Instructions

• Read carefully the assignment instructions contained within r-ind-2022s1.pdf.

• Create a single .R file (name this solution.R) containing your solutions to Questions 1--5. Only the code contained within this file will be assessed.

• It is important that you name your functions and variables as instructed, as your solutions will be auto-graded using R.

• The code in your script file should be able to be run without any errors.

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  • Use comments where appropriate;
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R Individual Assignment

Probability density distribution (pdf) of the Pareto distribution:

$$ f(x|\alpha,\beta)= \begin{cases} \frac{\alpha\beta^{\alpha}}{x^{\alpha +1}} & x\geq \beta \\ 0 & x<\beta \end{cases} $$

Deviance function:

$$ D(\alpha,\beta|X)= -2\sum\limits_{i = 1}^{n}{logf(X_i|\alpha,\beta)} $$

Cumulative distribution function (cdf) :

$$ F(x|\alpha,\beta)=Pr(X\leq x)= \int_{-\infty }^{x}{f(u|\alpha,\beta)du} $$

Question 3

Here is the derivation that I calculated for $\hat{\alpha}$ :

$$ \frac{\delta D}{\delta \alpha} = \frac{\delta}{\delta \alpha} \left [-2\sum_{i = 1}^{n}logf(X_i|\alpha,\beta) \right] $$

$$ = -2\frac{\delta}{\delta \alpha} \left [\sum_{i = 1}^{n}logf(X_i|\alpha,\beta) \right] $$

$$ =-2 \frac{\delta}{\delta\alpha} \left [\sum_{i=1}^{n}log \frac{\alpha\beta^{\alpha}}{X_i^{\alpha+1}} \right] $$

$$ = -2 \frac{\delta}{\delta\alpha} \left [\sum_{i=1}^{n}[log(\alpha\beta^{\alpha}) - logX_i^{\alpha+1}] \right] $$

$$ = -2\frac{\delta}{\delta\alpha} \left [[log(\alpha\beta^{\alpha})-logX_1^{\alpha + 1}] + ... + [log(\alpha\beta^{\alpha})-logX_n^{\alpha + 1}] \right] $$

$$ = -2\frac{\delta}{\delta\alpha} \left [nlog(\alpha\beta^{\alpha})-[log(X_1^{\alpha+1})+...+log(X_n^{\alpha+1}) \right] $$

$$ = -2n \frac{\delta}{\delta \alpha}[log(\alpha \beta^{\alpha})] +2 \frac{\delta}{\delta \alpha} [log(X_1^{\alpha}\cdot X_1) + ...+ log(X_n^{\alpha}\cdot X_n)] $$

$$ = -2n \frac{\delta}{\delta \alpha}[log\alpha + \alpha log\beta] +2 \frac{\delta}{\delta \alpha} \left[ [log(X_1^{\alpha})+ log(X_1)]+...+ [log(X_n^{\alpha})+ log(X_n)] \right] $$

$$ = -2n \left[ \frac{1}{\alpha} + log\beta \right] +2[logX_1+...+logX_n] $$

$$ = 2 \left[ [logX_1+...+logX_n] -nlog\beta -\frac{n}{\alpha} \right] $$

$$ = 2 \left[ \left[log \left(\frac{X_1}{\beta} \right)\right] +...+ \left[log \left(\frac{X_1}{\beta} \right)\right] -\frac{n}{\alpha} \right] $$

$$ =2\left[ \sum_{i = 1}^{n}log \left(\frac{X_i}{\beta}\right) -\frac{n}{\alpha} \right] $$

$\hat{\alpha}$ is $\alpha$ when $\frac{\delta D}{\delta\alpha}=0$ and $\beta = \hat{\beta}$

$$ 2\left[ \sum_{i = 1}^{n}log \left(\frac{X_i}{\beta}\right) -\frac{n}{\alpha} \right] = 0 $$

$$ \frac{n}{\alpha} = \sum_{i = 1}^{n}log \left(\frac{X_i}{\beta}\right) $$

$$ \alpha = \frac{n} {\sum_{i= 1}^{n}log\left(\frac{X_i}{\beta}\right)} = \hat{\alpha} $$

Question 4

find $F$ from integration of $f$

$$ F(x|\alpha,\beta)=\int_{-\infty}^{x}{f(u|\alpha,\beta)du} $$

$$ =\int_{-\infty}^{x}{\frac{\alpha\beta^{\alpha}}{x^{\alpha +1}}}du $$

$$ =\alpha\beta^\alpha lim_{t \to -\infty} \int_{t}^{x}{u^{-(\alpha+1)}}du $$

$$ =\alpha\beta^\alpha lim_{t \to \beta} \int_{t}^{x}{u^{-(\alpha+1)}}du $$

$$ =\alpha\beta \left [\frac{u^{-\alpha}}{-\alpha} \right]_{u = \beta}^{u = x} $$

$$ =-\beta^{\alpha} \left [u^{-\alpha} \right]_{u = \beta}^{u = x} $$

$$ = -\beta^{\alpha} \left[x^{-\alpha} - \beta^{-\alpha}\right] $$

$$ = 1 - \left(\frac{\beta}{x}\right)^{\alpha} $$