## Introduction

Rcpp 实现 Root-finding 算法。

## Notes

There are two questions on this assignment. The second requires you to have considered Lab 8 in the course material.

This assignment is worth 20% of the total grade.

Ensure your code is well-commented. You will be penalized for submitting code that is not well-commented.

Only the following files should be submitted:

Any .cpp files that form part of your solutions
A single R script file named in the format YourName assignment.R containing all of the R code required for this assignment (submit a single R script only!)
A single pdf document containing any other required information (e.g. descriptions of code implementations), named in the format YourName assignment.pdf.

## Root-finding algorithms

A root of a function `g(θ)` is a solution to the equation `g(θ) = 0` . Root-finding is an important computational problem that arises commonly across many disciplines. For example, in statistics, root-finding can be used in maximum likelihood estimation. Here you are required to find a maximum likelihood estimate for the Cauchy distribution with the help of root-finding algorithms implemented in C++ with Rcpp.

The Cauchy distribution with unknown location parameter `θ` and scale parameter set to `γ = 1` is a continuous probability distribution, with density function.

For n independent observations x1, x2, … , xn from this Cauchy distribution, the log likelihood is calculated.

Consider the following numeric vector in R storing observations generated from a Cauchy distribution with location parameter `θ` and scale parameter `γ= 1` :

```x <− c (12.262307, 10.281078, 10.287090, 12.734039,
11.731881,  8.861998, 12.246509, 11.244818,
9.696278, 11.557572, 11.112531, 10.550190,
9.018438, 10.704774,  9.515617, 10.003247,
10.278352,  9.709630, 10.963905, 17.314814)```

A researcher wishes to estimate the maximum likelihood estimate `θ` for the Cauchy distribution from which the data in x were generated.

A maximum likelihood estimate for `θ` can be found by differentiating the log likelihood, equating to zero, and solving for `θ` i.e. we can find a maximum likelihood estimate for the above Cauchy distribution by finding a root of `l(θ)` , the first derivative of the log likelihood. Taking the first derivative and equating to zero, gives the following equation.

A maximum likelihood estimate can be obtained by solving the equation `l(θ) = 0` for `θ` i.e. by finding a root of the function `l(θ)` . There are several numerical optimization algorithms available for finding the roots of continuous univariate functions such as `l(θ)` , including the:

Bisection method
Newton’s method
Secant Method

For this assignment, implement each of the above three root finding algorithms in C++ with Rcpp, to find the MLE of `θ` for the data x.

You will need to research the algorithm for each method. They are widely available from any online search or within the accompanying PDF (with pseudo-code). Call each function bisectRcpp, newtonRcpp, and secantRcpp, respectively. The function calls (from a wrapper using a cxxfunction) should take the form:

```bisectRcpp(x,a=*,b=*,n=1000,tol=0.00000001)
newtonRcpp(x,    b=*,n=100 ,tol=0.00000001) secantRcpp(x,a=*,b=*,n=1000,tol=0.00000001)```

where a and b represent initial values for the algorithms, n is the number of iterations in your algorithm, and tol is the tolerance – the level of accuracy for terminating before n is reached.

Each function should take the vector x as input from R (along with any other required inputs), and return the maximum likelihood estimate to R.
Ensure that all code submitted is well commented
Briefly describe each implementation in your submitted pdf document. This must be self-contained, i.e. do not refer to your commented code.
Evaluate and discuss the behaviour of each implemention with different starting values. Consider the following:

`bisectRcpp (a,b)= {(9,11), (0,30), (-10,50), (15,50)} newtonRcpp b= {11, 9, 13} secantRcpp (a,b)= {(10,12), (7,12), (8,13), (8,14)} where some will work and others will fail.`

Benchmark your implementations against each other by the following:

```# Benchmark the implementations against each other
benchmark ( bisectRcpp( x, a=9, b=11, n=1000, tol=0.00000001 ),
secantRcpp( x, a=8, b=11, n=1000, tol=0.00000001 ),
newtonRcpp( x, b=11, n=100, tol=0.00000001 ),
order = "relative" )```

Discuss their advantages and limitations. Comment on the suitability and efficiency of each implementation.

Note: the second derivative of the log likelihood is.

## Welford’s variance algorithm with Rcpp – with missing values

Modify Welford’s variance algorithm (Lab 8) such that it can handle missing values.

(本文出自 csprojectedu.com ，转载请注明出处)