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The following question is part (1/4) of a 2.30h written exam for the course "Probability and Statistics" in a school of engineering. So, although tricky and difficult (because the Professor is really demanding from his students), it should be solvable in a logical amount of time and with a logical amount of calculations.

Let $X_1, \ldots, X_n$ be a random sample (i.i.d. r.v.) from the exponential distribution $\exp(\lambda)$, where $\lambda$ is unknown. Let $M_n=\max\{X_1, \ldots, X_n\}$ with probability distribution function $$G(x)=(1-e^{-\lambda x})^{n}, \qquad x>0$$ and zero elsewhere.

Q1. Find the probability density function of $M_n$.

Q2. If $M_n$ is the only information that you have for $X_1,X_2,\ldots,X_n$, find the maximum likelihood estimator (MLE) $\hat{\lambda}_n$ of $\lambda$.

Q3. Using $(1+x)^n>1+nx$ (or any other way) prove that $\hat{\lambda}_n$ is consistent, i.e. that $P(| \hat{\lambda}_n-\lambda|>\epsilon)\longrightarrow0$, for $n\rightarrow \infty$

For Q1, I took the derivative of the cdf of $M_n$ which I found to be equal to $$g(x)=n\lambda e^{-\lambda x}(1-e^{-\lambda x})^{n-1}$$ (doublechecked with Wolfram|Alpha).

For Q2, I thought that the function I should maximize (with respect to $\lambda$) is $g(x)$ because that is my single observation from the sample of size $n$. If I understand the exercise correctly someone takes a sample of $n$ observations $X_1,X_2,\ldots X_n$ and tells me only their maximum $M_n$. Now, from this single information I have to calculate a MLE for $\lambda$. So, I will maximize the pdf of $M_n$ which is know my likelihood function, no? Is my mistake here?

However, if I took as $$L(x;\lambda)=g(x)$$ and $$l(x;\lambda)=\ln\left(L(x;\lambda)\right)=\ln\left(g(x)\right)=\ln(n)+\ln(\lambda)-\lambda x+(n-1)\ln(1-e^{-\lambda x})$$ Then, as usually, I calculated the derivative of $l(x;\lambda)$ and set it equal to $0$ $$\frac{d}{d\lambda}l(x;\lambda)=\frac{1}{\lambda}-x+(n-1)\frac{xe^{-\lambda x}}{1-e^{-\lambda x}}=0$$ which reduces to $$e^t=\frac{1-nt}{1-t}$$ where $t=\lambda x$. But I cannot solve this equation (called transcendental as someone told me).

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  • $\begingroup$ Is this an exam you are taking? Although, take a look at the 'self-study' tag. I think it's relevant to this question and you should add it to your post. $\endgroup$ – David Marx Mar 1 '14 at 15:36
  • $\begingroup$ @DavidMarx. Thanks for your comment. No, I am a private teacher who caters for students (see here: en.wikipedia.org/wiki/Frontistirio what I mean - no job too small) and usually I receive students from that engineering school or from other schools, to solve their exams (not on the time of the exam, but afterwards) in order to prepare them for the next exam. But during the last 1-2 years there is this Professor who gives the most difficult exams ever, one of which is this one!!! And they are not even math students... $\endgroup$ – Jimmy R. Mar 1 '14 at 15:49
  • $\begingroup$ From the language of your post, it sounds like you have already calculated the pdf for $M_n$. If this is the case, you should add it to your post. It's possible you have calculated the pdf incorrectly, and if not it will save others time by not having to derive it themselves. $\endgroup$ – David Marx Mar 1 '14 at 15:56
  • $\begingroup$ Yes, indeed ok I will post it. $\endgroup$ – Jimmy R. Mar 1 '14 at 15:57
  • $\begingroup$ See full answer there. $\endgroup$ – Did Nov 24 '14 at 10:08
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Since you are a tutor, any knowledge is always for a good cause. So I will provide some bounds for the MLE.

We have arrived at

$$(1-\lambda x_{(n)})e^{\lambda x_{(n)} } + \lambda n x_{(n)} - 1 = 0$$ with $x_{(n)}\equiv M_n$. So

$$(1-\hat \lambda x_{(n)})e^{\hat \lambda x_{(n)}} = 1-\hat \lambda x_{(n)}n $$ Assume first that $1-\hat \lambda x_{(n)} >0$. Then we must also have $1-\hat \lambda x_{(n)}n>0$ since the exponential is always positive. Moreover since $x_{(n)}, \hat \lambda > 0\Rightarrow e^{\hat \lambda x_{(n)}}>1$. Therefore we should have

$$\frac {1-\hat \lambda x_{(n)}n}{1-\hat \lambda x_{(n)}}>1 \Rightarrow \hat \lambda x_{(n)}>\hat \lambda x_{(n)}n$$ which is impossible. Therefore we conclude that

$$\hat \lambda >\frac 1{x_{(n)}},\;\; \hat \lambda = \frac c{x_{(n)}}, \;\; c>1$$

Inserting into the log-likelihood we get

$$\ell(\hat\lambda(c)\mid x_{(n)}) = \log \frac c{x_{(n)}} + \log n - \frac c{x_{(n)}} x_{(n)} + (n-1) \log (1 - e^{-\frac c{x_{(n)}} x_{(n)}})$$

$$= \log \frac n{x_{(n)}} + \log c - c + (n-1) \log (1 - e^{-c})$$

We want to maximize this likelihood with respect to $c$. Its 1st derivative is

$$\frac{d\ell}{dc}=\frac 1c -1 +(n-1)\frac 1{e^{c}-1}$$

Setting this equal to zero, we require that

$$e^{c}-1 - c\left(e^{c}-1\right)+(n-1)c =0$$

$$\Rightarrow \left(n-e^c\right)c = 1-e^c$$

Since $c>1$ the RHS is negative. Therefore we must also have $n-e^c <0 \Rightarrow c > \ln n$. For $n\ge 3$ this provides a tighter lower bound for the MLE, but it doesn't cover the $n=2$ case, so

$$\hat \lambda > \max \left\{\frac 1{x_{(n)}}, \frac {\ln n}{x_{(n)}}\right\}$$

Moreover (for $n\ge 3$) rearranging the 1st-order condition we have that

$$c= \frac{e^c-1}{e^c-n} > \ln n \Rightarrow e^c -1 > e^c\ln n -n\ln n $$

$$\Rightarrow n\ln n-1>e^c(\ln n -1) \Rightarrow c< \ln{\left[\frac{n\ln n-1}{\ln n -1}\right]}$$ So for $n\ge 3$ we have that

$$\frac 1{x_{(n)}}\ln n < \hat \lambda < \frac 1{x_{(n)}}\ln{\left[\frac{n\ln n-1}{\ln n -1}\right]}$$

This is a narrow interval, especially if $x_{(n)}\ge 1$. For example (truncated at 3d digit )

$$\begin{align} n=10 & &\frac 1{x_{(n)}}2.302 < \hat \lambda < \frac 1{x_{(n)}}2.827\\ n=100 & & \frac 1{x_{(n)}}4.605 < \hat \lambda < \frac 1{x_{(n)}}4.847\\ n=1000 & & \frac 1{x_{(n)}}6.907 < \hat \lambda < \frac 1{x_{(n)}}7.063\\ n=10000 & & \frac 1{x_{(n)}}9.210< \hat \lambda < \frac 1{x_{(n)}}9.325\\ \end{align}$$

Numerical examples indicate that the MLE tends to be equal to the upper bound, up to second decimal digit.

ADDENDUM: A CLOSED FORM EXPRESSION
This is just an approximate solution (it only approximately maximizes the likelihood), but here it is:
manipulating the 1st-order condition we want to have

$$\lambda = \frac 1{x_{(n)}}\ln \left[\frac {\lambda x_{(n)}n -1}{\lambda x_{(n)} -1}\right]$$

Now, one can show (see for example here) that

$$E[X_{(n)}] = \frac {H_n}{\lambda},\;\; H_n = \sum_{k=1}^n\frac 1k$$

Solving for $\lambda$ and inserting into the RHS of the implicit 1st-order condition, we obtain

$$\lambda = \frac 1{x_{(n)}}\ln \left[\frac {nH_n\frac {x_{(n)}}{E[X_{(n)}]} -1}{ H_n\frac {x_{(n)}}{E[X_{(n)}]} -1}\right]$$

We want an estimate of $\lambda$, given that $X_{(n)}=x_{(n)}$, $\hat \lambda \mid \{X_{(n)}=x_{(n)}\}$. But in such a case, we also have $E[X_{(n)}\mid \{X_{(n)}=x_{(n)}\}] =x_{(n)}$. this simplifies the expression and we obtain

$$\hat \lambda = \frac 1{x_{(n)}}\ln \left[\frac {nH_n -1}{ H_n -1}\right]$$

One can verify that this closed form expression stays close to the upper bound derived previously, but a bit less than the actual (numerically obtained) MLE.

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  • $\begingroup$ Aleco, impressive approach! $\endgroup$ – Jimmy R. Mar 3 '14 at 23:41
  • $\begingroup$ Thanks, although it leaves the main issue unresolved. Usually professors that want to show off and tough, don't make mistakes. Moreover this exercise asks also to prove consistency, which almost surely implies the existence of a closed form solution (although not necessarily)... The question still bugs me... maybe there is a probabilistic argument that provides a closed form for the MLE, since the mathematical approach doesn't ? Still thinking about it... $\endgroup$ – Alecos Papadopoulos Mar 4 '14 at 1:21
  • $\begingroup$ Yes, I do not believe that there is a mistake neither. A closed form (which should be between your bounds) is likely to exist. I am still thinking of it too. If I find the solutions (if they are distributed to the students) or if I come up with a solution (unlikely, I have given so much thought and nothing) I will let you know! Ta leme, Aleco! $\endgroup$ – Jimmy R. Mar 4 '14 at 1:27
  • $\begingroup$ See full answer there. $\endgroup$ – Did Nov 24 '14 at 10:08
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Q1. Trivial: differentiate $G$ to obtain $$f_{M_n}(x) = \lambda n e^{-\lambda x}(1-e^{-\lambda x})^{n-1}, \quad x > 0.$$

Q2. The likelihood of $\lambda$ given the single observation $M_n = x$ is $L(\lambda \mid x) = f_{M_n}(x)$, consequently the log-likelihood is $$\ell(\lambda \mid x) = \log \lambda + \log n - \lambda x + (n-1) \log (1 - e^{-\lambda x}).$$ Differentiation with respect to $\lambda$ gives $$\frac{d\ell}{d\lambda} = \frac{1}{\lambda} - x + \frac{(n-1) x e^{-\lambda x}}{1 - e^{-\lambda x}},$$ which we require to be zero; i.e., $$(1-\lambda x)e^{\lambda x} + n \lambda x - 1 = 0.$$ Such an equation does not, to the best of my knowledge, admit an elementary closed form solution for $\lambda$.

I would very much like to see what this professor's idea of $\hat\lambda_n$ is, because I can almost assure you that whatever he thinks it is, he is wrong.

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  • $\begingroup$ Thanks a lot. I have found the same result and cannot proceed. I am also infuriated with his exams because he does that repeatedly. Thanks for your time and your assurance. $\endgroup$ – Jimmy R. Mar 1 '14 at 16:28
  • $\begingroup$ If you find it infuriating, just imagine what his students are feeling.... $\endgroup$ – heropup Mar 1 '14 at 16:29
  • $\begingroup$ Hahaha, from what they tell me they are simply sad... He is showing off his knowledge to make an impression out of him, otherwise this is a ridiculous degree of difficulty for engineers (and probably also wrong as you say). $\endgroup$ – Jimmy R. Mar 1 '14 at 16:32
  • $\begingroup$ @Stefanos If by any chance you learn of his suggested solution, please let us know. I am very curious as well. $\endgroup$ – JohnK Mar 1 '14 at 16:39
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    $\begingroup$ If $\lambda x$ is small, you can Taylor expand $e^{\lambda x}\approx 1+\lambda x$ and solve the second degree equation to get an approximate MLE $\hat{\lambda}=n/x$. $\endgroup$ – Zen Mar 1 '14 at 17:33

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