By Wolfgang Karl Härdle, Zdeněk Hlávka

ISBN-10: 0387707840

ISBN-13: 9780387707846

ISBN-10: 0387735089

ISBN-13: 9780387735085

This ebook provides the instruments and ideas of multivariate info research. It demonstrates the appliance of straightforward calculus and uncomplicated multivariate equipment in actual lifestyles events, and lines greater than two hundred solved routines.

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**Additional resources for Multivariate Statistics - Exercises and Solutions**

**Example text**

How would you implement the resulting Let us consider the transformation X = µ + Σ 1/2 Y. We know that a linearly transformed normally distributed random vector is again normally distributed. From the rules for the mean and variance matrix of the linearly transformed random variable we know that EX = µ + Σ 1/2 EY = µ and Var X = Σ 1/2 Var Y (Σ 1/2 ) = Σ. 54 . 95 One then applies the above formula that linearly transforms Y into X. 2. Prove that if X ∼ Np (µ, Σ), then the variable U = (X − µ) Σ −1 (X − µ) has a χ2p distribution.

Let Y = X1 +X2 . , ﬁnd the transformation matrix A. Then compute Var (Y ). Clearly, Y = X1 + X2 = AX = (1, 1) X1 X2 and Var (AX) = E{(AX − EAX)(AX − EAX) } = A{E(X − EX)(X − EX) }A = A Var(X)A . Hence, Var (Y ) = AΣA = (1, 1) Σ 1 1 = (1, 1) 10 02 1 1 =3 Another possibility is to write Var (Y ) = Var (X1 + X2 ) = Var (X1 ) + 2 Cov(X1 , X2 ) + Var (X2 ) = 3. 4. Calculate the mean and the variance of the estimator βˆ = (X X )−1 X Y in a linear model Y = X β + ε, Eε = 0n , Var(ε) = σ 2 In . 23. It follows that E βˆ = (X X )−1 X EY = (X X )−1 X (X β + Eε) = β since we assume that Eε = 0n .

SMSlinregpull In this case, the inﬂuence of the outliers on the regression line does not seem to be too large. Deleting only one of the outliers would lead to much diﬀerent results. Moreover, such an outlier might inﬂuence the regression line so that it is not detectable using only the residuals. 12. Under what circumstances would you obtain the same coeﬃcients from the linear regression lines of Y on X and of X on Y ? 7: α = y − βx and β = sXY . sXX From the formula for the slope of the regression line, β, it follows that the slopes are identical if the variances of X and Y are equal, sXX = sY Y , or if the covariance between X and Y is equal to zero, sXY = 0.

### Multivariate Statistics - Exercises and Solutions by Wolfgang Karl Härdle, Zdeněk Hlávka

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