DRAFT. Edward Ngailo. Department of Mathematics, University of Dar es Salaam Department of Mathematics, Stockholm University

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1 On functions of a Wishart matrix and a normal vector with applications Edward Ngailo Department of Mathematics, University of Dar es Salaam Department of Mathematics, Stockholm University Second Network Meeting for Sida- and ISP-funded PhD Students in Mathematics DRAFT Stockholm February 2018

2 PAPER I: Discriminant Analysis in Small and Large Dimensions (jointly with Taras Bodnar, Stepan Mazur and Nestor Parolya) PAPER II: Higher Order Moments of the Estimated Tangency Portfolio Weights (jointly with Farrukh Javed and Stepan Mazur)

3 Outline 1 Motivation 2 Finite sample properties 3 Large dimensional asymptotics: Infinite samples 4 Theoretical part on portfolio theory 5 Empirical study 6 Conclusion

4 Motivation Wishart distribution is a multivariate analogue of the χ 2 distribution, [Rencher and Christensen, 2012]. Discriminant analysis is a multivariate statistical method for characterising the observations from two or more identified populations. The study considers two independent samples from the multivariate normal distributions. For linear discriminant function Σ 1 = Σ 2. Qudratic discriminant function the sample covariances of groups are not the same

5 Motivation We combine the information from both samples in order to estimate the common covariance Σ. S pl = 1 n 1 + n 2 2 [(n 1 1)S 1 + (n 2 1)S 2 ] The pooled sample covariance matrix, S pl has a p-dimensional Wishart distribution W p (n 1 + n 2 2, Σ) and is independent of the sample mean vectors x 1 and x 2. We study the product ( x (1) x (2)) S 1 pl This functions which depend on the product of an inverse Wishart matrix and a random normal vector are not comprehensively studied in literature, [Bodnar and Okhrin, 2011]. Our goal: to study the distribution properties of the product.

6 Motivation: Portfolio theory Harry Markowitz Markowitz is considered as the father of modern portfolio theory. He is the first person who gave the mathematical model for portfolio optimization and diversification. Figure: Source:(Wikipedia)

7 Motivation:Portfolio theory The products arises also in the portfolio theory where the vector of optimal portfolio weights is proportional to S 1 x Goal of portfolio theory is to determine efficient way of portfolio allocation. The MV optimization technique plays a central role in allocating the investments among different assets. The expression S 1 x is not comprehensively investigated in tangency portfolio, [Bodnar and Okhrin, 2011]. Our goal: To provide the higher order moments of the estimated sample weights of tangency portfolio.

8 Discriminant function coefficients The discriminant function coefficients are given by â = S 1 pl ( x (1) x (2)) â is a sample estimator of the population discriminant function coefficient vector Our interest is on a = Σ 1 pl (µ 1 µ 2 ) ( ˆθ = Lâ = LS 1 pl x (1) x (2)), L(k p)

9 Discriminant function coefficients Theorem The stochastic representation of ˆθ = Lâ is given by d ˆθ = (n 1 + n 2 2)ξ 1 LΣ 1 x x + T Σ 1 x ( ) LR x L T 1/2 t0, n 1 + n 2 p where R x = Σ 1 Σ 1 x x T Σ 1 / x T Σ 1 x; ξ χ ( ) ) 2 n 1 +n 2 p 1, x N p (µ 1 µ 2, Σ, and t n1 n2 0 t k (n 1 + n 2 p, 0 k, I k ). ξ, x and t 0 are mutually independent

10 Discriminant function coefficients Corollary The stochastic representation of ˆθ = l T â is given by ( ( ˆθ = d (n ) ) 1 + n 2 2) l T Σ 1 λ(p 1) (µ ξ 1 µ 2 ) + λ + n 1 + n 2 p u l T Σ 1 lz 0, where ξ χ 2 n 1 +n 2 p 1, z 0 N (0, 1), u F ( p 1, n 1 + n 2 p, (µ 1 µ 2 ) T R l (µ 1 µ 2 )/λ ) R l = Σ 1 Σ 1 ll T Σ 1 /l T Σ 1 l

11 Classification analysis: Error rates decision rules Assign new observation vector x to the first group if (µ 1 µ 2 ) Σ 1 x > 1 2 (µ 1 µ 2 ) Σ 1 (µ 1 + µ 2 ) holds and to the second group otherwise.( [Rencher, 1998] and [Rencher and Christensen, 2012]) Classification error rate is the probability of classifying the observation x into one group, while it comes from another one. ER p ( ) = 1 P(classify to the first group second group is true) P(classify to the second group first group is true) 2 ( = Φ ) with 2 = (µ 2 1 µ 2 ) Σ 1 (µ 1 µ 2 ),, where Φ(.) denotes the distribution function of the standard normal distribution.

12 Classification analysis: Error rates decision rules The decision rule based on sample: ( x (1) x (2) ) S 1 pl x > 1 2 ( x(1) x (2) ) S 1 pl ( x (1) + x (2) ) Theorem The error rate of the sample based decision rule: ˆd = ( x (1) x (2) ) S 1 pl The stochastic representation of ˆd is given by ( d ˆd = n 1 + n 2 2 ξ + ( n 1 + n 2 + ( x 1 2 ( x(1) + x (2) ) ( 1) i 1 λn i 2 2λn i ( λξ 2 + ( + λw 0 ) 2) p 1 ) λξ n 1 + n 2 p u 2 + ( + ) λw 0 ) 2 z 0, )

13 Classification analysis: Error rates decision rules ER s ( ) = 1 2 P(ˆd > 0 second group is true) P(ˆd 0 first group is true). We compare ER s ( ) to ER p ( ) Different values of sample size n for fixed dimension p.

14 ERp(Δ) ERs(Δ), n1 = n2 = 50 ERs(Δ), n1 = n2 = 100 ERs(Δ), n1 = n2 = 150 ERs(Δ), n1 = n2 = ERp(Δ) ERs(Δ), n1 = n2 = 50 ERs(Δ), n1 = n2 = 100 ERs(Δ), n1 = n2 = 150 ERs(Δ), n1 = n2 = p= p= p=75 ERp(Δ) ERs(Δ), n1 = n2 = 50 ERs(Δ), n1 = n2 = 100 ERs(Δ), n1 = n2 = 150 ERs(Δ), n1 = n2 = ERp(Δ) ERs(Δ), n1 = n2 = 50 ERs(Δ), n1 = n2 = 100 ERs(Δ), n1 = n2 = 150 ERs(Δ), n1 = n2 = p= Figure: ER ( ) and ER ( ) as functions of for p {10, 25, 50, 75}

15 Discriminant analysis under large dimensional asymptotics Theorem The asymptotic distribution of ˆθ = l T â is given by ( n1 + n 2 σγ 1 ˆθ 1 ) 1 c lt Σ 1 D (µ 1 µ 2 ) N (0, 1) for p/(n 1 + n 2 ) c [0, 1) as n 1 + n 2 with ( σγ 2 1 ( ) 2 = (1 c) 3 l T Σ 1 (µ 1 µ 2 ) +l T Σ 1 l(µ 1 µ 2 ) T Σ 1 (µ 1 µ 2 ) + λ(n 1 + n 2 )l T Σ 1 l1 {0} (γ) ) [ with help of [DasGupta, 2008] [Bodnar and Reiß, 2016] we get this results ]

16 Classification analysis in high dimension Theorem Let p γ and λn i b i for p/(n 1 + n 2 ) c [0, 1) as n 1 + n 2. It holds that ( p min(γ,1)/2 ˆd ( D N p n 1+n 2 2 γ n 1 +n 2 p 1 ( 1) i 1 c 1 c ) ( 1) i 1 2 p γ b i 2 2b i (b 1 + b 2 )1 {0} (γ), c 2 1 2(1 c) 3 (1,+ ) (γ) + 1 (c(b (1 c) b 2 )1 {0} (γ) + 1 [0,1] (γ)) for p/(n 1 + n 2 ) c [0, 1) as n 1 + n 2. )

17 Classification analysis in high dimension In the special case of n 1 = n 2, such that λn 1 = λn 2 = 2 we have, ( ) ER s ( ) = Φ h c ER p( ) ER s( ), c = 0.1 ER s( ), c = 0.5 ER s( ), c = 0.8 ER s( ), c = Figure: ER p ( ) and ER s ( ) for c {0.1, 0.5, 0.8, 0.95}.

18 Asymptotic True Asymptotic False Finite Sample Asymptotic True Asymptotic False Finite Sample Asymptotic True Asymptotic False Finite Sample Asymptotic True Asymptotic False Finite Sample

19 Asymptotic True Asymptotic False Finite Sample Asymptotic True Asymptotic False Finite Sample a) p = 50, n 1 = 25, n 2 = 475 b) p = 250, n 1 = 25, n 2 = 475 Figure: The kernel density estimator of the asymptotic distribution and standard normal for ˆθ as given in Theorem 4 for γ > 0 and c = {0.1, 0.5, 0.8, 0.95}.

20 Preliminary results: Portfolio theory Lemma l T A 1 z d = 1 u 1 (l T Σ 1 µ + ( λ + ) ) λ(k 1) n k + 2 u 3 l T Σ 1 lu 2 Theorem Then the r-th order moment of l T A 1 z is given by [ ] E (l T A 1 z) r = (l T Σ 1 µ) r + ( 1 + j m=1 1 (n k 1)...(n k 2r + 1) r/2 ( ) r (2j)! 2j 2 j j! j=1 ( ) j )c m m ( ) r 2j ( ) j l T Σ 1 µ λl T Σ 1 l

21 Preliminary results Corollary Then the r-th order central moment of l T A 1 z. ) r ] E [(l T A 1 z κ 1 = ( κ 1 ) r + r i=1 ( ) r [ ( 1) r i κ r i 1 E (l T A 1 z) i]. i Corollary Partial cases (a) The second order central moment of l T A 1 z is given by E[(l T A 1 z E[l T A 1 z]) 2 ] = d (0) 1 (lt Σ 1 µ) 2 + d (0) 2 lt Σ 1 l,

22 Preliminary results Corollary (b) The third order central moment of l T A 1 z is given by E[(l T A 1 z E[l T A 1 z]) 3 ] = d (1) 1 (lt Σ 1 µ) 3 + d (1) 2 lt Σ 1 µ l T Σ 1 l (c) The fourth order central moment of l T A 1 z is given by E[(l T A 1 z E[l T A 1 z]) 4 ] = d (3) 1 (lt Σ 1 µ) 4 + d (3) 2 (lt Σ 1 µ) 2 l T Σ 1 l + d (3) 3 (lt Σ 1 l) 2

23 Main results Tangent portfolio Portfolio (risky assets) on efficient frontier which holds in combination with risk-free asset, [Ingersoll, 1987]. Sample weights of tangent portfolio w TP = α 1 Σ 1 (µ r f 1). Figure: Source: (Wikipedia).

24 Main results Tangency portfolio weights are given by The sample estimators for µ and Σ w TP = α 1 Σ 1 (µ r f 1). (1) x = 1 n n j=1 x j and S = 1 n 1 n (x j x)(x j x) T. j=1 Replacing µ and Σ with x and S in(2) ŵ TP = α 1 S 1 (x r f 1).

25 Main results We are interested in θ = l T w TP = α 1 l T Σ 1 (µ r f 1), Then the sample estimator of θ is given by ˆθ = l T ŵ TP = α 1 l T S 1 (x r f 1).

26 Main results Theorem (a) the r-th order moment of ˆθ is given by µ r := E[ˆθ r ] (b) the r-th order central moment of ˆθ is given by Corollary µ r := E[(ˆθ µ 1 ) r ] The mean and the variance of ŵ TP are given by E[ŵ TP ] = n 1 n k 2 w TP Var[ŵ TP ] = d (0) 1 w TPw T TP + d (0) 2 Σ 1

27 Main results Corollary (a) Then the skewness ˆθ is given by Skewness[ˆθ] = µ 3 [ ] 3/2 = µ 3 ( d (0) 1 θ2 + d (0) 3/2 2 lt Σ l) 1, Var(ˆθ) µ 3 = d (1) 1 θ3 + d (1) 2 θlt Σ 1 l, (b) the kurtosis of ˆθ is expressed as Kurtosis[ˆθ] = µ 4 [ ] 2 = µ 4 ( d (0) 1 θ2 + d (0) 2 2 lt Σ l) 1. Var(ˆθ) µ 4 = d (3) 1 θ4 + d (3) 2 θ2 l T Σ 1 l + d (3) 3 (lt Σ 1 l) 2

28 Empirical study Data consists of 8 financial indexes from NASDAQ weekly log returns, 513 observations (specify period 30th April st April 2017), risk-free asset (3 WEEK TREASURY BILL, US), risk aversion, α = 50 The sample size n is set to be {50, 100, 250, 500}.

29 Table: Mean, Variance, Skewness and Kurtosis: NASDAQ RCMP IXTS IXCO TRAN INDS NBI n=50 Mean Variance Skewness Kurtosis n=100 Mean Variance Skewness Kurtosis n=250 Mean Variance Skewness Kurtosis

30 Conclusion We have studied the distributional properties of LDF coefficients via a stochastic representation Applied the LDF coefficients to approximate sample based decision rule error rate. Extended results to high dimension (i.e discriminant function coefficients and classification analysis) settings Higher order non-central and central moments have been derived. We provide expressions of mean,variance, skewness and kurtosis in closed form without confluent hypergeometric function. Theoretical results have been supported/motivated by financial indexes listed in NASDAQ stock exchange.

31 THANK YOU

32 References Bodnar, T. and Okhrin, Y. (2011). On the product of inverse wishart and normal distributions with applications to discriminant analysis and portfolio theory. Scandinavian Journal of Statistics, 38(2): Bodnar, T. and Reiß, M. (2016). Exact and asymptotic tests on a factor model in low and large dimensions with applications. Journal of Multivariate Analysis, 150: DasGupta, A. (2008). Asymptotic theory of statistics and probability. Springer Science & Business Media. Ingersoll, J. E. (1987). Theory of Financial Decision Making.

33 Bodnar, T. and Okhrin, Y. (2011). On the product of inverse wishart and normal distributions with applications to discriminant analysis and portfolio theory. Scandinavian Journal of Statistics, 38(2): Bodnar, T. and Reiß, M. (2016). Exact and asymptotic tests on a factor model in low and large dimensions with applications. Journal of Multivariate Analysis, 150: DasGupta, A. (2008). Asymptotic theory of statistics and probability. Springer Science & Business Media. Ingersoll, J. E. (1987). Theory of Financial Decision Making. Rowman & Littlefield Publishers. Rencher, A. C. (1998).

34 Multivariate statistical inference and applications, volume 338. Wiley-Interscience. Rencher, A. C. and Christensen, W. F. (2012). Methods of multivariate analysis. John Wiley & Sons.