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Linear Models代写 linear regression model代写 Qualifying Exam代写

2020-12-17 17:40 星期四 所属: 作业代写 浏览:168

Linear Models代写

Qualifying Exam: CAS MA575, Linear Models

Linear Models代写 where β = (β1, . . . , βp)T Rp is the regression coefficient to be estimated. Instead of using ordinary least squares

Boston University,  Fall 2016

1.Consider the linear regressionmodelLinear Models代写

Y = + e,

where β = (β1, . . . , βp)T Rp is the regression coefficient to be estimated. Instead of using ordinary least squares, an alternative is to consider the ridge estimator

β˜  = (XTX λIp×p)1XTY ,Linear Models代写

where λ 0 is a tuning parameter prespecified by the user and Ip×p is the identity matrix in Rp×p. Throughout this problem, assume that the design matrix X is deter- ministic and that the errors are independent normal random variables with mean zero and variance σ2 > 0.

  • FindE(β˜).  When is β˜an unbiased estimator of β?
  • Givena data, explain the behavior of β˜  when λ       , namely when an extremely large tuning parameter is
  • Findthe covariance matrix of β˜.Linear Models代写
  • Suppose σ2= 1 is known and one is interested in testing if β1 = 0. Is it possible to  achieve  this  by  using  the  ridge  estimator  β˜?   Explain  your  strategy  in  detail (including the form of your test statistic and the distribution that you will use to obtain the cut-off value).
  • Suppose σ2= 1 is known and one is interested in testing if β1 = β2 = = βp = 0. Is it possible to achieve this by using the ridge estimator β˜?  Explain your strategy in detail (including the form of your test statistic and the distribution that you will use to obtain the cut-off value).
Linear Models代写
Linear Models代写

2.Consider the followingdataLinear Models代写

Degree Response
Undergraduate 37.7, 32.5, 34.1
Master Linear Models代写 27.8, 22.7, 31.6, 36.5, 41.3
Doctoral 38.2, 44.6, 35.4, 33.7, 40.2

The associated R output of regressing the response on the degree variable is given below.

Call:

lm(formula = Y ~ Degree)

Residuals:

Min 1Q  Median 3Q Max

-9.280 -3.020 -0.380 2.933 9.320

Coefficients:Linear Models代写

Estimate Std. Error t value Pr(>|t|)

(Intercept) 38.420 2.441 15.737 2.2e-08 ***

DegreeMaster -6.440 3.453 -1.865 0.0917 .

DegreeUndergraduate -3.653 3.987 -0.916 0.3810Linear Models代写

Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1

Residual standard error: 5.459 on 10 degrees of freedom

Multiple R-squared: 0.2589, Adjusted R-squared: 0.1107

F-statistic: 1.747 on 2 and 10 DF, p-value: 0.2235Linear Models代写

Residual standard error: 5.459 on 10 degrees of freedom Multiple R-squared: 0.2589, Adjusted R-squared: 0.1107

F-statistic: 1.747 on 2 and 10 DF, p-value: 0.2235

Let

β1 = E(Response | Degree = Undergraduate)

β2 = E(Response | Degree = Master)

β3 = E(Response | Degree = Doctoral)

Throughout this problem, assume that the errors are independent normal random vari- ables with mean zero and variance σ2 > 0. Also assume that the errors are independent of the variable Degree.Linear Models代写

  • Findthe least squares estimate βˆ1  for β1.
  • Find the least squares estimate ξˆ for ξ = β2β1. Is it unbiased?
  • Supposeone is interested in the squared difference ζ = (β2 β1)2. Let ξˆ be as in part (b), is ξˆ2  an unbiased estimate of ζ?  Prov
  • Ifξˆ2  is not an unbiased estimate of ζ,  provide a way to correct for the bias and give an unbiased estimate of ζ.
  • Is it possible to provide an unbiased estimate of γ = β2β1 ? If yes, describe your strategy in achieving this. Note that γ can be interpreted as the absolute difference between Undergraduate and Master.
Linear Models代写
Linear Models代写

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