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I have the pdf $$f(y ; \theta) = \frac{1}{\theta} \exp( \frac{-y}{\theta}), \ y > 0$$ and I'm supposed to determine if the following two estimators are unbiased or not: $ \hat \theta = nY_{min} $ and $ \hat \theta = \frac{1}{n}\sum_{i=1}^n Y_i $. I'm running into some problems because when I try to find the expected value of $ Y_{min} $ and $ Y_i $, the integral is undefined. |
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Please see the following Wikipedia article and it shows that both the integral exist. Also, if you look lower in the article, you will also see that both the estimators are unbiased. $Y_{min}$ is also exponentially distributed with mean parameter $\frac {\theta} {n}$ Therefore, $E[nY_{min}]=n\times\frac {\theta} {n} = \theta$. Similarly, $E[\frac{1}{n}\sum_{i=1}^n Y_i] = \frac{n \theta}{n}=\theta$ I am sure both the integrals exists |
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The fact that the sample mean is an unbiased estimator is obtained combining these two facts: 1. The sample mean is an unbiased estimator of the population mean 2. The population mean is equal to theta |
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