An Information Based Methodology for the Change Point Problem Under the Non-central Skew t Distribution with Applications.

Transcription

1 An Information Based Methodology for the Change Point Problem Under the Non-central Skew t Distribution with Applications. Joint with Prof. W. Ning & Prof. A. K. Gupta. Department of Mathematics and Statistics Bowling Green State University

2 Outline Motivation & Literature Review 1 Motivation & Literature Review

3 Motivation Normal distribution is a classical model that has been used to fit data in many applications. For data sets with tails longer than normal, Student s t distribution is a more suitable alternative. The need for a distribution that can fit skewed and long-tailed data motivated researchers to investigate skew versions of Student s t distribution.

4 Long-tailed Distributions A distribution is said to be long-tailed if a larger share of the population rests within its tail than would under a normal distribution. A long-tailed distribution includes many values unusually far from the mean, and tends to have higher kurtosis than the normal distribution. The term long-tailed distributions has been used in the finance and insurance business for many years. Common examples of long-tailed distributions include the Cauchy distribution, the log-normal distribution and Student s t distribution.

5 Literature Review of the Skew Normal Distribution The skew normal distribution was first introduced by Roberts (1966) as an example of a weighted model, but he did not use the term skew normal". Azzalini (1985) formalized the skew normal distribution as a generalization of the normal distribution that can be used to model asymmetric data. Azzalini and Dalla Valle (1996) studied the multivariate skew normal distribution. Jamalizadeh, et al. (2009) studied the truncated skew normal distribution.

6 Azzalini s Skew Normal Distribution (1985) Definition A random variable z is said to have the standard skew normal distribution with shape parameter λ if its pdf is given by: g(z; λ) = 2φ(z)Φ(λz). A location- scale version of the pdf is given by: g(x; µ, σ, λ) = 2 σ φ(x µ σ )Φ(λx µ σ ), where φ( ) and Φ( ) are the standard normal distribution s pdf and cdf respectively.

7 Extended Skew Normal Distribution The following definition was proposed by Azzalini as a two parameter extension of his original definition. Definition A random variable Z λ,γ is said to have an extended standard skew normal distribution with parameters λ, γ R, denoted by Z λ,γ ESN(λ, γ), if its pdf is given by: φ SN (z; λ, γ) = φ(z)φ(λz + γ) γ Φ( ). 1+λ 2 The ESN distribution incorporates two shape parameters λ and γ.

8 Truncated Skew Normal Distribution The following truncated version of the extended skew normal distribution was studied by Jamalizadeh, et al. (2009). Definition A random variable Z is said to have a truncated skew normal distribution, denoted by: Z T(a,b);λ,γ TSN((a, b), λ, γ), with parameters λ, γ, a, b R, where a, b are not both infinite, if: Z T(a,b);λ,γ D = Zλ,γ (a < Z λ,γ < b), where D = means equal in distribution.

9 MGF of the Truncated Skew Normal Distribution The moment generating function of Z T(a,b),λ,γ is given by: M(s; a, b, λ, γ) = c(λ, γ, a, b) e s2 /2 { Φ 2 ( λs+γ 1+λ 2, b s; Φ 2 ( λs+γ 1+λ 2, a s; ) λ 1+λ 2 λ 1+λ 2 ) }, where c(λ, γ, a, b) = 1 γ Φ( ){Φ 1+λ 2 SN(b; λ, γ) Φ SN (a; λ, γ)} and Φ 2 (.,., δ) denotes the cdf of N 2 (0, 0, 1, 1, δ), the standard bivariate normal with correlation coefficient δ.

10 Literature Review of Skew t Distributions Skew t distributions received special attention after the introduction of the skew multivariate normal distribution in a paper by Azzalini and Dalla Valle (1996). Gupta (2003) defined the skew multivariate t distribution and skew Cauchy distribution. Azzalini and Capitanio (2002) defined a skew t variate as a scale mixture of skew normal and chi-squared variables. Huang, et al. (2007) studied generalized skew t distributions and used them in data analysis.

11 A non-central Skew t Distribution We propose the following non-central skew t distribution: Definition Let X be a skew normal variable with parameters (µ, σ, α). Let Y be a χ 2 variable with r degrees of freedom. Assume further that X, Y are independent, then the non-central skew t random variable, T, is defined by the transformation: T = X Y/r. The notation St r (µ, σ, α) will be used to denote the non-central skew t distribution.

12 The pdf of St r (µ, σ, α) If T St r (µ, σ, α), then the pdf of T can be expressed as a finite series involving moments of the TSN as follows: 4 f T (t) = rσγ( r 2 )2r/2 e( µ 2 2σ 2 + βγ2 4 ) Φ ( ) [ b 1 Φ SN ( γ 1 + â 2 { r (r ) ( β k 2 k=0 ) r+ 1 k 2 ] β ; â, b) 2 γ r k EZ k }, ( ) where β = 2rσ2, γ = tµ rσ 2 +t 2 σ 2 r, b = α tγβ σ 2 r µ, â = αt σ β r 2, and Z Z T( γ β, ), â, b 2

13 Moments Motivation & Literature Review If T St r (µ, σ, α), then ET = 0.5r Γ( r 1 2 ) Γ( r 2 ) [ µ + ] 2 π δσ, r > 1. The variance of T is given by: Var(T) = r [ µ 2 + σ 2 + 2µ ] (µ + σδ) r 2 2π { [ r ] 2 Γ( r 1 µ + 2 π δσ 2 ) } 2 Γ( r 2 ), where δ = α, and r > α2

14 Theoretical Properties I Lemma Let St r(µ, σ, α) denote the non-central skew t distribution and the vector (µ, σ, α, r) denote its parameter vector. (i) The random variable St r(µ, σ, α) reduces to the Student s t r variate if the parameter vector is chosen as (µ, σ, α, r) = (0, 1, 0, r). (ii) The random variable St r(µ, σ, α) reduces to the non-central Student s t r variate with non-centrality parameters µ, σ and r degrees of freedom if the parameter vector is chosen as (µ, σ, α, r) = (µ, σ, 0, r).

15 Theoretical Properties II The family of the non-central skew t distributions is closed under scalar multiplication. Theorem Let T St r (µ, σ, α). Let c be a nonzero real number. Let U = ct. Then U St r (cµ, c σ, γ), where: γ = αc 1 + α 2 (1 c 2 ).

16 Theoretical Properties III The relation between our distribution and the non-central skew F distribution is illustrated by: Theorem Let T St r(µ, 1, α), then T 2 SF 1,r (µ2, α), where SF 1,r (µ2, α) denotes the non-central skew F distribution with degrees of freedom (1, r), non-centrality parameter µ 2 and skewness coefficient α.

17 Asymptotic Properties I Theorem Let T St r(µ, σ, α). Then lim r T a.s. = SN(µ, σ, α). St'_r(2,2, 1) Vs SN(2,2, 1) PDF SN(2,2, 1) r=5 r=10 r=20 r=30 r= x

18 y Motivation & Literature Review Asymptotic Properties II Theorem Let X St r(0, 1, α). The limiting distribution of X as the shape parameter α is t r, where t r denotes Student s t distribution with r degrees of freedom. Density of ST_3(0,1,alpha) Vs t_ t_3 alpha =5 alpha=10 alpha=15 alpha=20 alpha= x

19 Example: Environmental Data This data set was obtained by measuring the concentration of the heavy metal Cadmium in soil. These measurements were obtained via a sampling process that is intended to measure the pollution of soil in certain locations in the USA. Measurements are given in milligrams of Cadmium per 1 kilogram of soil. Histogram of the Cd Data Cd concentration

20 Motivation & Literature Review Summary & Testing for Normality of the Cd Data Table : Summary of The Cadmium Concentration Data Min. Q1 Median Q3 Max. Mean SD Skewness Kurtosis Normal Q Q Plots of the Cadmium Concentration Data Sample Quantiles Theoretical Quantiles

21 Models Fitted to the Cadmium Data Table : Summary of the Models Fitted to the Cadmium Data Model µ σ α r Log L AIC SIC St r Truncated Normal Truncated Skew Normal

22 Density Curves Fitted to the Cadmium Data Fitted Density Curves to the Cd Data ST Normal SN Cd concentration Figure : Fitted density curves to the Cadmium data

23 Density Curves Fitted to the Tail of the Cadmium Data Fitted Density Curves to the Cd Data ST Normal SN Cd concentration Figure : Illustration of the tail of the fitted density curves to the Cadmium data

24 Guidelines for Data Analysis Using the non-central Skew t Distribution The following guidelines are based on simulation studies and model comparisons. We recommend using our non-central skew t distribution to study asymmetric data with long tails. It is a good strategy to start by fitting a given data set with the non-central skew t distribution. Based on the estimate of the parameters we can decide if a special case of our density would be sufficient. If the fitted value of the degrees of freedom is large enough, then the skew normal is sufficient to fit the data.

25 Motivation for the Change Point Problem In practice, data may follow a certain distribution up to a certain point until a dramatic change occurs due to some extraneous factors. After a certain point the data starts to follow a different distribution. A point after which the change in the distribution of the data occurs is commonly referred to as a change point. The change point problem is the problem of detecting the number of change points and estimating their locations.

26 Literature Review of the Change Point Problem Srivastava, et al. (1986) studied the multiple changes in the multivariate normal mean. Ning and Gupta (2009) studied the change point problem for the generalized lambda distribution. Ramanayake and Gupta (2010) considered the problem of detecting a change point in an exponential distribution with repeated values. Arellano-Valle, et al. (2013) presented a Bayesian approach to study the change point problem of the skew normal distribution.

27 General Change Point Problem Let x 1, x 2,..., x n be independent random variables with probability distribution functions F 1, F 2,..., F n, respectively. In general, the change point problem is to test the null hypothesis: H 0 : F 1 = F 2 = = F n (5.1) versus the alternative: H 1 : F 1 = = F k1 F k1 +1 = = F k2 F k2 +1 = = F kq F kq+1 = = F n, (5.2) where 1 < k 1 < k 2 < < k q < n, q is the unknown number of change points and k 1, k 2,, k q are the respective unknown positions that have to be estimated.

28 Change Point Problem of the non-central Skew t Let the distributions F 1, F 2,, F n belong to the family of non-central skew t distributions. Let θ R 4 be the parameter vector of the distribution. Then the change point problem of the non-central skew t distribution is to test the null hypothesis about the population parameters: H 0 : θ 1 = θ 2 = = θ n = θ (unknown), (5.3) versus the alternative hypothesis: H 1 : θ 1 = = θ k1 θ k1 +1 = = θ k2 θ k2 +1 = = θ kq 1 θ kq = = θ n, (5.4)

29 An Information Approach to the Change Point Problem Assuming that the change point of the data occurs in location k, we can compute the following information criteria for the model: SIC t (k) = 2logL(ˆθ t ) + tlog(n), where k indicates the location of the change point and t is the total number of parameters in the two models. We compute SIC(k) for k 0 k n k 0, where in practice we choose k 0 large enough so that the MLE can be computed accurately.

30 An Information Approach to the Change Point Problem (Cont d.) SIC(n) is computed under the assumption of no change in the distribution that the data comes from. We fail to reject H 0 if and reject H 0 if SIC(n) < min SIC(k), (5.5) k 0 k n k 0 SIC(n) > SIC(k), for some k 0 k n k 0. (5.6) The position of the change point can be estimated by ˆk such that SIC(ˆk) = min SIC(k). (5.7) k 0 k n k 0

31 Simulation Studies & Power Analysis: Case I: The Change Occurs in the Location Parameter Only. Table : Power of the MSICE change point detection as the location parameter µ changed from 0 to -2, σ = 1, α = 1, r = 3 and sample size n = 60 Location of the change point k Power = P(k 1 ˆk k + 1)

32 Case II: The Change Occurs in the Location and the Scale Parameters Simultaneously. Table : Power of the MSICE change point detection as the location parameter µ changed from 0 to -2, σ changes from 1 to 2,and α = 1, r = 3 and sample size n = 60 Location of the change point k Power = P(k 1 ˆk k + 1)

33 Case III: The Change Occurs in the Shape Parameter Only. Table : Power of the MSICE change point detection as the shape parameter α changed from 1 to -1, µ = 0, σ = 1, r = 3 and sample size n = 60 Location of the change point k Power = P(k 1 ˆk k + 1)

34 Applications: Distributions of emerging market stock returns often exhibit long-tails and asymmetry. Emerging markets are known to be more susceptible to the political scenario than developed markets, thus their indices tend to have more outliers and structural changes. we apply the MSICE change point detection method to three data sets obtained from The MSCI Global Equity indices". These indices are widely tracked global equity benchmarks. The goal is to detect change points that can indicate a change in the overall economy of the country.

35 Example: Chile s Weekly Return Prices. The data set lists the weekly returns of Chile starting on the November, 3, 1995 to November, 3, The change point in this data was studied in Arellano-Valle, et al. (2013) using the skew normal model. The weekly prices denoted by P t are typically dependent. To obtain an independent set of random variables we compute the rate of weekly returns using the following transformation which was proposed by Hsu (1979): R t = P t+1 P t P t, t = 1, 2,, 261.

36 Histogram for Chile s Weekly Return Rates. Chile's Weekly Return Rates Frequency Weekly Returen Rates Figure : Histogram for Chile s Weekly Return Rates.

37 Numerical Summary & Normal Q-Q Plot Table : Summary of Chile s Weekly Returns Data Min. Q1 Median Q3 Max. Mean SD Skewness Kurtosis Normal Q Q Plots of Chile's Weekly Return Rates Data Theoretical Quantiles Sample Quantiles

38 SIC Values of Chile s Weekly Returns Data SIC(k)Values for Chile's Stnadardized Weekly Return Rates Data SIC(k) k Figure : SIC(k) Values for Chile s Standardized Return Rates at each Location with a Vertical Line at the First Detected Change Point ˆk = 149.

39 Binary Segmentation of Chile s Weekly Returns Data SIC(k) Values for the First Segment of Chile's Weekly Returns Data SIC(k) Values for the Second Segment of Chile's Weekly Returns Data SIC(k) SIC(k) k k Figure : SIC(k) Values for Chile s Standardized Return Rates for the First Two Segments of the Data.

40 Detected Change Points for Chile s Weekly Returns We detect the change point ˆk = 149, which corresponds to the date September, 4, Using binary segmentation, two additional change points are detected: ˆk 1 = 141, and ˆk 2 = 238. These points correspond to the dates July, 10, 1998 and May, 19, The detected change points are explained by the tight monetary policies which were implemented in 1998 and the global economical downturn that occurred in 1999 in connection with the Asian and Russian financial crises. In March, 10, 2000 there was a collapse of a technology bubble that explains the change point detected in May, 19, 2000.

41 References on Skew Distributions Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist., 12, Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83(4): Genton, M. G. (2004). Skew elliptical distributions and their applications: A journey beyond normality. Chapman and Hall/CRC. Gupta, A.K. (2003). Multivariate skew t distribution. Statistics, 37, Lin, T.I. (2010) Robust mixture modeling using multivariate skew t distribution. Statistics and Computing, 20,

42 References on Change Point Analysis Arellano-Valle, R. B., Castro, L. M. and. Loschi, R. H. (2013). Change point detection in the skew-normal model parameters. Commun. Statist. Theory Meth., 42 (4), Chen, J., Gupta, A. K., and Pan, J. (2006). Information criterion and change point problem for regular models. Sankhya, 68, Chen, J. and Gupta, A. K. (2012). Parametric statistical change point analysis: with applications to genetics, medicine, and finance. Birkhauser Boston, 2nd Ed., Guan, Z. (2004). A semi-parametric change point model. Biometrika, 91, Gupta, A. K. and Chen, J. (1996). Change Detecting Point Problem for the changes non-central Skew of t Distributions

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