Geometric Skew-Normal Distribution

Transcription

1 Debasis Kundu Arun Kumar Chair Professor Department of Mathematics & Statistics Indian Institute of Technology Kanpur Part of this work is going to appear in Sankhya, Ser. B. April 11, 2014

2 Outline 1 Motivation

3 Outline 1 Motivation

4 Skew-Normal Distribution In recent time skew-normal distribution has received considerable attention. The skew-normal distribution of Azzalini can be obtained as follows: Suppose X and Y are two independent standard normal random variables. Then for < λ <, 1 = P(Y < λx) = 2 φ(x)φ(λx)dx.

5 Skew-Normal distribution: PDF Hence f(x;λ) = 2φ(x)Φ(λx), is a proper density function. Azzalini termed it as the skew-normal distribution. The location and scale parameter can be easily added to this distribution function. For example, the three-parameter skew normal distribution takes the following form: f(x;λ,µ,σ) = 2 σ φ ( x µ σ ) Φ ( ) λ(x µ) σ

6 Properties It is a very flexible distribution. The PDF can take variety of shapes. Here λ is the skewness parameter. It has a very nice physical interpretation as a hidden truncation model.

7 Issues Motivation 1 Estimation of the parameters is not a trivial issue. 2 The maximum likelihood estimators may not always exist. 3 It may not able to analyze heavy tail data. 4 It can be extended to the multivariate set up.

8 Power Normal Distribution Power normal distribution has been introduced by Gupta and Gupta (Test) in Power normal distribution has the following form for α > 0: F(x;α) = [Φ(x)] α. Hence the corresponding PDF becomes: f(x;α) = α[φ(x)] α 1 φ(x).

9 Properties In this case also the location and scale parameter can be easily introduced as follows: f(x;α,µ,σ) = α [ ( )] x µ α 1 ( ) x µ Φ φ σ σ σ It is a also very flexible distribution. The PDF can take variety of shapes. Here α is the skewness parameter.

10 Issues 1 Estimation of the parameters is not a trivial issue. 2 The maximum likelihood estimators exist, but they have to be obtained by solving three non-linear equations. 3 It may not able to analyze heavy tail data. 4 It can be extended to the multivariate set up.

11 Aim Motivation We want to introduce a new three-parameter distribution of which normal distribution is a special case. 1 It can be flexible enough to analyze skewed data. 2 The model should be able to analyze heavy tail data. 3 Estimation of the parameters should not be too difficult. 4 Natural multivariate extension should be possible.

12 Outline 1 Motivation

13 Definition Suppose N is a geometric random variable with parameter 0 < p 1, and it has the probability mass function P(N = n) = p(1 p) n 1, n = 1,2,... The random variables X 1,X 2,... are i.i.d. normal random variables with mean µ and variance σ 2, and N and X i s are independent. Then define a new random variable X as N X = d i=1 X i

14 Joint CDF The joint CDF of X and N is given by P(X x,n n) = = = p n P(X x,n = k) k=1 n P(X x N = k)p(n = k) k=1 n ( ) x kµ Φ σ (1 p) k 1 k k=1

15 Joint PDF The joint PDF of X and N is given by 1 f x,n (x,n) = σ 1 2πn e 2nσ 2(x nµ)2 p(1 p) n 1 if 0 < p < 1 1 σ 1 2π e 2σ 2(x µ)2 if p = 1

16 PDF and CDF of X The distribution function of X becomes: ( ) x kµ P(X x) = P(X x,n = k) = p Φ σ (1 p) k 1 k k=1 The PDF of X becomes: f X (x) = k=1 p σ k Φ k=1 ( x kµ σ k ) (1 p) k 1

17 Standard GSN When µ = 0 and σ = 1, we call it as the standard GSN. The random variable X is said to have a standard GSN, if it has the PDF ( p x f X (x) = Φ k )(1 p) k 1. k k=1 Shapes of the PDF of the standard GSN for different values of p are given below.

18 Shapes of the PDF of standard GSN p = p = 0.1 p =

19 Some Basic Features It is symmetric about zero. As p decreases the tail probabilities increase. As p tends to zero, the variance tends to.

20 Shapes of the PDF of GSN The PDF of GSN when µ = 1, p = 0.25, σ =

21 Shapes of the PDF of GSN The PDF of GSN when µ = 1, p = 0.5, σ =

22 Shapes of the PDF of GSN The PDF of GSN when µ = 3.5, p = 0.5, σ =

23 Shapes of the PDF of GSN The PDF of GSN when µ = -1, p = 0.25, σ =

24 Outline 1 Motivation

25 Moment Generating Function If X GSN(µ,σ,p), then the moment generating function of X becomes: M X (t) = E = [ ] [ ] E(e tx N) = E e Nµt+Nσ2 t 2 2 pe (µt+ σ2 t 2 ) 1 (1 p)e 2 ( µt+ σ2 t 2 ) 2

26 Mean, Variance, Moments The mean and variance become: E(X) = µ p V(X) = σ2 p +µ 2 (1 p) p 2 and skewness becomes γ 1 = (1 p)(µ3 (2p 2 p +2)+2µ 2 p 2 +µσ 2 (3 p)p) (σ 2 p +µ 2 (1 p)) 3/2. E(X m ) = p (1 p) n 1 c m (nµ,nσ 2 ). n=1 Here c m (nµ,nσ 2 ) = E(Y m ), where Y N(nµ,nσ 2 ).

27 Infinite Divisibility Consider the following random vector (R,T) when r = 1/n, R d = 1+nT i=1 where Y i s are i.i.d., Y i N(µ/n,σ 2 /n), and T follows a negative binomial NB(r, p) distribution with the probability mass function Y i, P(T = k) = Γ(k +r) k!γ(r) pr (1 p) k, k = 0,1,2,...

28 Infinite Divisibility The moment generating function of R for t (, ), is given by ( M R (t) = E e tr) ) = E (Ee t 1+nT i=1 Y i T = pe µt+σ2 t (1 p)e µt+σ2 t 2 2 1/n = (M X (t)) 1/n. where M X (t) is the moment generating function of GSN(µ,σ,p). Therefore, GSN law is infinitely divisible.

29 Geometric Stable X i s are i.i.d. GSN(µ,σ, p), and M is an independent GE(q), with 0 < q < 1, random variable. The moment generating function of M d X i = X becomes E ( e tx) = q = i=1 (1 q) m 1 m=1 Hence X GSN(µ,σ, pq). pqe µt+σ2 t (1 pq)e µt+σ2 t 2 2 pe µt+σ2 t (1 p)e µt+σ2 t 2 2. m

30 Interesting Decomposition Suppose X GSN(µ,σ,p), then X d = Y + Q Y i. i=1 Here Y N(µ,σ 2 ), and Q Poisson(λ). Further Y i Z i N(µZ i,σ 2 Z i ), where P(Z 1 = k) = (1 p)k ; k = 1,2,..., λ = lnp. λk

31 Conditional Distribution The conditional probability mass function of N given X = x, is P(N = n X = x) = The conditional expectations become (1 p) n 1 e 1 2σ 2 n (x nµ)2 / n k=1 (1 p)k 1 e 1 2σ 2 k (x kµ)2 /. k E(N X = x) = n=1 (1 p)n 1 e 1 2σ 2 n (x nµ)2 / n k=1 (1 p)k 1 e 1 2σ 2 k (x kµ)2 /, k and E(N 1 X = x) = n=1 (1 p)n 1 e 1 2σ 2 n (x nµ)2 /n 3/2 k=1 (1 p)k 1 e 1 2σ 2 k (x kµ)2 / k.

32 Outline 1 Motivation

33 Maximum Likelihood Estimators Suppose {x 1,...,x n } is a random sample of size n from GSN(µ, σ, p), the log-likelihood function becomes [ n n ( ] p l(µ,σ,p) = lnf X (x i ) = ln σ k φ xi kµ )(1 p) σ k 1. k i=1 i=1 k=1 The maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by maximizing the log-likelihood function with respect to the unknown parameters.

34 EM algorithm: Basic Idea Suppose, {(x 1,m 1 ),...,(x n,m n )} is a random sample of size n from (X, N). The log-likelihood function becomes l c (µ,σ,p) = nlnσ 1 2σ 2 n (x i m i µ) 2 +nlnp+ln(1 p) i=1 The MLEs of the unknown parameters are as follows: m i n (m i 1). i=1 µ = n i=1 x i n k=1 m, σ2 = 1 k n n i=1 (x i m i µ) 2 m i, p = n K +n, where K = n i=1 m i.

35 EM algorithm: E-step At the E -step, the pseudo log-likelihood function at the k-th stage can be formed by replacing the missing values with their expectations, and it is as follows; l (k) s (µ,σ,p) = nlnσ 1 2σ 2 ( n i=1 +nlnp +ln(1 p) x 2 i c (k) i 2µ n i=1 (d (k) i 1), n n x i +µ 2 here c (k) i and d (k) i can be obtained from E(N X = x) and E(N 1 X = x), by replacing x, µ, σ, p with x i, µ (k), σ (k), p (k), respectively. i=1 i=1 c (k) i )

36 EM algorithm: M-step The M -step can be obtained by maximizing l s (k) (µ,σ,p) with respect to the unknown parameters. Therefore, µ (k+1), σ (k+1), p (k+1), can be obtained as n µ (k+1) i=1 = x i, n j=1 c(k) j σ (k+1) = 1 n n n x i +(µ (k+1) ) 2 n i=1 x 2 i c(k) i 2µ (k+1) i=1 p (k+1) n = n i=1 d(k) i +n. i=1 c (k) i.

37 Asymptotic Distribution The asymptotic distribution of the MLEs can be obtained in a routine manner that is, if µ, σ and p denote the MLEs of µ, σ and p, respectively, then d n( µ µ, σ σ, p p) d N 3 (0,F 1 ), where denotes convergence in distribution, and the 3 3 matrix F is the expected Fisher information matrix.

38 Testing of Hypotheses Test I: H 0 : µ = 0 vs. H 1 : µ 0. The problem is of interest as it tests whether the distribution is symmetric or not. In this the MLEs of the unknown parameters can be obtained using the EM algorithm as before. Under the null hypothesis, the pseudo log-likelihood function becomes l (k) si (σ,p) = nlnσ 1 2σ 2 n i=1 x 2 i c (k) i +n ln p+ln(1 p) Therefore, σ (k+1), p (k+1), can be obtained as σ (k+1) = 1 n x 2 n i c(k) i and p (k+1) = i=1 n i=1 (d (k) i 1), n n i=1 d(k) i +n.

39 Testing of Hypotheses Test II: H 0 : p = 1 vs. H 1 : p < 1. The problem is of interest as it tests whether the distribution is normal or not. In this case under the null hypothesis the MLEs of µ and σ become n i=1 µ = x n i i=1, and σ = (x i µ) 2. n n In this case p is in the boundary under the null hypothesis, the standard results do not work. But using Theorem 3 of Self and Liang (1987), it follows that 2(l( µ, σ, p) l( µ, σ,1)) χ2 1.

40 Outline 1 Motivation

41 Definition Definition: Suppose N GE(p), {X i : i = 1,2,...,} are i.i.d. N m (µ,σ) random vectors, N and X i s are independently distributed. Define N X = d X i, then X is said to be m-variate geometric skew normal distribution with parameters p, µ and Σ, and it will be denoted by MGSN(m,p,µ,Σ). i=1

42 PDF Motivation The PDF of X becomes f X (x) = = p(1 p) k 1 (2π) m/2 Σ 1/2 k e p(1 p) k 1 φ m (x;kµ,kσ). k=1 k=1 1 2k (x kµ)t Σ 1 (x kµ) If µ = 0 and Σ = I, we say that X has standard MGSN distribution with PDF f X (x) = p(1 p) k 1 φ m (x;0,ki). k=1

43 Surface plot of BGSN The surface plot of BGSN

44 Surface plot of BGSN The surface plot of BGSN

45 Surface plot of BGSN The surface plot of BGSN

46 Surface plot of BGSN The surface plot of BGSN

47 Surface plot of BGSN The surface plot of BGSN

48 Marginals If X MGSN(m,p,µ,Σ), then X 1 MGSN(m 1,p,µ 1,Σ 11 ) and X 2 MGSN(m 2,p,µ 2,Σ 22 ).

49 Moment Generating Function If X MGSN(m,p,µ,Σ), then the moment generating function of X becomes; M X (t) = Ee ttx = E(Ee ttx N) = E [e N(µT t+ 1 2 tt Σt) ] = pe µt t+ 1 2 tt Σt 1 (1 p)e µt t+ 1 2 tt Σt.

50 Infinite Divisible Now we will show that similarly as the univariate case, MGSN distribution is also infinitely divisible. Consider the following random vector R, when r = 1 n. R d = 1+nT where Y i s are i.i.d., Y 1 N m ( 1 n µ, 1 n Σ) and T NB(r,p), as defined before. i=1 Y i,

51 Infinite Divisible: Contd. The moment generating function of R becomes ( ) M R (t) = E e tt R = E (e ) 1+nT i=1 t T Y i T [ = pe µt t+ 1 2 tt Σt 1 (1 p)e µt t+ 1 2 tt Σt = [M X (t)] r. Hence MGSN law is infinite divisible. ] r

52 Outline 1 Motivation

53 EM Algorithm In the multivariate case also the EM algorithm can be performed along the same line. Extensive simulations are needed to observe the performances of the maximum likelihood estimators and also the suggested EM algorithm.

54 Testing of Hypotheses Different testing of hypotheses also can be carried out along the same line. More work is needed along that line.

55 Outline 1 Motivation

56 MGSN law induces a Lévy process {(X(r),NB(r)),r 0}, which has the following stochastic representation NB(r) {(X(r),N(r));r 0} = d Y i +Z(r),r +NB(r) ;r 0, i=1 where Y i s are multivariate normal random variables, {Z(r) : r 0} is a normal Lévy process, and {NB(r);r 0} is a negative binomial Lévy process.

57 It will be interesting to look at the multivariate copula and study different properties of MGSN using copula properties.

58 Thank You