Question

Consider randomly selecting \({\rm{n}}\) segments of pipe and determining the corrosion loss (mm) in the wall thickness for each one. Denote these corrosion losses by \({{\rm{Y}}_{\rm{1}}}{\rm{,}}.....{\rm{,}}{{\rm{Y}}_{\rm{n}}}\). The article “A Probabilistic Model for a Gas Explosion Due to Leakages in the Grey Cast Iron Gas Mains” (Reliability Engr. and System Safety (\({\rm{(2013:270 - 279)}}\)) proposes a linear corrosion model: \({{\rm{Y}}_{\rm{i}}}{\rm{ = }}{{\rm{t}}_{\rm{i}}}{\rm{R}}\), where \({{\rm{t}}_{\rm{i}}}\) is the age of the pipe and \({\rm{R}}\), the corrosion rate, is exponentially distributed with parameter \({\rm{\lambda }}\). Obtain the maximum likelihood estimator of the exponential parameter (the resulting mle appears in the cited article). (Hint: If \({\rm{c > 0}}\) and \({\rm{X}}\) has an exponential distribution, so does \({\rm{cX}}\).)

Short answer

The maximum likelihood estimator is \({\rm{\hat \lambda = }}\frac{{\rm{n}}}{{\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {{{\rm{Y}}_{\rm{j}}}} {\rm{/}}{{\rm{t}}_{\rm{j}}}}}\).

Step-by-step solution

Define exponential function

A function that increases or decays at a rate proportional to its present value is called an exponential function.

Explanation

Given a set of random variables,

\({{\rm{Y}}_{\rm{i}}}{\rm{ = }}{{\rm{t}}_{\rm{i}}}{\rm{R}}\)

where\({{\rm{t}}_{\rm{i}}}\)is an integer and\({\rm{R}}\)is an exponentially distributed random variable with parameters\({\rm{\lambda }}\). The\({{\rm{Y}}_{\rm{i}}}\)cdf is,

\(\begin{array}{c}{{\rm{F}}_{{{\rm{Y}}_{\rm{i}}}}}{\rm{(y) = P}}\left( {{{\rm{Y}}_{\rm{i}}} \le {\rm{y}}} \right)\\{\rm{ = P}}\left( {{{\rm{t}}_{\rm{i}}}{\rm{R}} \le {\rm{y}}} \right)\\{\rm{ = P}}\left( {{\rm{R}} \le \frac{{\rm{y}}}{{{{\rm{t}}_{\rm{i}}}}}} \right)\\{\rm{ = }}{{\rm{F}}_{\rm{R}}}\left( {\frac{{\rm{y}}}{{{{\rm{t}}_{\rm{i}}}}}} \right)\\{\rm{ = 1 - exp}}\left\{ {{\rm{ - \lambda }}\frac{{\rm{y}}}{{{{\rm{t}}_{\rm{i}}}}}} \right\}\\{\rm{ = 1 - exp}}\left\{ {{\rm{ - }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{i}}}}}{\rm{y}}} \right\}{\rm{,y}} \ge {\rm{0}}\end{array}\)

For\({\rm{y < 0}}\), and is zero. As a result,\({{\rm{Y}}_{\rm{i}}}\); has a parametric exponential distribution.

\({{\rm{\lambda }}_{\rm{i}}}{\rm{ = }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{i}}}}}\)

Allow joint pdf or pmb for random variables \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\).

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right){\rm{, n,m}} \in {\rm{N}}\)

where\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)are unknown parameters. The likelihood function is defined as a function of parameters\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)where function f is a function of parameter. The maximum likelihood estimates (mle's), or values\(\widehat {{{\rm{\theta }}_{\rm{i}}}}\)for which the likelihood function is maximised, are the maximum likelihood estimates,

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{{\rm{\hat \theta }}}_{\rm{1}}}{\rm{,}}{{{\rm{\hat \theta }}}_{\rm{2}}}{\rm{, \ldots ,}}{{{\rm{\hat \theta }}}_{\rm{m}}}} \right) \ge {\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right)\)

As, \({\rm{i = 1,2, \ldots ,m}}\) for every \({{\rm{\theta }}_{\rm{i}}}\). Maximum likelihood estimators are derived by replacing \({{\rm{X}}_{\rm{i}}}\) with \({{\rm{x}}_{\rm{i}}}\).

Evaluating the maximum likelihood estimators

Because of the independence, the likelihood function becomes,

\(\begin{array}{c}{\rm{f}}\left( {{{\rm{y}}_{\rm{1}}}{\rm{,}}{{\rm{y}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{y}}_{\rm{n}}}{\rm{;\lambda }}} \right){\rm{ = }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{1}}}}}{\rm{exp}}\left\{ {{\rm{ - }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{1}}}}}{{\rm{y}}_{\rm{1}}}} \right\}{\rm{ \times }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{2}}}}}{\rm{exp}}\left\{ {{\rm{ - }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{2}}}}}{{\rm{y}}_{\rm{2}}}} \right\}{\rm{ \times \ldots \times }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{n}}}}}{\rm{exp}}\left\{ {{\rm{ - }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{n}}}}}{{\rm{y}}_{\rm{n}}}} \right\}\\{\rm{ = }}{{\rm{\lambda }}^{\rm{n}}}\prod\limits_{{\rm{i = 1}}}^{\rm{n}} {\frac{{\rm{1}}}{{{{\rm{t}}_{\rm{i}}}}}} {\rm{ \times exp}}\left\{ {{\rm{ - }}\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{i}}}}}{{\rm{y}}_{\rm{i}}}} \right\}\\{\rm{ = }}{{\rm{\lambda }}^{\rm{n}}}\left( {\prod\limits_{{\rm{i = 1}}}^{\rm{n}} {\frac{{\rm{1}}}{{{{\rm{t}}_{\rm{i}}}}}} } \right){\rm{ \times exp}}\left\{ {{\rm{ - }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{j}}}}}} {{\rm{y}}_{\rm{j}}}} \right\}\\{\rm{ = }}{{\rm{\lambda }}^{\rm{n}}}{\rm{ \times p \times exp}}\left\{ {{\rm{ - \lambda }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} } \right\}\end{array}\)

Where,\({\rm{p = }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{\rm{\lambda }}}{{{{\rm{t}}_{\rm{j}}}}}} \).

Look at the log likelihood function to determine the maximum.

\(\begin{array}{c}{\rm{lnf}}\left( {{{\rm{y}}_{\rm{1}}}{\rm{,}}{{\rm{y}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{y}}_{\rm{n}}}{\rm{;\lambda }}} \right){\rm{ = ln}}\left( {{{\rm{\lambda }}^{\rm{n}}}{\rm{ \times p \times exp}}\left\{ {{\rm{ - \lambda }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} } \right\}} \right)\\{\rm{ = n \times ln\lambda + lnp - \lambda }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} \end{array}\)

The maximum likelihood estimator is generated by taking the derivative of the log likelihood function in regard to\({\rm{\lambda }}\)and equating it to\({\rm{0}}\).

As a result, the derivative,

\(\begin{array}{c}\frac{{\rm{d}}}{{{\rm{d\lambda }}}}{\rm{f}}\left( {{{\rm{y}}_{\rm{1}}}{\rm{,}}{{\rm{y}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{y}}_{\rm{n}}}{\rm{;\lambda }}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{d\lambda }}}}\left( {{\rm{n \times ln\lambda + lnp - \lambda }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} } \right)\\{\rm{ = }}\frac{{\rm{n}}}{{\rm{\lambda }}}{\rm{ + 0 - }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} \\{\rm{ = }}\frac{{\rm{n}}}{{\rm{\lambda }}}{\rm{ - }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} \end{array}\)

As a result, solving equation provides the maximum likelihood estimator.

\(\begin{array}{c}\frac{{\rm{n}}}{{{\rm{\hat \lambda }}}}{\rm{ - }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} {\rm{ = 0}}\\\frac{{\rm{n}}}{{{\rm{\hat \lambda }}}}{\rm{ = }}\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {\frac{{{{\rm{y}}_{\rm{j}}}}}{{{{\rm{t}}_{\rm{j}}}}}} \end{array}\)

for\({\rm{\hat \lambda }}\). Hence, the maximum likelihood estimator is,

\({\rm{\hat \lambda = }}\frac{{\rm{n}}}{{\sum\limits_{{\rm{j = 1}}}^{\rm{n}} {{{\rm{Y}}_{\rm{j}}}} {\rm{/}}{{\rm{t}}_{\rm{j}}}}}\).