ) λ (y i θ)]i(y () θ) For ay fixed λ > 0, the log likelihood is maximized by maximizig θ Hece ˆθ = Y (), ad the profile log likelihood is log(l(λ y () )) = log(λ) λ (y i y () ) is maximized by ˆλ = (y i y () ) Hece the MLE ( ) (ˆθ, ˆλ) = Y (), (Y i Y () ) = (Y (), Y Y () ) Let D = (Y i Y () ) = ˆλ The for, ( ) D D, χ ( ), α/ χ ( ),α/ (03) is a 00( α)% CI for λ, while (Y () ˆλ[(α) /( ) ], Y () ) (04) is a 00 ( α)% CI for θ See Ma, Schafer, ad Sigpurwalla (974, p 76) If θ is kow ad T = (Y i θ), the a 00( α)% CI for λ is ( ) T T, (05) χ, α/ χ,α/ 0 The F Distributio If Y has a F distributio, Y F(ν, ν ), the the pdf of Y is f(y) = Γ( ν +ν ) Γ(ν /)Γ(ν /) ( ) ν / ν ν 93 y (ν )/ ) (ν +ν )/ ( + ( ν ν )y

where y > 0 ad ν ad ν are positive itegers ad E(Y ) = ν ν, ν > ( ) ν (ν + ν ) VAR(Y ) = ν ν (ν 4), ν > 4 E(Y r ) = Γ(ν +r )Γ( ν r ) Γ(ν /)Γ(ν /) ( ν ν ) r, r < ν / Suppose that X ad X are idepedet where X χ ν ad X χ ν The W = (X /ν ) (X /ν ) F(ν, ν ) ( ) r Notice that E(Y r ) = E(W r ν ) = ν E(X r )W(X r ) If W t ν, the Y = W F(, ν) 03 The Gamma Distributio If Y has a gamma distributio, Y G(ν, λ), the the pdf of Y is f(y) = yν e y/λ λ ν Γ(ν) where ν, λ, ad y are positive The mgf of Y is ( ) ν ( ) ν /λ m(t) = t = λt λ for t < /λ The characteristic fuctio ( ) ν c(t) = iλt E(Y ) = νλ VAR(Y ) = νλ E(Y r ) = λr Γ(r + ν) Γ(ν) 94 if r > ν (06)

Che ad Rubi (986) show that λ(ν /3) < MED(Y ) < λν = E(Y ) Empirically, for ν > 3/, ad MED(Y ) λ(ν /3), MAD(Y ) λ ν 483 This family is a scale family for fixed ν, so if Y is G(ν, λ) the cy is G(ν, cλ) for c > 0 If W is EXP(λ) the W is G(, λ) If W is χ p, the W is G(p/, ) Some classical estimators are give ext Let [ w = log y geometric mea() where geometric mea() = (y y y ) / = exp[ log(y i)] The Thom s estimator (Johso ad Kotz 970a, p 88) is ˆν 05( + + 4w/3 ) w Also 05000876 + 064885w 0054474w ˆν MLE w for 0 < w 0577, ad ˆν MLE 889899 + 9059950w + 09775374w w(77978 + 968477w + w ) for 0577 < w 7 If W > 7 the estimatio is much more difficult, but a rough approximatio is ˆν /w for w > 7 See Bowma ad Sheto (988, p 46) ad Greewood ad Durad (960) Fially, ˆλ = Y /ˆν Notice that ˆβ may ot be very good if ˆν < /7 Several ormal approximatios are available The Wilso Hilferty approximatio says that for ν > 05, Y /3 N Hece if Y is G(ν, λ) ad ( (νλ) /3 ( ), (νλ)/3 9ν 9ν α = P[Y G α ], 95 ] )

the G α νλ [ z α 9ν + 9ν ] 3 where z α is the stadard ormal percetile, α = Φ(z α ) Bowma ad Sheto (988, p 0) iclude higher order terms Notice that [ ] f(y) = I(y > 0)exp y + (ν )log(y) λ ν Γ(ν) λ is a P REF Hece Θ = (0, ) (0, ), η = /λ, η = ν ad Ω = (, 0) (, ) If Y,, Y are idepedet G(ν i, λ) the Y i G( ν i, λ) If Y,, Y are iid G(ν, λ), the T = Y i G(ν, λ) Sice f(y) = Γ(ν) exp[(ν )log(y)]i(y > 0) [ ] λ exp ν λ y, Y is a P REF whe ν is kow If ν is kow, the the likelihood The log likelihood L(β) = c exp λν [ λ ] y i log(l(λ)) = d ν log(λ) λ y i Hece or y i = νλ or d ν log(l(λ)) = dλ λ + y i λ ˆλ = y/ν 96 set = 0,

This solutio is uique ad d ν log(l(λ)) = dλ λ y i λ 3 λ=ˆλ = ν ˆλ νˆλ ˆλ 3 Thus Y is the UMVUE, MLE ad MME of νλ if ν is kow = ν ˆλ < 0 04 The Geeralized Gamma Distributio If Y has a geeralized gamma distributio, Y GG(ν, λ, φ), the the pdf of Y is f(y) = φyφν λ φν Γ(ν) exp( yφ /λ φ ) where ν, λ, φ ad y are positive This family is a scale family with scale parameter λ if φ ad ν are kow If φ ad ν are kow, the E(Y k ) = λk Γ(ν + k φ ) Γ(ν) f(y) = φyφν Γ(ν) if k > φν (07) [ ] I(y > 0) exp λφν λ φ yφ, which is a oe parameter expoetial family Notice that W = Y φ G(ν, λ φ ) If Y,, Y are iid GG(ν, λ, φ) where φ ad ν are kow, the T = Y φ i G(ν, λ φ ), ad T r is the UMVUE of E(T) r = λ φrγ(r + ν) Γ(ν) for r > ν 05 The Geeralized Negative Biomial Distributio If Y has a geeralized egative biomial distributio, Y GNB(µ, κ), the the pmf of Y is ( ) κ ( Γ(y + κ) κ f(y) = P(Y = y) = κ ) y Γ(κ)Γ(y + ) µ + κ µ + κ 97

for y = 0,,, where µ > 0 ad κ > 0 This distributio is a geeralizatio of the egative biomial (κ, ρ) distributio with ρ = κ/(µ + κ) ad κ > 0 is a ukow real parameter rather tha a kow iteger The mgf is [ ] κ κ m(t) = κ + µ( e t ) for t < log(µ/(µ + κ)) E(Y ) = µ ad VAR(Y ) = µ + µ /κ If Y,, Y are iid GNB(µ, κ), the Y i GNB(µ, κ) Whe κ is kow, this distributio is a P REF If Y,, Y are iid GNB(µ, κ) where κ is kow, the ˆµ = Y is the MLE, UMVUE ad MME of µ 06 The Geometric Distributio If Y has a geometric distributio, Y geom(ρ) the the pmf of Y is f(y) = P(Y = y) = ρ( ρ) y for y = 0,,, ad 0 < ρ < The cdf for Y is F(y) = ( ρ) y + for y 0 ad F(y) = 0 for y < 0 Here y is the greatest iteger fuctio, eg, 77 = 7 To see this, ote that for y 0, y F(y) = ρ ( ρ) y = ρ i=0 ( ρ) y + ( ρ) by Lemma 0a with = 0, = y ad a = ρ E(Y ) = ( ρ)/ρ VAR(Y ) = ( ρ)/ρ Y NB(, ρ) Hece the mgf of Y is for t < log( ρ) Notice that m(t) = ρ ( ρ)e t f(y) = ρexp[log( ρ)y] 98

is a P REF Hece Θ = (0, ), η = log( ρ) ad Ω = (, 0) If Y,, Y are iid geom(ρ), the T = Y i NB(, ρ) The likelihood ad the log likelihood Hece L(ρ) = ρ exp[log( ρ) y i ], log(l(ρ)) = log(ρ) + log( ρ) d dρ log(l(ρ)) = ρ ρ y i y i set = 0 or ( ρ)/ρ = y i or ρ ρ y i = 0 or ˆρ = + y i This solutio is uique ad d log(l(ρ)) = dρ ρ y i ( ρ) < 0 Thus ˆρ = + Y i is the MLE of ρ The UMVUE, MLE ad MME of ( ρ)/ρ is Y 07 The Gompertz Distributio If Y has a Gompertz distributio, Y Gomp(θ, ν), the the pdf of Y is ν ] f(y) = νe exp[ θy θ ( eθy ) 99

for θ 0 where ν > 0 ad y > 0 The parameter θ is real, ad the Gomp(θ = 0, ν) distributio is the expoetial (/ν) distributio The cdf is ν ] F(y) = exp[ θ ( eθy ) for θ 0 ad y > 0 For fixed θ this distributio is a scale family with scale parameter /ν 08 The Half Cauchy Distributio If Y has a half Cauchy distributio, Y HC(µ, σ), the the pdf of Y is f(y) = πσ[ + ( y µ σ ) ] where y µ, µ is a real umber ad σ > 0 The cdf of Y is F(y) = π arcta(y µ σ ) for y µ ad is 0, otherwise This distributio is a right skewed locatioscale family MED(Y ) = µ + σ MAD(Y ) = 07305σ 09 The Half Logistic Distributio If Y has a half logistic distributio, Y HL(µ, σ), the the pdf of Y is f(y) = exp( (y µ)/σ) σ[ + exp( (y µ)/σ)] where σ > 0, y µ ad µ are real The cdf of Y is F(y) = exp[(y µ)/σ] + exp[(y µ)/σ)] for y µ ad 0 otherwise This family is a right skewed locatio scale family MED(Y ) = µ + log(3)σ MAD(Y ) = 067346σ 300

00 The Half Normal Distributio If Y has a half ormal distributio, Y HN(µ, σ ), the the pdf of Y is f(y) = (y µ) exp( ) π σ σ where σ > 0 ad y µ ad µ is real Let Φ(y) deote the stadard ormal cdf The the cdf of Y is F(y) = Φ( y µ σ ) for y > µ ad F(y) = 0, otherwise E(Y ) = µ + σ /π µ + 0797885σ VAR(Y ) = σ (π ) 0363380σ π This is a asymmetric locatio scale family that has the same distributio as µ + σ Z where Z N(0, ) Note that Z χ Hece the formula for the rth momet of the χ radom variable ca be used to fid the momets of Y MED(Y ) = µ + 06745σ MAD(Y ) = 0399096σ Notice that [ f(y) = I(y µ)exp ( ] π σ σ)(y µ) is a P REF if µ is kow Hece Θ = (0, ), η = /(σ ) ad Ω = (, 0) W = (Y µ) G(/, σ ) If Y,, Y are iid HN(µ, σ ), the T = (Y i µ) G(/, σ ) If µ is kow, the the likelihood [ ] L(σ ) = c σ exp ( σ ) (y i µ), 30

ad the log likelihood Hece log(l(σ )) = d log(σ ) σ d d(σ ) log(l(σ )) = (σ ) + (σ ) or (y i µ) = σ or Thus This solutio is uique ad (σ ) ˆσ = d (y i µ) d(σ ) log(l(σ )) = (y i µ) (σ ) 3 = σ =ˆσ ˆσ = (y i µ) (y i µ) set = 0, (ˆσ ) ˆσ ( ˆσ ) 3 = ˆσ < 0 (Y i µ) is the UMVUE ad MLE of σ if µ is kow If r > / ad if µ is kow, the T r is the UMVUE of E(T r ) = r σ r Γ(r + /)/Γ(/) Example 53 shows that (ˆµ, ˆσ ) = (Y (), (Y i Y () ) ) is MLE of (µ, σ ) Followig Pewsey (00), a large sample 00( α)% cofidece iterval for σ is ( ) ˆσ χ ( α/), ˆσ (08) χ (α/) while a large sample 00( α)% CI for µ is (ˆµ + ˆσ log(α) Φ ( + ) ( + 3/ ), ˆµ) (09) If µ is kow, the a 00( α)% CI for σ is ( ) T χ ( α/), T (00) χ (α/) 30

0 The Hypergeometric Distributio If Y has a hypergeometric distributio, Y HG(C, N C, ), the the data set cotais N objects of two types There are C objects of the first type (that you wish to cout) ad N C objects of the secod type Suppose that objects are selected at radom without replacemet from the N objects The Y couts the umber of the selected objects that were of the first type The pmf of Y is ( C N C ) y)( f(y) = P(Y = y) = y ( N ) where the iteger y satisfies max(0, N + C) y mi(, C) The right iequality is true sice if objects are selected, the the umber of objects y of the first type must be less tha or equal to both ad C The first iequality holds sice y couts the umber of objects of secod type Hece y N C Let p = C/N The ad VAR(Y ) = E(Y ) = C N = p C(N C) N N N = p( p)n N If is small compared to both C ad N C the Y BIN(, p) If is large but is small compared to both C ad N C the Y N(p, p( p)) 0 The Iverse Gaussia Distributio If Y has a iverse Gaussia distributio, Y IG(θ, λ), the 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 θ t θ λ 303

for t < λ/(θ ) See Datta (005) ad Schwarz ad Samata (99) for additioal properties The characteristic fuctio is E(Y ) = θ ad Notice that f(y) = φ(t) = exp [ λ θ ( VAR(Y ) = θ3 λ is a two parameter expoetial family If Y,, Y are iid IG(θ, λ), the θ it λ )] [ λ λ π eλ/θ y3i(y > 0)exp θ y λ ] y Y i IG(θ, λ) ad Y IG(θ, λ) If λ is kow, the the likelihood ad the log likelihood Hece or y i = θ or L(θ) = c e λ/θ exp[ λ θ y i ], log(l(θ)) = d + λ θ λ θ d λ log(l(θ)) = + λ dθ θ θ 3 ˆθ = y y i y i set = 0, This solutio is uique ad d λ log(l(θ)) = 3λ y i dθ θ 3 θ 4 304 θ=ˆθ = λ ˆθ 3 3λˆθ ˆθ 4 = λ ˆθ 3 < 0

Thus Y is the UMVUE, MLE ad MME of θ if λ is kow If θ is kow, the the likelihood [ ] L(λ) = c λ / λ (y i θ) exp, θ ad the log likelihood Hece or Thus log(l(λ)) = d + log(λ) λ θ d dλ log(l(λ)) = λ θ This solutio is uique ad is the MLE of λ if θ is kow ˆλ = θ y i (y i θ) (y i θ) y i y i d log(l(λ)) = dλ λ < 0 ˆλ = θ (Y i θ) Y i (y i θ) y i set = 0 Aother parameterizatio of the iverse Gaussia distributio takes θ = λ/ψ so that f(y) = [ λ π e λψ ψ y3i[y > 0] exp y λ ], y where λ > 0 ad ψ 0 Here Θ = (0, ) [0, ), η = ψ/, η = λ/ ad Ω = (, 0] (, 0) Sice Ω is ot a ope set, this is a parameter full expoetial family that is ot regular If ψ is kow the Y is a P REF, but if λ is kow the Y is a oe parameter full expoetial family Whe ψ = 0, Y has a oe sided stable distributio with idex / See Bardorff Nielse (978, p 7) 305

03 The Iverted Gamma Distributio If Y has a iverted gamma distributio, Y INV G(ν, λ), the the pdf of Y is f(y) = y ν+ Γ(ν) I(y > 0) ( ) λ exp ν λ y where λ, ν ad y are all positive It ca be show that W = /Y G(ν, λ) This family is a scale family with scale parameter τ = /λ if ν is kow If ν is kow, this family is a parameter expoetial family If Y,, Y are iid INVG(ν, λ) ad ν is kow, the T = the UMVUE of for r > ν λ rγ(r + ν) Γ(ν) Y i G(ν, λ) ad T r is 04 The Largest Extreme Value Distributio If Y has a largest extreme value distributio (or Gumbel distributio), Y LEV (θ, σ), the the pdf of Y is f(y) = σ exp( (y θ θ ))exp[ exp( (y σ σ ))] where y ad θ are real ad σ > 0 The cdf of Y is F(y) = exp[ exp( ( y θ σ ))] This family is a asymmetric locatio scale family with a mode at θ The mgf m(t) = exp(tθ)γ( σt) for t < /σ E(Y ) θ + 0577σ, ad VAR(Y ) = σ π /6 64493σ MED(Y ) = θ σ log(log()) θ + 03665σ 306

ad MAD(Y ) 0767049σ W = exp( (Y θ)/σ) EXP() Notice that f(y) = σ eθ/σ e y/σ exp [ e θ/σ e y/σ] is a oe parameter expoetial family i θ if σ is kow If Y,, Y are iid LEV(θ, σ) where σ is kow, the the likelihood L(σ) = c e θ/σ exp[ e θ/σ e yi/σ ], ad the log likelihood log(l(θ)) = d + θ σ eθ/σ e yi/σ Hece or or or d dθ log(l(θ)) = σ eθ/σ σ e θ/σ e yi/σ =, e θ/σ = ( ˆθ = log e y i/σ, e y i/σ e y i/σ set = 0, ) Sice this solutio is uique ad is the MLE of θ d log(l(θ)) = dθ σ eθ/σ ( ˆθ = log e yi/σ < 0, e Y i/σ 307 )

05 The Logarithmic Distributio If Y has a logarithmic distributio, the the pmf of Y is f(y) = P(Y = y) = θ y log( θ) y for y =,, ad 0 < θ < This distributio is sometimes called the logarithmic series distributio or the log-series distributio The mgf m(t) = log( θet ) log( θ) for t < log(θ) Notice that f(y) = E(Y ) = θ log( θ) θ log( θ) y exp(log(θ)y) is a P REF Hece Θ = (0, ), η = log(θ) ad Ω = (, 0) If Y,, Y are iid logarithmic (θ), the Y is the UMVUE of E(Y ) 06 The Logistic Distributio If Y has a logistic distributio, Y L(µ, σ), the the pdf of Y is f(y) = where σ > 0 ad y ad µ are real The characteristic fuctio of Y is F(y) = exp( (y µ)/σ) σ[ + exp( (y µ)/σ)] + exp( (y µ)/σ) = exp((y µ)/σ) + exp((y µ)/σ) This family is a symmetric locatio scale family The mgf of Y is m(t) = πσte µt csc(πσt) for t < /σ, ad the chf is c(t) = πiσte iµt csc(πiσt) where csc(t) is the cosecat of t E(Y ) = µ, ad 308

MED(Y ) = µ VAR(Y ) = σ π /3, ad MAD(Y ) = log(3)σ 0986 σ Hece σ = MAD(Y )/log(3) The estimators ˆµ = Y ad S = (Y i Y ) are sometimes used Note that if q = F L(0,) (c) = ec q the c = log( + e c q ) Takig q = 9995 gives c = log(999) 76 To see that MAD(Y ) = log(3)σ, ote that F(µ + log(3)σ) = 075, F(µ log(3)σ) = 05, ad 075 = exp(log(3))/( + exp(log(3))) 07 The Log-Cauchy Distributio If Y has a log Cauchy distributio, Y LC(µ, σ), the the pdf of Y is f(y) = πσy[ + ( log(y) µ σ ) ] where y > 0, σ > 0 ad µ is a real umber This family is a scale family with scale parameter τ = e µ if σ is kow It ca be show that W = log(y ) has a Cauchy(µ, σ) distributio 08 The Log-Logistic Distributio If Y has a log logistic distributio, Y LL(φ, τ), the the pdf of Y is f(y) = φτ(φy)τ [ + (φy) τ ] where y > 0, φ > 0 ad τ > 0 The cdf of Y is F(y) = + (φy) τ for y > 0 This family is a scale family with scale parameter φ if τ is kow 309

MED(Y ) = /φ It ca be show that W = log(y ) has a logistic(µ = log(φ), σ = /τ) distributio Hece φ = e µ ad τ = /σ Kalbfleisch ad Pretice (980, p 7-8) suggest that the log-logistic distributio is a competitor of the logormal distributio 09 The Logormal Distributio If Y has a logormal distributio, Y LN(µ, σ ), the the pdf of Y is ( ) (log(y) µ) f(y) = y πσ exp σ where y > 0 ad σ > 0 ad µ is real The cdf of Y is ( ) log(y) µ F(y) = Φ σ for y > 0 where Φ(y) is the stadard ormal N(0,) cdf This family is a scale family with scale parameter τ = e µ if σ is kow ad For ay r, E(Y ) = exp(µ + σ /) VAR(Y ) = exp(σ )(exp(σ ) )exp(µ) E(Y r ) = exp(rµ + r σ /) MED(Y ) = exp(µ) ad exp(µ)[ exp( 06744σ)] MAD(Y ) exp(µ)[ + exp(06744σ)] Notice that f(y) = π σ exp( µ σ ) I(y 0)exp y [ + µ ] σ (log(y)) σ log(y) is a P REF Hece Θ = (, ) (0, ), η = /(σ ), η = µ/σ ad Ω = (, 0) (, ) Note that W = log(y ) N(µ, σ ) Notice that f(y) = π σ [ ] I(y 0)exp y σ(log(y) µ) 30

is a P REF if µ is kow, If Y,, Y are iid LN(µ, σ ) where µ is kow, the the likelihood [ ] L(σ ) = c σ exp (log(y σ i ) µ), ad the log likelihood Hece log(l(σ )) = d log(σ ) σ d d(σ ) log(l(σ )) = σ + (σ ) (log(y i ) µ) (log(y i ) µ) set = 0, or (log(y i) µ) = σ or ˆσ = (log(y i) µ) Sice this solutio is uique ad (σ ) d d(σ ) log(l(σ )) = (log(y i) µ) (σ ) 3 = σ =ˆσ (ˆσ ) ˆσ (ˆσ ) 3 = (ˆσ ) < 0, ˆσ = (log(y i) µ) is the UMVUE ad MLE of σ if µ is kow Sice T = [log(y i) µ] G(/, σ ), if µ is kow ad r > / the T r is UMVUE of If σ is kow, E(T r ) = r σ rγ(r + /) Γ(/) f(y) = [ I(y 0)exp( )exp( µ µ ] π σ y σ (log(y)) σ )exp σ log(y) 3

is a P REF If Y,, Y are iid LN(µ, σ ), where σ is kow, the the likelihood [ ] L(µ) = c exp( µ σ )exp µ log(y σ i ), ad the log likelihood Hece or log(y i) = µ or log(l(µ)) = d µ σ + µ σ log(y i ) d µ log(l(µ)) = + log(y i) dµ σ σ This solutio is uique ad ˆµ = log(y i) d log(l(µ)) = dµ σ < 0 Sice T = log(y i) N(µ, σ ), ˆµ = log(y i) is the UMVUE ad MLE of µ if σ is kow Whe either µ or σ are kow, the log likelihood log(l(σ )) = d log(σ ) σ Let w i = log(y i ) the the log likelihood is log(l(σ )) = d log(σ ) σ set = 0, (log(y i ) µ) (w i µ), 3

which has the same form as the ormal N(µ, σ ) log likelihood Hece the MLE (ˆµ, ˆσ) = W i, (W i W) Hece iferece for µ ad σ is simple Use the fact that W i = log(y i ) N(µ, σ ) ad the perform the correspodig ormal based iferece o the W i For example, a the classical ( α)00% CI for µ whe σ is ukow is where S W S W (W t, α, W + t, α ) S W = ˆσ = (W i W), ad P(t t, α) = α/ whe t is from a t distributio with degrees of freedom Compare Meeker ad Escobar (998, p 75) 030 The Maxwell-Boltzma Distributio If Y has a Maxwell Boltzma distributio, Y MB(µ, σ), the the pdf of Y is (y µ) e σ f(y) = (y µ) σ 3 π where µ is real, y µ ad σ > 0 This is a locatio scale family E(Y ) = µ + σ Γ(3/) = µ + σ π [ Γ( 5 VAR(Y ) = σ ) ( ) ] ( Γ(3/) = σ 3 8 ) Γ(3/) π MED(Y ) = µ + 5387σ ad MAD(Y ) = 046044σ This distributio a oe parameter expoetial family whe µ is kow Note that W = (Y µ) G(3/, σ ) 33

If Z MB(0, σ), the Z chi(p = 3, σ), ad for r > 3 The mode of Z is at σ E(Z r ) = r/ σ rγ(r+3) Γ(3/) 03 The Negative Biomial Distributio If Y has a egative biomial distributio (also called the Pascal distributio), Y NB(r, ρ), the the pmf of Y is ( ) r + y f(y) = P(Y = y) = ρ r ( ρ) y y for y = 0,, where 0 < ρ < The momet geeratig fuctio [ ρ m(t) = ( ρ)e t for t < log( ρ) E(Y ) = r( ρ)/ρ, ad ] r VAR(Y ) = r( ρ) ρ Notice that ( ) r + y f(y) = ρ r exp[log( ρ)y] y is a P REF i ρ for kow r Thus Θ = (0, ), η = log( ρ) ad Ω = (, 0) If Y,, Y are idepedet NB(r i, ρ), the Y i NB( r i, ρ) 34

If Y,, Y are iid NB(r, ρ), the T = Y i NB(r, ρ) If r is kow, the the likelihood ad the log likelihood Hece or or r ρr ρ y i = 0 or L(p) = c ρ r exp[log( ρ) y i ], log(l(ρ)) = d + r log(ρ) + log( ρ) This solutio is uique ad d r log(l(ρ)) = dρ ρ ρ ρ ρ r = y i, ˆρ = r r + y i d r log(l(ρ)) = dρ ρ ( ρ) y i y i set = 0, y i < 0 Thus r ˆρ = r + Y i is the MLE of ρ if r is kow Notice that Y is the UMVUE, MLE ad MME of r( ρ)/ρ if r is kow 35

03 The Normal Distributio If Y has a ormal distributio (or Gaussia distributio), Y N(µ, σ ), the the pdf of Y is ( ) (y µ) f(y) = exp πσ σ where σ > 0 ad µ ad y are real Let Φ(y) deote the stadard ormal cdf Recall that Φ(y) = Φ( y) The cdf F(y) of Y does ot have a closed form, but ( ) y µ F(y) = Φ, σ ad Φ(y) 05( + exp( y /π) ) for y 0 See Johso ad Kotz (970a, p 57) The momet geeratig fuctio is m(t) = exp(tµ + t σ /) The characteristic fuctio is c(t) = exp(itµ t σ /) E(Y ) = µ ad VAR(Y ) = σ E[ Y µ r ] = σ r r/ Γ((r + )/) π for r > If k is a iteger, the E(Y k ) = (k )σ E(Y k ) + µe(y k ) See Stei (98) ad Casella ad Berger (00, p 5) MED(Y ) = µ ad MAD(Y ) = Φ (075)σ 06745σ Hece σ = [Φ (075)] MAD(Y ) 483MAD(Y ) This family is a locatio scale family which is symmetric about µ Suggested estimators are Y = ˆµ = Y i ad S = S Y = 36 (Y i Y )

The classical ( α)00% CI for µ whe σ is ukow is S Y S Y (Y t, α, Y + t, α ) where P(t t, α) = α/ whe t is from a t distributio with degrees of freedom If α = Φ(z α ), the where z α m c o + c m + c m + d m + d m + d 3 m 3 m = [ log( α)] /, c 0 = 5557, c = 080853, c = 00038, d = 43788, d = 08969, d 3 = 000308, ad 05 α For 0 < α < 05, z α = z α See Keedy ad Getle (980, p 95) To see that MAD(Y ) = Φ (075)σ, ote that 3/4 = F(µ + MAD) sice Y is symmetric about µ However, ( ) y µ F(y) = Φ σ ad 3 4 = Φ ( µ + Φ (3/4)σ µ So µ + MAD = µ + Φ (3/4)σ Cacel µ from both sides to get the result Notice that [ f(y) = πσ exp( µ σ )exp + µ ] σ y σ y is a P REF Hece Θ = (0, ) (, ), η = /(σ ), η = µ/σ ad Ω = (, 0) (, ) If σ is kow, [ ] [ f(y) = exp exp( µ µ ] πσ σ y σ )exp σ y 37 σ )

is a P REF Also the likelihood ad the log likelihood Hece or µ = y i, or This solutio is uique ad L(µ) = c exp( µ σ )exp [ µ σ log(l(µ)) = d µ σ + µ σ ] y i y i d µ log(l(µ)) = + y i dµ σ σ ˆµ = y d log(l(µ)) = dµ σ < 0 set = 0, Sice T = Y i N(µ, σ ), Y is the UMVUE, MLE ad MME of µ if σ is kow If µ is kow, [ ] f(y) = exp πσ σ(y µ) is a P REF Also the likelihood [ L(σ ) = c σ exp σ ad the log likelihood Hece ] (y i µ) log(l(σ )) = d log(σ ) σ d dσ log(l(σ )) = σ + (σ ) 38 (y i µ) (y i µ) set = 0,

or σ = (y i µ), or ˆσ = This solutio is uique ad d d(σ ) log(l(σ )) = (σ ) (y i µ) (y i µ) (σ ) 3 = σ =ˆσ = (ˆσ ) < 0 Sice T = (Y i µ) G(/, σ ), ˆσ = (Y i µ) (ˆσ ) ˆσ (ˆσ ) 3 is the UMVUE ad MLE of σ if µ is kow Note that if µ is kow ad r > /, the T r is the UMVUE of E(T r ) = r σ rγ(r + /) Γ(/) 033 The Oe Sided Stable Distributio If Y has a oe sided stable distributio (with idex /, also called a Lévy distributio), Y OSS(σ), the the pdf of Y is ( ) σ f(y) = σ exp πy 3 y for y > 0 ad σ > 0 This distributio is a scale family with scale parameter σ ad a P REF Whe σ =, Y INVG(ν = /, λ = ) where INVG stads for iverted gamma This family is a special case of the iverse Gaussia IG distributio It ca be show that W = /Y G(/, /σ) This distributio is eve more outlier proe tha the Cauchy distributio See Feller (97, p 5) ad Lehma (999, p 76) For applicatios see Besbeas ad Morga (004) 39

If Y,, Y are iid OSS(σ) the T = Y i G(/, /σ) The likelihood L(σ) = ( ( f(y i ) = )σ / σ exp πy 3 i ad the log likelihood Hece or or log(l(σ)) = log ( This solutio is uique ad Hece the MLE of d dσ log(l(σ)) = σ ) + πy 3 i log(σ) σ = σ ˆσ = y i, y i y i set = 0, d dσ log(l(σ)) = σ < 0 ˆσ = ), Notice that T / is the UMVUE ad MLE of /σ ad T r is the UMVUE for r > / Y i r Γ(r + /) σ r Γ(/) y i y i 30

034 The Pareto Distributio If Y has a Pareto distributio, Y PAR(σ, λ), the the pdf of Y is f(y) = λ σ/λ y +/λ where y σ, σ > 0, ad λ > 0 The mode is at Y = σ The cdf of Y is F(y) = (σ/y) /λ for y > σ This family is a scale family with scale parameter σ whe λ is fixed for λ < E(Y r ) = E(Y ) = σr rλ σ λ for r < /λ MED(Y ) = σ λ X = log(y/σ) is EXP(λ) ad W = log(y ) is EXP(θ = log(σ), λ) Notice that f(y) = [ ] I[y σ] exp σλ y λ log(y/σ) is a oe parameter expoetial family if σ is kow If Y,, Y are iid PAR(σ, λ) the T = log(y i /σ) G(, λ) If σ is kow, the the likelihood [ ] L(λ) = c λ exp ( + λ ) log(y i /σ), ad the log likelihood log(l(λ)) = d log(λ) ( + λ ) log(y i /σ) 3

Hece or log(y i/σ) = λ or d log(l(λ)) = dλ λ + λ ˆλ = log(y i/σ) log(y i /σ) set = 0, This solutio is uique ad d dλ log(l(λ)) = λ log(y i/σ) λ 3 = λ=ˆλ ˆλ ˆλ ˆλ 3 = ˆλ < 0 Hece ˆλ = log(y i/σ) is the UMVUE ad MLE of λ if σ is kow If σ is kow ad r >, the T r is the UMVUE of E(T) r = λ rγ(r + ) Γ() If either σ or λ are kow, otice that f(y) = [ ( )] log(y) log(σ) y λ exp I(y σ) λ Hece the likelihood L(λ, σ) = c λ exp [ ( ) ] log(yi ) log(σ) I(y () σ), λ ad the log likelihood is [ log L(λ, σ) = d log(λ) ( ) ] log(yi ) log(σ) I(y () σ) λ 3

Let w i = log(y i ) ad θ = log(σ), so σ = e θ The the log likelihood is [ ( ) ] wi θ log L(λ, θ) = d log(λ) I(w () θ), λ which has the same form as the log likelihood of the EXP(θ, λ) distributio Hece (ˆλ, ˆθ) = (W W (), W () ), ad by ivariace, the MLE (ˆλ, ˆσ) = (W W (), Y () ) Let D = (W i W : ) = ˆλ where W () = W : For >, a 00( α)% CI for θ is (W : ˆλ[(α) /( ) ], W : ) (0) Expoetiate the edpoits for a 00( α)% CI for σ A 00( α)% CI for λ is ( ) D D, (0) χ ( ), α/ χ ( ),α/ This distributio is used to model ecoomic data such as atioal yearly icome data, size of loas made by a bak, et cetera 035 The Poisso Distributio If Y has a Poisso distributio, Y POIS(θ), the the pmf of Y is f(y) = P(Y = y) = e θ θ y for y = 0,,, where θ > 0 The mgf of Y is m(t) = exp(θ(e t )), ad the characteristic fuctio of Y is c(t) = exp(θ(e it )) E(Y ) = θ, ad VAR(Y ) = θ Che ad Rubi (986) ad Adell ad Jodrá (005) show that < MED(Y ) E(Y ) < /3 33 y!

Pourahmadi (995) showed that the momets of a Poisso (θ) radom variable ca be foud recursively If k is a iteger ad ( 0 0) =, the k ( ) k E(Y k ) = θ E(Y i ) i i=0 The classical estimator of θ is ˆθ = Y The approximatios Y N(θ, θ) ad Y N( θ, ) are sometimes used Notice that f(y) = e θ y! exp[log(θ)y] is a P REF Thus Θ = (0, ), η = log(θ) ad Ω = (, ) If Y,, Y are idepedet POIS(θ i ) the Y i POIS( θ i) If Y,, Y are iid POIS(θ) the T = Y i POIS(θ) The likelihood L(θ) = c e θ exp[log(θ) ad the log likelihood y i ], log(l(θ)) = d θ + log(θ) y i Hece or y i = θ, or d dθ log(l(θ)) = + θ ˆθ = y y i set = 0, This solutio is uique ad d dθ log(l(θ)) = y i < 0 θ 34

uless y i = 0 Hece Y is the UMVUE ad MLE of θ Let W = Y i ad suppose that W = w is observed Let P(T < χ d (α)) = α if T χ d The a exact 00 ( α)% CI for θ is ( χ w ( α ), χ w+( α ) ) for w 0 ad ( ) 0, χ ( α) for w = 0 036 The Power Distributio If Y has a power distributio, Y POW(λ), the the pdf of Y is f(y) = λ y λ, where λ > 0 ad 0 < y The cdf of Y is F(y) = y /λ for 0 < y MED(Y ) = (/) λ W = log(y ) is EXP(λ) Notice that Y beta(δ = /λ, ν = ) Notice that f(y) = λ I (0,](y)exp [( λ ] )log(y) = λ [ ] y I (0,](y)exp ( log(y)) λ is a P REF Thus Θ = (0, ), η = /λ ad Ω = (, 0) If Y,, Y are iid POW(λ), the The likelihood T = log(y i ) G(, λ) [ ] L(λ) = λ exp ( λ ) log(y i ), 35

ad the log likelihood Hece or log(y i) = λ, or log(l(λ)) = log(λ) + ( λ ) log(y i ) d log(l(λ)) = dλ λ log(y i) λ ˆλ = log(y i) This solutio is uique ad d dλ log(l(λ)) = λ log(y i) λ 3 = ṋ λ + ˆλ ˆλ 3 = ˆλ < 0 Hece ˆλ = log(y i) is the UMVUE ad MLE of λ If r >, the T r is the UMVUE of A 00( α)% CI for λ is ( E(T) r = λ rγ(r + ) Γ() T χ, α/, T χ,α/ 037 The Rayleigh Distributio ) set = 0, λ=ˆλ (03) If Y has a Rayleigh distributio, Y R(µ, σ), the the pdf of Y is [ f(y) = y µ exp ( ) ] y µ σ σ 36

where σ > 0, µ is real, ad y µ See Cohe ad Whitte (988, Ch 0) This is a asymmetric locatio scale family The cdf of Y is [ F(y) = exp ( ) ] y µ σ for y µ, ad F(y) = 0, otherwise E(Y ) = µ + σ π/ µ + 5334σ VAR(Y ) = σ (4 π)/ 04904σ MED(Y ) = µ + σ log(4) µ + 774σ Hece µ MED(Y ) 655MAD(Y ) ad σ 30MAD(Y ) Let σd = MAD(Y ) If µ = 0, ad σ =, the 05 = exp[ 05( log(4) D) ] exp[ 05( log(4) + D) ] Hece D 0448453 ad MAD(Y ) 0448453σ It ca be show that W = (Y µ) EXP(σ ) Other parameterizatios for the Rayleigh distributio are possible Note that f(y) = [ σ (y µ)i(y µ)exp ] σ(y µ) appears to be a P REF if µ is kow If Y,, Y are iid R(µ, σ), the T = (Y i µ) G(, σ ) If µ is kow, the the likelihood ad the log likelihood L(σ ) = c exp σ [ σ ] (y i µ), log(l(σ )) = d log(σ ) σ 37 (y i µ)

Hece d d(σ ) log(l(σ )) = σ + σ or (y i µ) = σ, or ˆσ = This solutio is uique ad d d(σ ) log(l(σ )) = (y i µ) set = 0, (y i µ) (σ ) (ˆσ ) ˆσ (ˆσ ) 3 = (ˆσ ) < 0 (y i µ) (σ ) 3 = σ =ˆσ Hece ˆσ = (Y i µ) is the UMVUE ad MLE of σ if µ is kow If µ is kow ad r >, the T r is the UMVUE of E(T r ) = r σ rγ(r + ) Γ() 038 The Smallest Extreme Value Distributio If Y has a smallest extreme value distributio (or log-weibull distributio), Y SEV (θ, σ), the the pdf of Y is f(y) = σ exp(y θ θ )exp[ exp(y σ σ )] where y ad θ are real ad σ > 0 The cdf of Y is F(y) = exp[ exp( y θ σ )] This family is a asymmetric locatio-scale family with a loger left tail tha right 38

E(Y ) θ 0577σ, ad VAR(Y ) = σ π /6 64493σ MED(Y ) = θ σ log(log()) MAD(Y ) 0767049σ Y is a oe parameter expoetial family i θ if σ is kow If Y has a SEV(θ, σ) distributio, the W = Y has a LEV( θ, σ) distributio 039 The Studet s t Distributio If Y has a Studet s t distributio, Y t p, the the pdf of Y is f(y) = Γ( p+ ) y p+ ( + (pπ) / Γ(p/) p ) ( ) where p is a positive iteger ad y is real This family is symmetric about 0 The t distributio is the Cauchy(0, ) distributio If Z is N(0, ) ad is idepedet of W χ p, the Z ( W p )/ is t p E(Y ) = 0 for p MED(Y ) = 0 VAR(Y ) = p/(p ) for p 3, ad MAD(Y ) = t p,075 where P(t p t p,075 ) = 075 If α = P(t p t p,α ), the Cooke, Crave, ad Clarke (98, p 84) suggest the approximatio where t p,α p[exp( w α p ) )] w α = z α(8p + 3) 8p +, z α is the stadard ormal cutoff: α = Φ(z α ), ad 05 α If 0 < α < 05, the t p,α = t p, α This approximatio seems to get better as the degrees of freedom icrease 39

040 The Topp-Leoe Distributio If Y has a Topp Leoe distributio, Y TL(ν), the pdf of Y is f(y) = ν( y)(y y ) ν for ν > 0 ad 0 < y < The cdf of Y is F(y) = (y y ) ν for 0 < y < This distributio is a P REF sice f(y) = ν( y)i (0,) (y)exp[( ν)( log(y y ))] MED(Y ) = (/) /ν, ad Example 7 showed that W = log(y Y ) EXP(/ν) The likelihood L(ν) = c ν (y i yi ) ν, ad the log likelihood Hece log(l(ν)) = d + log(ν) + (ν ) d dν log(l(ν)) = ν + log(y i yi ) log(y i y i ) set = 0, or + ν log(y i yi ) = 0, or ˆν = log(y i yi ) Hece This solutio is uique ad ˆν = d log(l(ν)) = dν ν < 0 log(y i Y i ) = log(y i Yi ) is the MLE of ν If T = log(y i Y i ) G(, /ν), the T r is the UMVUE of E(T r ) = ν r Γ(r + ) Γ() for r > I particular, ˆν = T is the MLE ad UMVUE of ν for > 330

04 The Trucated Extreme Value Distributio If Y has a trucated extreme value distributio, Y TEV(λ), the the pdf of Y is f(y) = ( ) λ exp y ey λ where y > 0 ad λ > 0 The cdf of Y is [ ] (e y ) F(y) = exp λ for y > 0 MED(Y ) = log( + λlog()) W = e Y is EXP(λ) Notice that f(y) = [ ] λ ey I(y 0)exp λ (ey ) is a P REF Hece Θ = (0, ), η = /λ ad Ω = (, 0) If Y,, Y are iid TEV(λ), the T = (e Y i ) G(, λ) The likelihood L(λ) = c λ exp [ λ ] log(e y i ), ad the log likelihood log(l(λ)) = d log(λ) λ log(e y i ) Hece d log(l(λ)) = dλ λ + log(ey i ) λ set = 0, 33

or log(ey i ) = λ, or ˆλ = log(ey i ) This solutio is uique ad d dλ log(l(λ)) = λ log(ey i ) λ 3 = ṋ λ ˆλ ˆλ 3 = ˆλ < 0 Hece ˆλ = log(ey i ) is the UMVUE ad MLE of λ If r >, the T r is the UMVUE of A 00( α)% CI for λ is ( E(T) r = λ rγ(r + ) Γ() T χ, α/, T χ,α/ 04 The Uiform Distributio ) λ=ˆλ (04) If Y has a uiform distributio, Y U(θ, θ ), the the pdf of Y is f(y) = θ θ I(θ y θ ) The cdf of Y is F(y) = (y θ )/(θ θ ) for θ y θ This family is a locatio-scale family which is symmetric about (θ + θ )/ By defiitio, m(0) = c(0) = For t 0, the mgf of Y is m(t) = etθ e tθ (θ θ )t, 33

ad the characteristic fuctio of Y is c(t) = eitθ e itθ (θ θ )it E(Y ) = (θ + θ )/, ad MED(Y ) = (θ + θ )/ VAR(Y ) = (θ θ ) /, ad MAD(Y ) = (θ θ )/4 Note that θ = MED(Y ) MAD(Y ) ad θ = MED(Y ) + MAD(Y ) Some classical estimators are ˆθ = Y () ad ˆθ = Y () 043 The Weibull Distributio If Y has a Weibull distributio, Y W(φ, λ), the the pdf of Y is f(y) = φ λ yφ e yφ λ where λ, y, ad φ are all positive For fixed φ, this is a scale family i σ = λ /φ The cdf of Y is F(y) = exp( y φ /λ) for y > 0 E(Y ) = λ /φ Γ( + /φ) VAR(Y ) = λ /φ Γ( + /φ) (E(Y )) E(Y r ) = λ r/φ Γ( + r φ ) for r > φ MED(Y ) = (λlog()) /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 /φ ), σ = /φ) distributio Notice that f(y) = φ [ ] λ yφ I(y 0)exp λ yφ is a oe parameter expoetial family i λ if φ is kow 333

If Y,, Y are iid W(φ, λ), the T = Y φ i G(, λ) If φ is kow, the the likelihood ad the log likelihood L(λ) = c λ exp [ λ y φ i ], log(l(λ)) = d log(λ) λ y φ i Hece or yφ i = λ, or d log(l(λ)) = dλ λ + yφ i λ ˆλ = yφ i This solutio was uique ad d dλ log(l(λ)) = λ yφ i λ 3 = ṋ λ ˆλ ˆλ 3 = ˆλ < 0 Hece ˆλ = Y φ i is the UMVUE ad MLE of λ If r >, the T r is the UMVUE of E(T r + ) ) = λrγ(r Γ() set = 0, λ=ˆλ MLEs ad CIs for φ ad λ are discussed i Example 98 334

044 The Zeta Distributio If Y has a Zeta distributio, Y Zeta(ν), the the pmf of Y is f(y) = P(Y = y) = y ν ζ(ν) where ν > ad y =,, 3, Here the zeta fuctio ζ(ν) = y ν for ν > This distributio is a oe parameter expoetial family for ν >, ad VAR(Y ) = E(Y ) = y= ζ(ν ) ζ(ν) ζ(ν ) ζ(ν) [ ζ(ν ) ζ(ν) for ν > 3 E(Y r ζ(ν r) ) = ζ(ν) for ν > r + This distributio is sometimes used for cout data, especially by liguistics for word frequecy See Lidsey (004, p 54) 045 Complemets May of the distributio results used i this chapter came from Johso ad Kotz (970a,b) ad Patel, Kapadia ad Owe (976) Bickel ad Doksum (007), Castillo (988), Cohe ad Whitte (988), Cramér (946), DeGroot ad Schervish (00), Ferguso (967), Hastigs ad Peacock (975), Keedy ad Getle (980), Kotz ad va Dorp (004), Leemis (986), Lehma (983) ad Meeker ad Escobar (998) also have useful results o distributios Also see articles i Kotz ad Johso (98ab, 983ab, 985ab, 986, 988ab) Ofte a etire book is devoted to a sigle distributio, see for example, Bowma ad Sheto (988) Abuhassa ad Olive (007) discuss cofidece itervals for the two parameter expoetial, half ormal ad Pareto distributios 335 ]