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Can anyone solve for $t$ the equation: $$ e^t=\frac{1-nt}{1-t} $$

with $n \in \mathbb N$ (known) and $t>0$. Online solvers give an answer only for specific values of $n$, but I need a general formula for $t$.

Thanks.

egreg
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Jimmy R.
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  • the best you can do is for small t and expand maybe. But why do you need a formula? – Chinny84 Feb 11 '14 at 20:22
  • These kinds of equations are generally referred to as transcendental equations and they basically never have closed-form solutions, unless a special function has been developed specifically for that purpose. The best you are going to be able to get is some approximate solutions. – Kevin Driscoll Feb 11 '14 at 20:23
  • @Chinny84 Thanks for your response. I need a formula for t because it is needed in the subsequent question of the exercise. This equation is the derivative (set equal to 0) of a function for which I need to specify the argmax and then use it for further calculations. – Jimmy R. Feb 11 '14 at 20:30
  • @KevinDriscoll. Thanks for your response. – Jimmy R. Feb 11 '14 at 20:32
  • @Stefanos if you give the entire question included the derivative to give a bit of context. Since I doubt the excerise would call for you to solve this analytically. If indeed the question leads you to this equation, it would be not to solve it but use the functional form maybe..but show the rest of the question then we can help you :) – Chinny84 Feb 11 '14 at 20:38
  • @Chinny84. Okay, there it is. I have a sample $X_1 ,\ldots,X_n$ from the exponential distribution $\exp(\lambda)$ and the only information I know is $M_n=\max{X_1,\ldots,X_n}$. I have to find the maximum likelihood estimator of $\lambda$ based only(!) on $M_n$. So I took the probability density function of $M_n$, which is equal to $n\lambda\exp(-\lambda x)\cdot(1-\exp(-\lambda x))^{(n-1)}$, for $x>0$ and I tried to maximize it. I believe that in that case the probability function of $M_n$ is equal to the likelihood function itself. So I differentiated and that is the result. – Jimmy R. Feb 11 '14 at 20:47
  • @Chinny84. In the susequent question I have to show that this maximum likelihood estimator is consistent, so I need a formula for it. Thanks – Jimmy R. Feb 11 '14 at 20:51
  • "I have to show that this maximum likelihood estimator is consistent, so I need a formula for it" Not quite, fortunately, as shown in the answer to your other question. – Did Nov 24 '14 at 10:17

1 Answers1

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In this special case, you can guess t = 0 as a solution.

This solution is, moreover, independent from n.

Perhaps the solution can be found by expanding the right side of the equation. It is a geometric series (up to a constant).

The expansion of the right side is

$$1 + (1-n) (t+t^2+t^3+t^4+...)$$

Peter
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  • Thanks, true, sorry I have forgotten to mention that t must be positive. The expantion is probably a good idea. – Jimmy R. Feb 11 '14 at 20:39
  • This is consistent that he only found a solution for certain n i.e. 0<n<1 since the right hand side is $1+t+t^{2} ...$. – Chinny84 Feb 11 '14 at 20:57
  • It also found solutions for n's larger than 1. I expanded also the left side, but I did not come any further. Thanks, though for the expantion idea and the formula for the right side. Probably I have a mistake in the derivation of this equation. – Jimmy R. Feb 11 '14 at 21:04
  • Or, perhaps, it is not important to solve the equation to prove the consistency. – Peter Feb 11 '14 at 21:05
  • Do you think that the likelihood function (that should be maximized) is the probability density function of the ordered statistic $M_n$? According to the exercise the maximum $M_n$ is the only information that we get from the sample. @Peter Thanks for your responses. – Jimmy R. Feb 11 '14 at 21:10