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Problem VI . S … nonlinear (10 points)

1. Describe in your own words how and when the nonlinear regression can be used (it is sufficient to describe the following: nonlinear regression model, estimation of unknown regression coefficients, expression of uncertainty of estimates of regression coefficients and fitted values, statistical tests of hypotheses about regression coefficients, identifiability of regression coefficients and choice of regression function). It is not necessary to provide derivations and proofs, brief overview is sufficient.
2. See values (xi, yi) in the attached fileregrese1.csv. We want to fit the theoretical functional dependence, which in this case is a sinusoid, i.e. the function

$f(x)=a+b\cdot\sin(cx+d)\,.$

Plot a graph of observed values and fitted function (such graph has to meet usual requirements) and provide brief interpretation. It is not necessary to do regression diagnostics.

Hint: Do not forget to correctly solve the identifiability problem of this model by suitable restrictive conditions on possible values of parameter c.

3. See values (xi, yi) in the attached fileregrese2.csv. We want to fit the theoretical functional dependence, which, in this case, is an exponential function, i.e. the function

$f(x)=a+\mathrm{e}^{bx+c}\,.$

Provide values of estimates of all regression coefficients including corresponding standard errors.

Hint: Try to check (by the means of graphical methods) whether the assumption of homoskedasticity holds and if necessary, use White's estimate (sandwich estimate) of covariance matrix to compute standard errors correctly.

4. See values (xi, yi) in the attached fileregrese3.csv. We want to fit the theoretical functional dependence, which in this case is a hyperbolic function, i.e. the function

$f(x)=a+\frac{1}{bx+c}\,.$

Plot a graph of observed values (in the form of error bars) and fitted function and provide brief interpretation. Perform regression diagnostics.

Bonus:   See values (xi, yi) in the attached fileregrese4.csv. We want to fit the theoretical functional dependence, which in this case is too complicated to be expressed in analytical form. Try to fit regression splines (with suitably chosen knots and suitable degree). Plot a graph of observed values and fitted function.

It is recommended to use statistical software R for all computations. Sample R script (with comments in code explaining syntax of \emph{R} programming language) may be helpful (in Czech only). .