Linear Models代写 linear regression model代写 Qualifying Exam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 = Xβ + e,

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

β˜  = (X^TX + λI_p×p)⁻¹X^TY ,Linear Models代写

where λ 0 is a tuning parameter prespecified by the user and I_p×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 = = β_p = 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).

2.Consider the followingdataLinear Models代写

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 = β₂− β₁. 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.

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