Weighted Exponential Distribution and Process

Transcription

1 Weighted Exponential Distribution and Process Jilesh V Some generalizations of exponential distribution and related time series models Thesis. Department of Statistics, University of Calicut, 200

2 Chapter 5 Weighted Exponential Distribution and Process 5. Introduction Different methods may be used to introduce a shape parameter to an exponential model and result different generalizations of weighted exponential distributions. For example, the gamma distribution, and the generalized exponential distribution are different weighted versions of the exponential distribution. Gupta and Kundu (2009) introduced a generalized form of exponential distribution termed as Weighted Exponential Distribution, denoted by WE(α). They used the idea of Azzalini (985) to introduce a shape parameter to an exponential distribution which results a new class of weighted exponential distributions. Suppose X and X 2 are two independent and identically distributed random variables, with the probability density function f Y (y) and cumulative distribution function (CDF) F Y (y). Then for any α > 0, consider a new random variable X = X given that αx > X 2. Then the PDF of the new 68

3 random variable X is f(x) = P (αx > X 2 ) f Y (x)f Y (αx), x > 0. (5..) Weighted Exponential distribution of Gupta and Kundu (2009) is obtained by choosing f Y (x) as the exponential density function and F Y (x) as the corresponding distribution function. Then the density function of weighted exponential distribution is given by f(x) = α + α e x ( e αx ), α > 0, x > 0. (5..2) The graph of the WE (α) for different values of α is given in Figure 5.. The characteristic function of the random variable X d W E(α) is ψ X (t) = ( it) ( ) (5..3) it +α From the characteristic function (5..3), it is clear that the WE distribution is the distribution of the convolution of two independent but non- identically distributed exponential random variables. That is, random variable following WE (α) can be is can be represent as X d E + δe 2, (5..4) where E i, i =, 2 are independent standard exponential random variables and δ = +α. Another representation of the WE(α) can be done using beta transformation. That is, suppose U has a Beta(α, β) distribution, then for any c > 0 consider a new random variable V such that U = e cv, which has the probability density function f V (v) = Γ(a + b) Γ(a)Γ(b) ce acv ( e cv) b, v > 0. (5..5) 69

4 f(x) x Figure 5.: Shapes of the density function (5..2) for α=5 (black), 2 (red), (blue),.5 (green). Therefore, the probability density function of WE() can be obtained as a special case by taking a=, b=2 and c=α. Note that when α 0, the distribution tends α to gamma distribution and when α to exponential distribution. f X (x; α) is always log-concave. The probability density function is always unimodal and the mode is located at the point ln ( ) α+ α. Jayakumar and Jilesh (200) introduce an autoregressive process with WE(α) as marginals. In Section 2 a first order autoregressive process with WE(α) distribution as marginal is introduced. The properties of such process and its higher order extension also discussed. A generalization of WE(α) distribution and process is given in Section 3. In Section 4, We introduce a new distribution with support on real line using weighted exponential distribution and related time series are discussed in Section 5. A generalization is discussed in Section 6. A study has been done about weighted Weibull distribution in Section 7. 70

5 5.2 First order Autoregressive Process with WE(α) as marginals (WEAR()) Consider the first order autoregressive AR() process, X n = ρx n + ɛ n, 0 < ρ <, (5.2.) where {ɛ n } is a sequence of independent and identically distributed random variables. In terms of characteristic functions, we have, Ψ Xn (t) = Ψ Xn (ρt)ψ ɛn (t) which gives Ψ ɛn (t) = Ψ Xn (t) Ψ Xn. Using the characteristic function (5..3), we can represent (ρt) the innovation random variable as 0 with probability ρ 2 δe n with probability ρ( ρ) ɛ n = E 2n with probability ρ( ρ) Z n with probability ( ρ) 2 (5.2.2) where E in, i=,2 are standard exponential random variables and Z n random variable. is a WE(α) But Similarly we the innovation variable ɛ n also can be written as Ψ ɛn (t) = ( iρt) ( ) iρt +α ( it) ( ). (5.2.3) it +α [ ( iρt) ( it) = ρ + ( ρ) it ]. (5.2.4) Therefore, we obtain the distribution of the innovation random variable ɛ n as the 7

6 convolution of two independent tailed exponential random variables discussed in Littlejohn (994). That is, ɛ n is distributed as the convolution of two independent random variables ET n and ET 2n defined as 0 with probability ρ ET n = δe n with probability ρ (5.2.5) and 0 with probability ρ ET 2n = E 2n with probability ρ (5.2.6) where δ = +α and E n and E 2n are two independent standard exponential random variables with characteristic function. Similarly, we can write the random variable it ɛ n as ɛ n d I E n + δi 2 E 2n, (5.2.7) where I i s i=,2 are Bernoulli random variables with P (I i = ) = ρ, and E in, i=,2 are independent standard exponential random variables. Another representation for ɛ n can be given by writing where p = ρ 2, p 2 = ( ρ)( δρ) δ ( iρt) ( ) iρt +α ( it) ( ) = p it + p 2 ( it) + p ( 3 ) (5.2.8) +α it +α and p 3 = ( ρ)(ρ δ). Clearly 0 < p δ i <, i=,2,3 and p + p 2 + p 3 =. Therefore, the ɛ n can be also represented as 0 with probability p ɛ n = E n with probability p 2 δe 2n with probability p 3 (5.2.9) where δ, E n and E 2n are as defined above. Using Gupta and Kundu (2009) it can be shown that the moments of innovation 72

7 sequence ɛ n are E(ɛ n ) = ( ρ)( + δ) and V (ɛ n ) = ( ρ 2 )( + δ 2 ). Higher order cumulants are k r = Γ(r)( ρ r )( + δ r ), for integers r > 2. Theorem The AR() process (5.2.) is strictly stationary Markovian with WE(α) as marginal distribution if and only if ɛ n is distributed as (5.2.2) or it is the convolution of two independent tailed exponential random variables defined as in (5.2.5), (5.2.6), provided X d 0 W E(α) and {X n } is independent of ɛ n for all n. Proof:The proof follows by mathematical induction. Remark If X 0 is distributed arbitrarily, then also the process is asymptotically Markovian with WE(α) distribution. Proof:We have from (5.2.), X n = ρ n X 0 + n k=0 ρk ɛ n k. Using the characteristic function we can write it as On substituting (5.2.3), we can see that n Ψ Xn t = Ψ X0 (ρ n t) Ψ ɛn k (ρ k t) (5.2.0) k=0 Ψ Xn t ( it)( it +α ) Hence it follows that even if X 0 is arbitrarily distributed, the process is asymptotically stationary Markovian with WE marginals. Remark The model (5.2.) is defined for all values of ρ (0, ). The lag-k autocorrelation is given by ρ k = Corr(X n, Xn k) = ρ k ; k = 0,,... and the correlations are always positive. The joint distribution of observations (X n, X n+ ) can be given in terms of char- 73

8 acteristics function as Ψ Xn,X n+ (s, s 2 ) = ( iρs 2 )( iρδs 2 ) ( is iρs 2 )( iδ(s + ρs 2 ))( is 2 )( iδs 2 ) (5.2.) The above joint characteristic function is not symmetric in s and s 2, Therefore the process is not time reversible. When s = s 2 = s, we get the transform of the sum X n + X n+ as, E ( e is(x n+x n+ ) ) = ( iρs)( iρδs) ( is)( iδs) ( ( + ρ)is)( δ( + ρ)is) (5.2.2) A simple quantification of the sample path behavior is given by P (X n > X n ), which is related to the average length of down run sequences. Calculation of P (X n > X n ) follows from (5.2.2) as, P (X n > X n ) = ρ 2 P (ρx n > X n ) + ρ( ρ)p (ρx n + δe n > X n ) + ρ( ρ)p (ρx n + E 2n > X n ) + ( ρ) 2 P (( ρ)x n < Z n ) (5.2.3) = ρ 2 P (( ρ)x n < 0) + ρ( ρ)p (( ρ)x n < δe n ) + ρ( ρ)p (( ρ)x n < E 2n ) + ( ρ) 2 P (( ρ)x n < Z n ) = ρ 2 I + ρ( ρ)i 2 + ρ( ρ)i 3 + ( ρ) 2 I 4 (5.2.4) Note that I = P (( ρ)x n < 0) = 0 and I 2 = P (( ρ)x n < δe n ) (5.2.5) ( = P E n > ρ ) x f(x)dx (5.2.6) x δ = α + e ( ( ( ρ) +)x δ e αx) dx (5.2.7) α = α + α 0 { } + + ( ρ) + + α δ δ ( ρ) (5.2.8) 74

9 = ( ) ( δ ( ρ) + ( ρ) δ δ ) (5.2.9) + + α Similarly we obtain I 3 = P (( ρ)x n < E 2n ) = δ (( ρ) + ) ( ) (5.2.20) ( ρ) + δ and I 4 = P (( ρ)x n < Z n ) = J + J 2 (5.2.2) where ( ) 2 [ ] α + J = α α +, (5.2.22) [( ρ) + (α + )][( ρ) + ] ( ) [ ] α + J 2 = α + α + (5.2.23) ( + ( ρ)( + α)) (( + α) + ( ρ)( + α)) On substituting the values I n, i=,2,3,4 in (5.2.4) we obtain P (X n > X n ). Let T r = X + X X r, which can be defined as the time of r th event in a point process which starts with an event at the origin. Now we shall consider the regression behavior of the WEAR() model. Study of the regression of the model is in effect forecasting of the process. As stated in Jose et al. (2008) the practical implication of regression will be in the statistical analysis of direction-dependent data, since the WEAR process is not time reversible. Regression is linear, since E(X n /X n ) = ρx + ( ρ)( + δ). Further more the conditional variance is constant. The simulated sample path of the process can be seen in Figure

10 Figure 5.2: Sample path of the process (5.2.) for ρ =.25 and α = 3, k th order AR process with WE(α) as marginals (WEAR(k)) k th order WEAR(k) is given by the model ρ X n + ɛ n with probability p X n = ρ 2 X n 2 + ɛ n with probability p 2... ρ k X n k + ɛ n with probability p k (5.2.24) where 0 < p r <, k r= p r =, r={,2,...,k} and {X n, n } are marginally WE distributed. If all ρ i s are equal and using the characteristic function approach as above we obtain the distribution of the innovation sequence as the convolution of the same variables defined in (5.2.5) and (5.2.6). 76

11 Another autoregressive model of interest which is free from the zero defect is ɛ n with probability p X n = ρx n + ɛ n with probability p (5.2.25) Dewald and Lewis (985) studied the autoregressive process given by the equation (5.2.25) with Laplace distribution as marginals. Here we discuss the autoregressive process of structure (5.2.25) with weighted exponential distribution as marginals. Theorem The first order autoregressive process (5.2.25) with X 0 d W E(α) is stationary with weighted exponential distribution as marginals if and only if ɛ n = Z n +U n +V n, where U n and V n are two independent tailed exponential random variables and Z n is weighted exponential distributed with parameter pδ. proof: In terms characteristic function, the equation (5.2.25) becomes ψ ɛ (t) = ψ Xn (t) p + ( p)ψ Xn (ρt) (5.2.26) If we assume that {X n } is stationary with weighted exponential marginal distribution, then (5.2.26) implies ( iρt)( iρδt) ψ ɛ (t) = ( it)( iδt) ( iρt)( ipρδt) ( ) ( ) = ρ + ( ρ) ρ + ( ρ) ( it) ( δit) ( iρt)( ipρδt) Therefore we can write ɛ n = Z n + U n + V n, where U n and V n are two independent tailed exponential random variables and Z n is weighted exponential distributed with parameter pδ. Converse can be proved by mathematical induction, assuming X n d W E(α). Thus {X n } is a stationary process with weighted exponential marginal distribution. 77

12 ɛ n can also be represent as ɛ n d I E n + δi 2 E 2n + ρ (E 3n + pδe 4n ), (5.2.27) where E in, i =, 2, 3, 4 are standard exponential random variables. Next we introduced a generalization of weighted exponential distribution and discuss related time series models. 5.3 A Generalization of Weighted Exponential Distribution Here we introduce a Generalized Weighted Exponential distribution (GWE) as a generalization of weighted exponential distribution. A random variable X is said to be GWE distributed with parameters α and τ, if its characteristic function is given by ( τ Ψ(t) = )) ( it) (, α > 0, τ > 0. (5.3.) it +α We denote the distribution with the characteristic function (5.3.) as GWE(α, τ). Note that when τ = we obtain the WE(α) distribution and for α = 0, τ, (5.3.) is the characteristic function of a Gamma(, 2τ) distributed random variable, where Gamma(a, b) means a gamma distributed random variable with characteristic function ( at) b. The distribution with characteristic function (5.3.) arise as the distribution of the τ-fold convolution of independent WE(α) random variables. From (5.3.), we have ( ) ( ) τ τ Ψ(t) = it it, α > 0, τ > 0. (5.3.2) +α 78

13 Therefore X can be represented as X = G + G 2, (5.3.3) where G d Gamma(, τ) and G 2 d Gamma(δ, τ) distributed random variables. Now we discuss the autoregressive process with generalized weighted exponential distribution as marginals First order Generalized Weighted Exponential GWEAR() Process For the process defined in (5.2.), where X n d GW E(α, τ) using the characteristic function, we obtain the characteristic function of innovation sequence as [ ] τ [ ] τ Ψ ɛ (t) = ρ + ( ρ) ρ + ( ρ) (5.3.4) it iδt Therefore ɛ n as the τ fold convolution of tailed exponentials defined in (5.2.5) and (5.2.6). Therefore we can represent the innovation sequence as ɛ n = E n + E 2n, where E n and E 2n are the τ fold convolutions of tailed exponential variables. Similarly, we can represent ɛ n as the τ fold convolution the random variable defined in the right hand side of (5.2.9) GWEARMA(,) Process Consider the Autoregressive moving average model with GWE marginals defined by X n = ρx n + ζɛ n + ɛ n (5.3.5) 79

14 Assuming stationarity and when ζ =, we obtain in terms of characteristic function ) 2 Ψ ɛ (t) =. Therefore we obtain the distribution of the innovation sequence ( ΨX (t) Ψ X (ρt) as the distribution of τ fold convolution of tailed exponential random variables Weighted Exponential distribution on real line For any positive random variable X with density f(x) can be extended symmetrically to real line with density h(x) = f( x ), x R. A similar symmetrization of the 2 density (5..2) gives rise to a weighted exponential distribution on real line. Jilesh and Jayakumar (200a) Definition A random variable with support on the real line is said to follow the Double Weighted Exponential distribution with parameters α and λ, denoted by X d DW E(α, λ) if its probability density function (PDF) is given by f(x) = 2 (α + ) λe ( λ x e αλ x ), x R, α > 0, λ > 0. (5.4.) α For various derivations it would seem convenient to consider a location parameter also, that is for a location parameter θ, such that < θ <, the above density becomes, f(x) = 2 (α + ) λe ( λ x θ e αλ x θ ), x R, α, λ > 0 (5.4.2) α Without loss of generality assume λ = and for brevity call it as DWE(α). The probability density function of the DWE(α) is bimodal with modes located on both sides of the origin at a distance ±log ( ) +α α. As α, the DWE(α) converges to the standard Laplace distribution. For α =, we obtain a symmetric extension of 80

15 f(x) x Figure 5.3: Shapes of the density function of DWE(α)for α=.5(red),.5(black),5(blue) the generalized exponential distribution of Gupta and Kundu (999) to the real line. The distribution function takes the form F (x) = α+ 2α eλx ( α+ 2α e λx ( ) eαλx α+ ) e αλx α+ for x < 0 for x 0 (5.4.3) The moment generating function of DWE(α) distribution is given by M(t) = ( t 2 )( δ 2 t 2 ), (5.4.4) where δ =. Consequently, the cumulant generating function log (M(t)) is given +α by log(m(t)) = log( t 2 ) log( δ 2 t 2 ) (5.4.5) 8

16 5.4. Moments and Related Measures cumulants The n th cumulant of a DWE(α) random variable X, denoted κ n, is defined as the coefficient of tn n! in the Taylor expansion (about t=0) of the cumulant generating function of X. Formula (5.4.5) for the cumulant generating function generate the cumulants of DWE(α) in a straightforward manner. Indeed, using the Taylor expansion of log(-z) about z=0, we have log( t 2 ) = k= t 2k k!. (5.4.6) Thus for the DWE(α) random variable X, we have 0 if n is odd κ n (x) = 2(n )!( + δ 2n ) if n is even (5.4.7) Moments By writing Taylor expansion of the moment generating function (5.4.4) ( ) ( ) M(t) = t 2k δt 2k = k= k=0 l=0 k=0 ( k ) (2k)! δ 2l t 2k (2k)! (5.4.8) (5.4.9) we obtain the n th central moments of the DWE(α) random variable X, as 0 if n is odd µ n (x) = ( n ) 2 l=0 δ2l n! if n is even (5.4.0) In particular, E(X) = 0 and V (X) = 2( + δ 2 ). Instead of the random variable X is distributed with the probability density function (5.4.), if it is distributed according 82

17 to (5.4.2) we obtain E(X) = θ, V (X) = 2( + δ 2 ) and Coefficient variation as CV = 2( + δ2 ). (5.4.) θ Note that mean and variance involve different parameters as in the case of normal, Laplace distributions. Coefficient of Skewness and Kurtosis For a distribution of an random variable X with a finite third moment and standard deviation greater than zero, the coefficient of skewness is a measure of symmetry defined by γ = µ 3. (5.4.2) (µ 2 ) 3 2 The coefficient of skewness for DWE(α) random variable is zero as in the case of any symmetric distribution with a finite third moment. For a random variable with finite fourth the excess kurtosis is defined as γ 2 = µ 4 (µ 2 ) 2 3 (5.4.3) For DWE(α) random variable the excess kurtosis is given by γ 2 = 6 ( + δ2 + δ 4 ) ( + δ 2 ) 2 3. (5.4.4) Representations. Mixture of Normal Distribution Any DWE(α) random variable can be represent as a Gaussian random variable with mean zero and stochastic variance which has an weighted exponential distribution, results the following proposition. 83

18 Proposition A standard DWE(α) random variable has the representation X d 2W Z (5.4.5) where Z is a standard normal random variable, that is Z d N(0, ) and W is a WE(α) distributed with moment generating function M W (t) = ( t)( δ 2 t). Proof Let Z be a standard normal random variable with characteristic function φ Z (t) = e t2 2, < t <. To obtain the characteristic function of the product 2W Z, conditioning on W, we obtain Ψ(t) = E(e it 2W Z ) ( ) = E E(e it 2W Z W ) ( = E φ Z (t ) 2W ) ( ) = E e t2 W = M W ( t 2 ) 2. Relation to Laplace Distribution = ( + t 2 ) ( + δ 2 t 2 ). (5.4.6) The moment generating function (5.4.4) is the product of the two moment generating functions of the independent and non-identically distributed Laplace random variable. Proposition A standard DWE(α) random variable has the representation X d L () + L 2 (δ) (5.4.7) where L i (ξ), i=,2 means a Laplace random variable with characteristic function Ψ(t) = +ξ 2 t 2. 84

19 Proof Since for independent random variables the product of moment generating functions corresponds to their sum, we can derive the representation directly using (5.4.4). Remark Another related representations are possible by considering the fact that, a standard Laplace random variable L have the representation L d E E 2, where E is are independent and identically distributed standard exponential variables. The characteristic function (5.4.6) can also be decomposed as ( + t 2 ) ( + δ 2 t 2 ) = 2 ( it) ( iδt) + 2 ( + it) ( + iδt) (5.4.8) The right hand side of (5.4.8) is the characteristic function of the product IW, where I is a discrete random variable takes the values ± with probabilities, while W is a 2 weighted exponential random variable with moment generating function M(t) = ( it) ( iδt). (5.4.9) Therefore the DWE(α) random variable is the mixture of weighted exponential random variables. Proposition A standard DWE(α) random variable admits the representation X d IW (5.4.20) where W is weighted exponential with moment generating function (5.4.9) and I takes values ± with probabilities 2. A probability distribution with characteristic function Ψ(t) is infinitely divisible if, for any integer n, we have Ψ(t) = (Φ n (t)) n, where Φ n (t) is another characteristic 85

20 function. In other words, an random variable Y with characteristic function Ψ(t) has the representation Y d n i= X i, for some random variables X i. For more details about infinite divisibility and related concepts see, Steutal and Van Harn (2004). The characteristic function of DWE(α) can be factorize as [ ( ) ( ) ( + t 2 )( + δ 2 t 2 ) = n ( ) n ( ) ] n n n it + it iδt + iδt = (Φ n (t)) n For each integer n, the function Φ n (t) is the characteristic function of E n () E 2n () + E 3n (δ) E 4n (δ), where E in (ξ), i=,2,3,4 are independent exponentials with characteristic function iξt. Consequently, the DWE(α) is infinitely divisible. 5.5 Time series models with Double Weighted Exponential as Marginals (DWEAR Process) Let {X n, n } be a sequence of random variables defined by the autoregressive equation as defined in (5.2.) with ρ < and {ɛ n } be a sequence of independent and identically distributed random variables, Assume that {X n } is stationary with DWE (α) as marginal distribution having characteristic function (5.4.6). From (5.2.), we have the characteristic function of 86

21 the innovation sequence {ɛ n } as Ψ ɛn (t) = ( + ρ2 t 2 )( + δ 2 ρ 2 t 2 ) ( + t 2 )( + δ 2 t 2 ) = ρ 4 + ( ρ2 ) (ρ 2 + ) ρ2 2 δ 2 + it + ( ρ2 ) (ρ 2 + ) ρ2 2 δ 2 it + ( + ) ρ2 2 δ 2 iδt + 2 ( + ρ2 δ 2 ) + iδt (5.5.) (5.5.2) (5.5.3) Hence the innovation sequence {ɛ n } of the first order autoregressive process (5.2.) is the convex mixture of random variables given by ɛ n = 0 with probability ρ 4 ( E n () with probability ( ρ2 ) ρ 2 + ρ2 2 ( E 2n () with probability ( ρ2 ) ρ 2 + ρ2 2 ( ) E 3n (δ) with probability + ρ2 2 δ 2 ( ) E 4n (δ) with probability + ρ2, 2 δ 2 δ 2 ) δ 2 ) (5.5.4) where E in (ζ) means exponential random variable with characteristic function iζt. Also it can be verified that, if X 0 d DW E(α) and {ɛ n } be a sequence of independent and identically distributed convex mixture of exponential random variables given by (5.5.4), the first order autoregressive process (5.2.) is stationary with DWE (α) marginal distribution. Thus we have the following theorem. Theorem Let {ɛ n } be a sequence of independent and identically distributed random variables defined as in (5.5.4). Then the first order autoregressive process of structure (5.2.) with X 0 d DW E(α) defines a stationary autoregressive process with DWE(α) distribution. 87

22 We call the process defined in (5.2.) with X 0 d DW E(α) and ɛ n as in (5.5.4) as the first order Double Weighted Exponential autoregressive (DWEAR()) process. Let X 0 d DW E(α) and for n=,2,... the sequence {X n } can be written as X n = ρx n with probability ρ 4 ( ρx n E n () with probability ( ρ2 ) ρ 2 + ρ2 2 ( ρx n + E 2n () with probability ( ρ2 ) ρ 2 + ρ2 2 ( ) ρx n + E 3n (δ) with probability + ρ2 2 δ 2 ( ) ρx n E 4n (δ) with probability + ρ2. 2 δ 2 δ 2 ) δ 2 ) (5.5.5) Another representation ( for ɛ n is obtained by writing Ψ ɛn (t) as Ψ ɛn (t) = ρ 2 + ( ρ2 ) 2 it + ( ) ( ρ2 ) ρ 2 + ( ρ2 ) 2 + it 2 iδt + ( ) ρ2 ). 2 + iδt Hence we obtain where U n and V n as ɛ n d U n + V n, (5.5.6) U n = 0 with probability ρ 2 E n () with probability ( ρ2 ) 2, E 2n () with probability ( ρ2 ) 2, (5.5.7) and V n = 0 with probability ρ 2 E 3n (δ) with probability ( ρ2 ) 2, E 4n (δ) with probability ( ρ2 ) 2. (5.5.8) The moments of the innovation sequence ɛ n is give as E(ɛ n )=0 and V (ɛ n ) = 2( ρ 2 )( + δ) other moments will be obtained by using of (5.4.7). Also we can write ɛ d I L n + δi 2 L 2n, (5.5.9) 88

23 where L in, i=,2 are independent and identically distributed standard Laplace random variables and I i, i=,2 are Bernoulli random variables with P (I = ) = ( ρ 2 ). Theorem The AR() process (5.2.) is strictly stationary Markovian with DWE(α) as marginal distribution if and only if ɛ n is distributed as (5.5.4), (or the distribution of the convolution of two independent random variables defined as in (5.5.7), (5.5.8)), provided X d 0 DW E(α) and {X n } is independent of ɛ n for all n. Proof:The proof follows by mathematical induction. Remark If X 0 is distributed arbitrarily, then also the process is asymptotically Markovian with DWE (α) distribution. Proof:We have from (5.2.), X n = ρ n X 0 + n k=0 ρk ɛ n k. Using the characteristic function we can write it as n Ψ Xn t = Ψ X0 (ρ n t) Ψ ɛn k (ρ k t) (5.5.0) On substituting the characteristic function of DWE (α), we can see that Ψ Xn t k=0 ( + t 2 )( + δ 2 t 2 ) Hence it follows that even if X 0 is arbitrarily distributed, the process is asymptotically stationary Markovian with DWE marginals. Remark The model (5.2.) is defined for all values of ρ <. autocorrelation is given by Corr(X n, X n k ) = ρ k ; k = 0,,.... The lag-k The joint distribution of observations (X n, X n+ ) can be given in terms of char- 89

24 acteristics function as Ψ Xn,X n+ (s, s 2 ) = ( + ρ 2 s 2 2)( + δ 2 ρ 2 s 2 2) ( + (s + ρs 2 ) 2 )( + δ 2 (s + ρs 2 ) 2 )( + s 2 2)( + δ 2 s 2 2) (5.5.) The above joint characteristic function is not symmetric in s and s 2, Therefore the process is not time reversible k th order AR process with DWE as marginals (DWEAR(k)) k th order DWEAR(k) is given by the model ρ X n + ɛ n with probability p X n = ρ 2 X n 2 + ɛ n with probability p 2... ρ k X n k + ɛ n with probability p k (5.5.2) where 0 < p r <, k r= p r =, r={,2,...,k} and {X n, n } are marginally DWE distributed. If all ρ i s are equal and using the characteristic function approach as above we obtain the distribution of the innovation sequence variables defined in (5.5.4) or as distribution of the convolution of (5.5.7) and (5.5.8). 90

25 5.6 A Generalization of Double Weighted Exponential Distribution Here we introduce a Generalized form of DWE distribution (GDWE). A random variable X is said to be GDWE distributed with parameters α and τ, if its characteristic function is given by ( ) τ Ψ(t) =, α > 0, τ > 0. (5.6.) ( + t 2 ) ( + δ 2 t 2 ) We denote the distribution with the characteristic function (5.6.) as GDWE(α, τ). Note that when τ = we obtain the DWE(α) distribution and for values τ, (5.6.) is the characteristic function of the convolution of two independent but non identically distributed generalized Laplace random variable of Mathai (2000). The distribution with characteristic function (5.6.) arise as the distribution of the τ-fold convolution of independent DWE(α) random variables. From (5.6.), we have Ψ(t) = ( ) τ ( ) τ ( ) τ ( ) τ, α > 0, τ > 0. (5.6.2) it + it iδt + iδt Therefore X can be represented as the convolution of four independently distributed random variables as X = G G 2 + G 3 G 4, (5.6.3) where G i d Gamma(, τ), i=,2 and G i d Gamma(δ, τ), i=3,4 distributed random variables. Now we discuss the autoregressive process with generalized weighted exponential distribution as marginals. 9

26 5.6. First order Generalized Double Weighted Exponential GDWEAR() Process For the process defined in (5.2.), where X n d GDW E(α, τ), using the characteristic function we can see that the innovation sequence is distributed as ɛ n d U n + V n, where U n and V n are the independent τ convolution of U n and V n defined in (5.5.7) and (5.5.8) respectively. Similarly, we can represent ɛ n as the τ fold convolution the random variable defined in the right hand side of (5.5.4) GDWEARMA(,) Process Consider the Autoregressive moving average model with GWE marginals defined by X n = θx n + ζɛ n + ɛ n (5.6.4) Assuming stationarity and when ζ =, we obtain in terms of characteristic function ) 2 Ψ ɛ (t) =. Therefore we obtain the distribution of the innovation sequence ( ΨX (t) Ψ X (θt) as the distribution of random variable defined as above with τ = Weighted Weibull distribution A random variable X with positive support is said to follow Weighted Weibull distribution with parameter α, β > 0, denoted by X d W W (α, β) if the probability 92

27 f(x) f(x) x x Figure 5.4: Shapes of the density functions of Weighted Weibull distribution density function of X is given by f(x; α, β) = (α + )β x β e xβ ( e αxβ ), α > 0, β > 0, x > 0. (5.7.) α This model can be obtained from two independent identically distributed Weibull random variables exactly the same way Azzalini (985) obtained the skew-normal distribution from two independent and identically distributed normal distributions. On substituting f Y (x) = e xβ and F Y (x) = e xβ in (5..) we will get (5.7.). This model can be obtained as a hidden truncation model as it was observed by Arnold and Beaver (2000a) in case of skew-normal distribution. This distribution can also considered as a special case of distributions generated from beta family as done by Famoye et al (2005). The k th moment is given by 93

28 E(X k (α + )β ( ) = x (k+β ) e xβ e αxβ) α 0 (α + )β = y ( k β e y e αy) dy, where y = x β α 0 ( ) ( ) (α + ) k = Γ α β + (α + ) α+ β (5.7.2) but it is not possible to write an explicit expression for the k th moment. The median of (5.7.) satisfies the equation ( ) e xβ e xβ (α + ) = α 2 (5.7.3) An approximate value for the mode is given by x mode ( ) 2β β β(α + ) (5.7.4) The distribution function corresponding to (5.7.) is given by F α,β (x) = α + α [ e xβ ] α + ( e xβ ) (5.7.5) and the hazard function is γ(t) = βt β ( e αtβ ) α+ e αtβ (5.7.6) 5.7. Maximum Likelihood Estimation of the Parameters In this section we discuss the maximum likelihood estimation of the parameters of the distribution. Let x, x 2,..., x n be a random sample from a population following 94

29 WE(α, β) distribution. Then the log likelihood function is given by logl = nlog(α + ) + nlog(β) nlog(α) + (β ) + n i= ( ) log e αxβ i n log(x i ) i= n i= x i β (5.7.7) On differentiating (5.7.7) with respect to α and β we obtain the normal equations as n n β + log(x i ) β i= n i= n α + n n α + x β i + αβ i= i= x β i e αxβ i ( e αxβ i ) = 0 (5.7.8) n ( x β i e αxβ i ) = 0 (5.7.9) e αxβ i But it is not possible to write an explicit form for he estimate of the parameters but using softwares we can do this as done in the next section Data Analysis In this Section we fit the model two data sets. The Data set is a simulated data where the values are as given below. Data set :.58, 0.43, 0.50, 2.44, 3.8,.95, 0.78, , 3.54,.42,3.2, 2.5,.9, 3.46, 2.54, 0.63, 2.92, 2.35, 0.54, 3.42, 0.3, 0.84, 0.49,.48,.92,4.25, 4.55, 2.37, 2.3,.4, 0.63,.26, 0.39,., 0.76,.35, 0.68, 0.89,.89,3.95,.32, 2.4,.58, 3.82, 2.25, 0.28,.29, 2.29, 0.70,0.69, 2.42, 2.80,.58,.2, 3.6, 0.86, 0.72, 0.23, 2.7, 0.56, 2.4, 2.43,.42, 2.76,.99, 0.92, 2.76, 2.9, 0.44,.02,.4, 2.32, 0.92,.5,.65,.65,.96, 0.60, 0.3, 3.85, 3.70, 2.32, 0.58, 0.68,2.72, 3.75, 0.43,.58,.56, 3.6,.20, 2.09, 2.02, 3.6, 3.07, 0.52,.58,.73. The maximum likelihood estimates are obtained as ˆα = and ˆβ =.792. See the Figure 5.5 for the fitness of the model to the data. 95

30 Density x Figure 5.5: WW fit to simulated data The Data set 2 is real data set reported by Murthy et al (2004) the data values are the failure times of 50 items. The data values are given below. The estimates obtained as ˆα = and β = , see Figure 5.6. Data set 2: 0.036,0.058,0.06,0.074,0.078,0.086, 0.02,0.03,0.4, 0.6, 0.48, 0.83, 0.92, 0.254,0.262, 0.379,0.38,0.538,0.570,0.574, 0.590,0.68, 0.645,0.96,.228,.600, 2.006,2.054,2.804,3.058, 3.076,3.47,3.625,3.704, 3.93, 4.073,4.393, 4.534, 4.893,6.274, 6.86,7.896, 7.904,8.022,9.337, 0.94,.02, 3.88,4.73,

31 Density x Figure 5.6: WW fit to Data set 2 97