Let Y = Xβ + ε be the linear model where X be an n × p matrix with orthonormal columns (columns of X are orthogonal to each other and each column has length 1)
Let be the least-squares estimate of β, and let be the ridge regression estimate with tuning parameter λ.
Prove that for each j, .
Note: The ridge regression estimate is given by:
The least squares estimate is given by:
Ridge regression estimate is obtained here for a particular choice of Data matrix X.
A metric space (X, d) is totally bounded if, given ε>0, there exists a finite subset = of X, called an ε-net, such that for each x∈X there exists ∈ such that d(x,) < ε. Prove that if Y is a subset of a totally bounded space X then, given ε>0, the subset Y has an ε-net and therefore Y is also totally bounded. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to...
Let be a sample (size n=1) from the exponential distribution, which has the pdf , where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 1122 T(X) We were unable to transcribe this image
The random vector Y = (Y1, ..., Yn)T is such that Y = Xβ + ε, where X is an n × p full-rank matrix of known constants, β is a p-length vector of unknown parameters, and ε is an n-length vector of random variables. A multiple linear regression model is fitted to the data. (a) Write down the multiple linear regression model assumptions in matrix format. (b) Derive the least squares estimator β^ of β. (c) Using the data:...
Linear statistical models For ridge regression, we choose parameter estimators b which minimise where is a constant penalty parameter. Show that these estimators are given by 7n i=1 We were unable to transcribe this imageWe were unable to transcribe this image 7n i=1
1.Given the Multiple Linear regression model as Y-Po + β.X1 + β2X2 + β3Xs + which in matrix notation is written asy-xß +ε where -έ has a N(0,a21) distribution + + ßpXo +ε A. Show that the OLS estimator of the parameter vector B is given by B. Show that the OLS in A above is an unbiased estimator of β Hint: E(β)-β C. Show that the variance of the estimator is Var(B)-o(Xx)-1 D. What is the distribution o the...
Let X and Y be a first countable spaces. Prove that f:XY is continuous if whenever xnx in X then f(xn )f(x) in Y We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be a sample (size n = 1) from the exponential distribution, which has the pdf where is an unknown parameter. Let's define a statistic as . Is a sufficient statistic for ? We were unable to transcribe this imagef(x: λ) = Xe We were unable to transcribe this imageT(X) = 11>2 T(X) We were unable to transcribe this image
Let n, and let n be a reduced residue. Let r = odd(). Prove that if r = st for positive integers s and t, then old(t) = s. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let yp(y) be the C(2) inverse demand function facing a monopoly, where y++ is its rate of output, and let yC(y) be the C(2) total cost function of the monopoly. Assume that p(y)>0, p'(y)<0, and C'(y)>0 for all y++, and that a profit maximizing rate of output exists. Total revenue is therefore given by R(y)=p(y)y. Given that question uses an inverse demand function, the elasticity of demand, namely (y), is defined as (y)= 1/p'y p(y)/y. Why is (y)<0? Prove that...
Let be the orthogonal group of (2 x 2)-matrices over , and let be the subset of . a) Show that is a subgroup of . b) Show that is a normal subgroup of **abstract algebra 02(R) We were unable to transcribe this imageA (R) = {(8) E O2R): a, b E R We were unable to transcribe this image(a(R),.) We were unable to transcribe this image(R):ܠ We were unable to transcribe this image