Point estimation is the application of an estimator to the data in order to learn about a certain population parameter.

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UMVUE for pareto distribution

Let $X_1,..X_n$ random sample with $f(x;\theta,a)=\frac{\theta}{a}(\frac{a}{x})^{(\theta+1)}I_{(a,\infty)}(x),a>0,\theta>0$. Find the UMVUE for $\theta$ when $a$ is fixed. My attempt ...
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35 views

Cramer-Rao Lower Bound

Let $X_1,..,X_n$ be an iid sample of $N(0,\sigma^2)$. Find an unbiased estimator of $\sigma^2$ and its lower bound. I found that $$\hat{\sigma}^2 = \sum_{i=1}^{n} X_i^2$$ is an unbiased ...
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23 views

Unbiased estimator and variance

A random sample of n people are asked whether they are against smoking or not. Suppose x are against smoking. What is the distribution of the random variable X (number of those against smoking). State ...
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46 views

How many samples needed to approximate true mean?

If I have some arbitrary random variable with true mean $\mu$ how many samples from its distribution do I need to take such that the empirical mean $x$ approximates $\mu$ within an error of less than ...
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16 views

Factorization Theorem for sufficient statistics

I have a question on factorization theorem for sufficient statistics. I understand that to state whether or not a sufficient is statistic, you must factor the likelihood function of the pdf into two ...
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223 views

Shrunken $r$ vs unbiased $r$: estimators of $\rho$

There has been some confusion in my head about two types of estimators of the population value of Pearson correlation coefficient. A. Fisher (1915) showed that for bivariate normal population ...
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1answer
56 views

Self Study: ML Parameter Estimates — do I need numerical maximization?

I have a particular PDF with two parameters, specified as: $$\alpha \beta e^{-\beta x}(1 - e^{-\beta x})^{\alpha - 1}, \alpha > 0, \beta > 0, x_i > 0$$ Given a random iid sample $(x_1, ...
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14 views

Determining clustering words

I'm looking for an alternative to PMI for the following problem: I have a set of $n$ classes of text corpuses, and I'm trying to find the keywords that differentiate the corpuses from each other. For ...
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41 views

Problem estimating the parameters of a Weibull distribution based on a survival table

Some help would be appreciated. I'm trying to estimate the parameters of a Weibull distribution from a life survival table, by doing a regression. So I've linearized the cdf of the Weibull. 8400 ...
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32 views

A Proof of Tukey's Inequality

Suppose that $W_1,W_2,...,W_n$ are uncorrelated unbiased estimators of a parameter $\theta$. Consider $W=\sum_{i=1}^na_iW_i$ such that $E(W)=\theta$ and $Var(W_i)=\sigma^2_i$, where the $a_i$'s are ...
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149 views

Sufficiency or Insufficiency

Consider a random sample $\{X_1,X_2,X_3\}$ where $X_i$ are i.i.d. $Bernoulli(p)$ random variables where $p\in(0,1)$. Check if $T(X)=X_1+2X_2+X_3$ is a sufficient statistic for $p$. Firstly, ...
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1answer
83 views

Calculating means and confidence intervals when most data points are 0

I am looking at data set that has four groups. In each group, the data is mostly, 99+% of time, composed of zeros, but, when it is not zero it can be any float number (e.g., 0.01 to 921.2, with most ...
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19 views

Prevalence estimates based on randomized sample of clinical data

This is probably one of the more straight forward questions on here but here it is: I want to use a random number generator to sample X number of charts to look for the # occurrences of Y event. So ...
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47 views

Shape uncertainty of a 3D point cloud

Given a point cloud of a 3D object, how to calculate the shape uncertainty in this discrete sample set? and what factors maximize or minimize this uncertainty?
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1answer
50 views

Is there an article/book reviewing different methods for constructing posterior point/interval estimates?

Given a one-dimensional posterior distribution it is often the case that you want to calculate a point estimate and a credible interval for the corresponding parameter. There are, of course, many ways ...
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243 views

Why is the geometric median called the $L_1$ estimator?

My question is simply, why is the geometric median called the $L_1$ estimator? This always reminds of $L_p$ spaces but the distance being minimized in the geometric median's definition isn't $L_1$ but ...
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34 views

How to find conditional distribution for Rao-Blackwellizing an estimator?

Let's say I have an unbiased estimator $u(\underline x)$ for function $v(\theta)$ where $\theta$ is a parameter of the distribution of $x$, and $T(\underline x)$ which is a sufficient statistic for ...
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129 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates

I've struck an unpleasant problem with the noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ ...
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78 views

Biased but consistent estimator for the mean of Gaussian distribution?

$(X_1,X_2,\ldots,X_n)$ is a random sample from $\mathrm{N}(θ, 1)$. We know sample mean is a unbiased estimator that is consistent. What would be a biased but consistent estimator for θ? Would it be ...
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84 views

Comparing two estimators for accuracy using empirical bootstrap

I am trying to figure out the proper way to compare the quality of two estimators of a parameter based on data. The basic approach I've taken is to compute the MSE of the empirical bootstrap ...
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36 views

EM Algorithm - Expectation w.r.t. a subset of current parameters

Suppose I want to make inference on a parameter vector $\theta $=$(\theta_{1},\theta_{2},\theta_{3})$ and I have some missing data $Y_{mis}$. I would like to use the EM algorithm to find the mode of ...
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1answer
228 views

Show that $\mathbb{E}(g(T-p)) < \mathbb{E}(g(S-p))$ for any convex function $g$ if $T$ and $S$ are estimators of $p$

The more detailed question. I'm kinda having some trouble starting out with answering this question. My initial approach would be to $g(x)= x^2$ since that is a convex function and find the ...
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2answers
97 views

Why we shouldn't be obsessed with unbiasedness

In my Bayesian statistics class, my professor makes the remark that we should not be obsessed with unbiased estimator. First: I understand this statement in the sense of trading biasedness for ...
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70 views

Estimation of unknown vector's amplitude with Gaussian noise

I have the following model: y = P v + n Where y is the vector of observations, v is a unit vector and n is a Gaussian random noise whose covariance matrix is the identity matrix. P is a positive ...
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132 views

Calculating point estimates from model-averaged parameters

I'm using an IT-approach and multi-model inference with some count data. I have used model averaging to obtain parameter estimates from several GLMMs with Poisson-lognormal errors (Poisson family ...
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1answer
2k views

Confidence Interval for a Random Sample Selected from Gamma Distribution

Working on a homework question and having some trouble... Any help would be greatly appreciated. Based on a sample 1.23, 0.36, 2.13, 0.91, 0.16, 0.12 from the GAM$(2,\theta)$ distribution, find an ...
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299 views

How to use a point estimator on an interval?

I got part a and part b just fine, but I am confused on how to do part (c), (d), and (e). I don't really understand what the question is asking, nor do I understand how to do that sort of interval ...
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71 views

Complex-valued estimator of Real Parameter?

I was in statistics class today and the professor wrote "consistent estimator". The first thing I thought of was that the range of the estimator should be a subset of the set of plausible population ...
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82 views

Yet another “Bayesian vs Maximum Likelihood” question

In the fully Bayesian approach, the predictive distribution is: $$ P( Y|X ) = \int P(\theta | X ) P( Y | \theta ) d\theta $$ When the integral is difficult to compute, we might resort to the Maximum ...
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46 views

Why do matching algorithms use point estimates rather than intervals?

I'm developing a matching algorithm, and I am wondering why websites choose using point estimates, e.g. "You match 60% with person X". Given that there will be most likely be missing data, rather ...
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188 views

Do we have different assumptions for point estimate and interval estimate?

The procedure for interval estimate and point estimate appears to be based on the central limit theorem. Is it a correct assertion.
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Invariance property of MLE: what is the MLE of $\theta^2$ of normal, $\bar{X}^2$?

Invariance property of MLE: if $\hat{\theta}$ is the MLE of $\theta$, then for any function $f(\theta)$, the MLE of $f(\theta)$ is $f(\hat{\theta})$. Also, $f$ must be a one-to-one function. The ...
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1answer
194 views

Prove/counter example: A minimax decision rule is always Bayes wrt some proper prior

Not sure whether the claim is true or false. If claim is true, intuitively, it might have something to do with "least favorable priors", but am not able to figure out the connection. If claim is ...
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1answer
40 views

What does “I” represent in this context?

I'm trying to work on a problem which contains a symbol that I don't recall seeing before - I. I assume it has some special significance but I'm having a hard time looking it up. Relevant portion of ...
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467 views

Extract data points from moving average?

Is it possible to extract data points from moving average data? In other words, if a set of data only has simple moving averages of the previous 30 points, is it possible to extract the original data ...
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891 views

Inference for the skeptical (but not math-averse) reader

I just watched a lecture on statistical inference ("comparing proportions and means"), part of an intro to stats online course. The material made as little sense to me as it always does (by now I ...
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1answer
124 views

Given MCMC samples, what are the options for estimating posterior of parameters?

Markov chain Monte Carlo (MCMC) is a class of algorithms for sampling from probability distributions. In the end, given a parametric model explaining the data, MCMC could also be used for parameter ...
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223 views

Determine an unknown number of real world locations from GPS-based reports

I'm working on some software which should determine real world locations (f.e. speed cams) from several GPS-based reports. An user will be driving when reporting a location, thus the reports a very ...
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1answer
328 views

Estimated error variance $\sigma^2$ for MCMC estimation in a high-dimensional space

Let $f$ be a function such that: $$f~:~(x,~\theta)\in\mathbb{R}^{3}\times\mathbb{R}^{12} \rightarrow f(x,~\theta)\in\mathbb{R}^3$$ My observations $y$ are noisy values taken by the function $f(\cdot ...
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3answers
798 views

How to describe the differences in skewed data with same median but statistically different distribution?

I am comparing length of stay after laparoscopic and open appendectomy in over 160000 patients. LOS is typically a skewed variable so I use the median and interquartile range and ranksum test to ...
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1answer
263 views

Why do we rely on the standard error? [duplicate]

Possible Duplicate: Difference between standard error and standard deviation I cannot come to terms with the fact that the standard error is used to describe the accuracy of a point ...
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346 views

Sufficient, Complete Sufficient, UMVUE, Rao-Blackwell, Admissible. What are ties between these?

I am taking stat inference course. I have some trouble understanding some these terms: Sufficient Statistics: a stat that does not depend on the parameter, say $\Sigma X$ for normal distribution ...
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1answer
62 views

How to estimate 5 percent breaking point of a sample of ropes?

The breaking strengths in pounds of five specimens of rope were 660, 460, 540, 580, and 550. How can I estimate the point at which only 5 percent of such specimens would be expected to break?
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2answers
2k views

How to find an unbiased estimator?

Suppose $X_1, X_2, ...,X_n$ are samples from a uniform discrete distribution with probability 1/3 on each of the points $\theta-1, \theta, \theta+1$, where $\theta\in\mathbb{Z}.$ From "Theory of ...
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842 views

Is the theory of minimum variance unbiased estimation overemphasized in graduate school?

Recently I was very embarrassed when I gave an off the cuff answer about minimum variance unbiased estimates for parameters of a uniform distribution that was completely wrong. Fortunately I was ...
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1answer
37 views

Improving an estimate of mean with observations of sign

Suppose $x_i$ are drawn i.i.d. from a $p$-variate Gaussian, $\mathcal{N}\left(\mu,\Sigma\right)$. Suppose one observes $x_1,x_2,\ldots,x_n$. One also observes $s_{n+1},s_{n+2},\ldots,s_{n+m},$ where ...