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I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the samples are similar to the following sets \begin{align*} &S_{1}=\{0.1,0.3,0.3,0.7\}\\ &S_{2}=\{0.3,0.3,0.9\}\\ &S_{3}=\{0.1,0.1,0.3,0.3,0.5,0.7,0.9\}\\ &S_{4}=\{0.3,0.5\}\\ &...\\ &S_{i}=\{0.3\}\\ &...\\ &S_{j}=\{0.1,0.1\}\\ &...\\ &S_{N-3}=\{0.5,0.5,0.7,0.7,0.7,0.7,0.9,0.9\}\\ &S_{N-2}=\{0.3,0.3,0.5\}\\ &S_{N-1}=\{0.5,0.5,0.7\}\\ &S_{N}=\{0.1,0.5,0.7,0.9,0.9,0.9,0.9\} \end{align*} Each of them is a sample (containing observation(s)), related to a separate Beta distributed random variable. For estimating the corresponding Beta random variables, I take advantage of the well-known estimators, introduced in general resources such as Wikipedia: \begin{align*} &\text{Method of Moments}:\\ &\bar{x}=\frac{1}{N}\sum_{i=1}^{N}X_i\\ &\bar{v}=\frac{1}{N-1}\sum_{i=1}^{N}{(X_i-\bar{x})^2}\\\\ &\hat{\alpha}=\bar{x}\left(\frac{\bar{x}(1-\bar{x})}{\bar{v}}-1\right),\text{if }\bar{v}<\bar{x}(1-\bar{x})\\ &\hat{\beta}=(1-\bar{x})\left(\frac{\bar{x}(1-\bar{x})}{\bar{v}}-1\right),\text{if }\bar{v}<\bar{x}(1-\bar{x})\\\\ &\text{Maximum Likelihood Estimator}\\ &\hat{G}_X=\prod_{i=1}^N{X_i^{\frac{1}{N}}}\\ &\hat{G}_{1-X}=\prod_{i=1}^N{(1-X_i)^{\frac{1}{N}}}\\\\ &\hat{\alpha}\approx\frac{1}{2}+\frac{\hat{G}_X}{2\left(1-\hat{G}_{X}-\hat{G}_{1-X}\right)}\text{if }\hat{\alpha}>1\\ &\hat{\beta}\approx\frac{1}{2}+\frac{\hat{G}_{1-X}}{2\left(1-\hat{G}_{X}-\hat{G}_{1-X}\right)}\text{if }\hat{\beta}>1 \end{align*} Most of the mentioned random variables are estimated by the above techniques. However, in (non-common) situations like $S_i$ or $S_j$, in which the variance of the sample is zero, the well-known estimators fail in estimation (having zero in denominator).

Briefly speaking, the question is how to estimate the parameters of the Beta distribution, when the sample variance is zero.

hossayni
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  • What do you think a sample with variance zero looks like? – Did Jul 28 '15 at 12:56
  • And what is the probability that a Beta distribution produces three times the same value? – Did Jul 28 '15 at 12:58
  • The point is that in situations where assuming a Beta distribution is sensible, this will never happen. The situation you describe seems especially ill-suited to Beta distributions. – Did Jul 28 '15 at 13:01
  • Sorry but your question asks about one-point samples. If you have a different situation in mind, please modify the question. – Did Jul 28 '15 at 13:12
  • The question is edited and the comments are removed. – hossayni Jul 28 '15 at 14:25

1 Answers1

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There's a systematic reason why the estimators don't work for zero variance. The mode is at $(\alpha-1)/(\alpha+\beta-2)$; you can use arbitrarily high values of $\alpha$ and $\beta$ to attain a given mode, and the higher you make them, the higher the probability density at the mode. So there's no pair of parameters with maximum likelihood in this case. You'd need a prior that decays sufficiently quickly at high $\alpha$ and $\beta$ values to make any sense out of such a situation.

joriki
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  • I agree that estimators like MLE are not inherently proposed for zero-varianced samples. I also, have tried to provide such estimations like what you proposed. However, those estimations were very tastes-base. I liked to find another estimator that can propose some systematic estimation. My project is a scientific one and tastes-base estimations are not suggested for scientific papers.\begin{align}&\text{I apologize, because, my time is rigidly restricted and I am leaving my lab. So, kindly thanks for your reply, in advance.}\end{align} – hossayni Jul 28 '15 at 15:01
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    @hossayni: I'm afraid everything you can do in this situation will be "taste-based" (nice rhyme :-). You simply don't have any information to make an informed estimate. You could take a look at this answer for another example of a minimum of information being needed to overcome initial ignorance. My suggestion would be: a) Rethink whether this setup is the best possible. b) If the answer is yes, and you really, really need these estimates, try to come up with a sensible prior (e.g. from earlier trials). – joriki Jul 28 '15 at 15:07
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    P.S.: I don't know how you managed to change the font in that comment, but it looks like it messed up the formatting -- the comments are bleeding into the sidebar (at least in my browser). – joriki Jul 28 '15 at 15:10
  • Sorry for delay considering the massive scientific burden on my shoulder these days. Thank you very much for the answer. However, I hope that among several miscellaneous existing estimators, one of them can assist me in these zero-variance cases. – hossayni Jul 30 '15 at 13:02
  • About the initial ignorance, it is not always the problem. For example in the cases which my observation set is $S_j={0.1,0.1}$ (2 observations) the estimator cannot estimate whereas they are able to estimate the observation set $S_4={0.3,0.5}$ (again 2 observation). It means that (at least in those cases) the problem is more distribution of the observed samples rather than ignorance itself. – hossayni Jul 30 '15 at 13:12
  • @hossayni: Yes, I didn't mean to imply otherwise. – joriki Jul 30 '15 at 13:13
  • About the style, since it almost did not occurred meaningful changes in my display I did not understand. But, now, unfortunately there is no option for edit. – hossayni Jul 30 '15 at 13:14