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Suppose $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ are independent exponential r.v., where the density of $X_i$ is $f_i(x)= \lambda_i\theta \exp(-\lambda_i \theta x_i)$ for $x\geq0$, while the density of $Y_i$ is $g_i(x) = \lambda_i \exp(-\lambda_i x_i)$ for $x\geq0$.

Find the MLE of theta (based on $X_1,\ldots,X_n,Y_1,\ldots,Y_n$).

Find the MLEs of $\lambda_i$ for each $i$.

Michael Hardy
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MGF01
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  • The problem is that I don`t know how to find each lamda_i, and I for theta I don´t know if I need to multiply both densit functions, or if I need the MLEs of the lamda_i´s – MGF01 Feb 18 '18 at 20:44

1 Answers1

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$$ L = \prod_{i=1}^n\lambda_i\theta \exp(-\lambda_i \theta x_i)\lambda_i \exp(-\lambda_i y_i) = \theta^n \left( \prod_{i=1}^n \lambda_i^2\right) \exp\left( -\theta \sum_{i=1}^n \lambda_i x_i \right) $$ \begin{align} \log L & = n\log\theta + 2\sum_{i=1}^n \log\lambda_i - \theta\sum_{i=1}^n \lambda_i x_i \\[10pt] \frac\partial {\partial\theta} \log L & = \frac n \theta -\sum_{i=1}^n \lambda_i x_i = 0 \text{ if and only if } \theta = \frac n {\sum_{i=1}^n \lambda_i x_i} \tag 1 \\[10pt] \frac \partial {\partial\lambda_i} \log L & = \frac 2 {\lambda_i} -\theta x_i = 0 \text{ if and only if } \lambda_i = \frac 2 {\theta x_i} \tag 2 \end{align} Maybe tomorrow I'll say something about solving $(1)$ and $(2)$ for $\theta$ and $\lambda_i.$

Michael Hardy
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