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The probability density function of the exponential distribution is defined as

$$ f(x;\lambda)=\begin{cases} \lambda e^{-\lambda x} &\text{if } x \geq 0 \\ 0 & \text{if } x<0 \end{cases} $$

Its likelihood function is

$$ \mathcal{L}(\lambda,x_1,\dots,x_n)=\prod_{i=1}^n f(x_i,\lambda)=\prod_{i=1}^n \lambda e^{-\lambda x}=\lambda^ne^{-\lambda\sum_{i=1}^nx_i} $$

To calculate the maximum likelihood estimation I solved the equation

$$ \frac{d\ln\left(\mathcal{L}(\lambda,x_1,\dots,x_n)\right)}{d\lambda}\overset{!}{=}0 $$

for $\lambda$.

$$ \begin{align} \frac{d\ln\left(\mathcal{L}(\lambda,x_1,\dots,x_n)\right)}{d\lambda} &= \frac{d\ln\left(\lambda^ne^{-\lambda\sum_{i=1}^nx_i}\right)}{d\lambda} \\ &= \frac{d\ln\left(n\ln(\lambda)-\lambda\sum_{i=1}^n x_i\right)}{d\lambda} \\ &= \frac{n}{\lambda}-\sum_{i=1}^n x_i \end{align} $$

Finally we get $$\lambda = \frac{n}{\sum_{i=1}^n x_i}$$

I hope this is correct this far.

Where I am more uncertain is the proof for consistency.

I understand that to be consistent is in this case equivalent to to converge in probability to $\lambda$. So I have a hinch, that something like

$$ \lim_{n\to\infty}\mathbb{P}\left(\mathcal{L}(\lambda,x_1,\dots,x_n)-\lambda\right)=0 $$

will lead me to a solution.

Am I correct this far? If yes, how can I solve this? A hint would be great.


Update:


Using Didier Piau's and cardinal's hints I will try to show the consistensy by proving that $\frac{1}{\Lambda_n}\to\frac{1}{\lambda}$ for $n\to\infty$ where

$$ \Lambda_n=\frac{n}{\sum\limits_{k=1}^nX_k} $$

Since $E(X_1)=E\left(\lambda e^{-\lambda x_1}\right)=\frac{1}{\lambda}$ and all $X_i,i=1,2,\dots$ are independent the strong law of great numbers implies that

$$ P\left(\lim\sup_{n\to\infty}\left|\frac{1}{\Lambda_n}-\frac{1}{\lambda}\right|=0\right)=P\left(\lim\sup_{n\to\infty}\left|\frac{\sum\limits_{k=1}^nX_k}{n}-\frac{1}{\lambda}\right|=0\right)=1 $$

is true which implies convergence almost everywhere. This implies convergence in probability of $\Lambda_n$ against $\lambda$ which is equivalent to consistence.

Is this proof correct?

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2 Answers 2

up vote 4 down vote accepted

The computation of the MLE of $\lambda$ is correct.

The consistency is the fact that, if $(X_n)_{n\geqslant1}$ is an i.i.d. sequence of random variables with exponential distribution of parameter $\lambda$, then $\Lambda_n\to\lambda$ in probability, where $\Lambda_n$ denotes the random variable $$ \Lambda_n=\frac{n}{\sum\limits_{k=1}^nX_k}. $$ Thus, one is asked to prove that, for every positive $\varepsilon$, $\mathrm P(|\Lambda_n-\lambda|\geqslant\varepsilon)\to0$ when $n\to\infty$.

In the case at hand, it might be easier to prove the stronger statement that $\frac1{\Lambda_n}\to\frac1\lambda$ almost surely when $n\to\infty$. Hint: Law of large numbers.

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(+1) I think your answer and my comment were (nearly) simultaneous. I've deleted mine. –  cardinal Jan 23 '12 at 1:30
    
First, thank you for the explanation. So you mean I have to show, that $P\left(\lim\sup_{n\to\infty}\left|\frac{1}{\Lambda_n}-\frac{1}{\lambda}\right|=‌​0\right)=1$? (Strong law of great numbers.) Correct? –  Aufwind Jan 23 '12 at 1:49
    
Weak law, if I'm not mistaken. You need to show convergence in probability, not almost sure convergence. –  Michael Hardy Jan 23 '12 at 1:52
    
Aufwind: Yes, if you know this, you know that $\Lambda_n\to\lambda$ almost surely, hence you know that $\Lambda_n\to\lambda$ in probability, which is what you want. –  Did Jan 23 '12 at 1:53
    
@DidierPiau I tried another approach. Would you say, that is sufficient? Thanks for any of your efforts! –  Aufwind Jan 23 '12 at 5:55

$\hat\lambda= \frac{n}{\sum_{i=1}^n x_i}$ to be consistent estimator of $\lambda$ it should be Asymptotically

  1. Unbiased,
  2. and it's variance goes to zero.

Using $E\left\{ x\right\}=\frac{1}{\lambda}$ and $E\left\{ x^2\right\}=\frac{2}{\lambda^2}$ and the fact that $x_i$ are iid, we have

Condition 1: $\lim_{n\rightarrow \infty} E\{\hat\lambda - \lambda\}=0$ $$ \begin{aligned} E\left\{ \frac{n}{\sum_{i=1}^n x_i} \right\} - \lambda= \frac{n}{E\left\{\sum_{i=1}^n x_i \right\}} - \lambda= \frac{n}{nE\left\{ x\right\}} - \lambda= \frac{n}{nE\left\{ x\right\}} - \lambda= \lambda - \lambda=0 \end{aligned} $$

Condition 2: $\lim_{n\rightarrow \infty}E\left\{\left(\hat\lambda - E\{\hat\lambda\}\right)^2\right\}=0 $ $$ \begin{aligned} E\left\{\left(\hat\lambda - E\{\hat\lambda\}\right)^2\right\}&=E\left\{\hat\lambda^2\right\} - E\left\{\hat\lambda\right\}^2=E\left\{\frac{n^2}{(\sum_{i=1}^n x_i)(\sum_{i=1}^n x_i)}\right\} -\lambda^2 \\ &= E\left\{\frac{n^2}{\sum_{i,j=1}^n x_ix_j}\right\} -\lambda^2= E\left\{\frac{n^2}{\sum_{i=j} x_ix_j +\sum_{i\neq j}^n x_ix_j}\right\} -\lambda^2 \\ &= \frac{n^2}{nE\left\{x^2\right\}+ n(n-1) E\left\{x\right\}^2} -\lambda^2 = \frac{n}{\frac{2}{\lambda^2}+ \frac{n-1}{\lambda^2}} -\lambda^2 =\frac{n}{n+1}\lambda^2-\lambda^2\\ &\Rightarrow \lim_{n\rightarrow \infty} \frac{n}{n+1}\lambda^2-\lambda^2 =0 \end{aligned} $$

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