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

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Rao-Blackwellization of Gibbs Sampler

I am currently estimating a stochastic volatility model with Markov Chain Monte Carlo methods. Thereby, I am implementing Gibbs and Metropolis sampling methods.Assuming I take the mean of the ...
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18 views

Are there examples of non exponential family distributions with sufficient statistics?

In Casella Berger's Statistical Inference, they observe that it is rare to find 'a sufficient statistic with dimension smaller than the sample' (section 6.2.1). Although rare, are there examples of ...
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1answer
26 views

How best to summarize a predictive discrete distribution in a single number?

I have generated a predictive distribution for a future discrete observable outcome, and would like to generate a single value $p$ which we would most likely encounter when we perform the experiment ...
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1answer
20 views

How to define a study area for RIpley's L analysis?

I have a spatial point pattern which spans across multiple districts in the country. Now I want to examine its pattern using Ripley's L function. I know that Ripley's L function is used for ...
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24 views

Find point estimate, bias and variance

I am kind of new to Statistics and I stuck at this question. Lets say in a consumer survey, 250 people out of a representative sample of 450 people say that they prefer product A to product B. Let p ...
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12 views

Does Cramer-Rao Inequality needs the parameter space to be an open subset of the real line?

I came across a lecture on Cramer-Rao lower bounds for an unbiased estimator and the visiting professor remarked that for CR inequality to be valid one of the regularity conditions we need is that the ...
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19 views

What is λ in the context of Spatial Point Statistics?

What is a small lambda λ in the context of spatial point statistics and how do I calculate it for a given set of point? Specifically I am trying to calculate it for different sets of observations of ...
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30 views

Does an Efficient Unbiased Estimator exist for a function of a parameter of an exponential family distribution?

Say I have an i.i.d. sequence sequence $X_1,\ldots X_n \sim \text{Bernoulli}(p)$, and I am interested in estimating $p^2$. Let $T$ denote $\sum_{i=1}^n X_i$. It turns out that the mle $\bar X^2 = \...
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1answer
37 views

estimating a parametric function

I am working on this problem and am stumped. Can anyone take a look at it? $X_1, \ldots ,X_n$ are distributed Bernoulli$(p)$ where $p$ is unknown. Consider the parametric function $\tau(p)=(p+(1-p)...
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1answer
43 views

Practical situation in which the posterior mean is prefered to the MAP

Sometimes experts for which we design models are interested in having a point estimate and in practical situations, they always say me "give us the most probable parameter value". And whether the ...
3
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1answer
40 views

Estimating accurately the mean of an autocorrelated bounded integer time series

I have a bounded integer time series $X_{1:\infty}$ ($1\leq X_k\leq M$), and I want to estimate the mean $$ s = \lim_{L\to\infty} \frac{1}{L}\sum_{k=1}^L X_k. $$ I'm assuming it exists and that $X_k$ ...
4
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1answer
57 views

Cramér-Rao Lower Bound for Exponential Families

I am having a problem with applying the Cramér-Rao inequality to identify the lower bound for the variance of an unbiased estimator and hoped that you guys could help me. The problem is the following: ...
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1answer
60 views

How to get the maximum likelihood estimator of $U(\theta,\theta +1)$?

I know how to find the MLE for $U(0,\theta)$ but I am in trouble with this one. let $X_1,\dots,X_n$ be a random sample from $U(\theta,\theta +1)$. Consider the following three estimators for $\...
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0answers
49 views

Does the UMVUE have to be a minimal sufficient statistic?

I'm studying point estimation and I have found this question that seems pretty tricky to me. If $T$ is a minimal sufficient statistic for $\theta$ with $E(T) = \tau(\theta)$, can you say that $T$ ...
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27 views

Observed Fisher Info as an estimator of Expected Info

When I construct an asymptotic confidence interval for a parameter $\theta$, taken from a sample iid distributed with a generic pdf/pmf, I usually implement the mle $\hat\theta$ instead of $\theta$, ...
3
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2answers
122 views

Asymptotic properties of MLEs

Are there any relationship between the asymptotic properties of MLEs (assuming that the regularity conditions hold)? I mean, once I know that the MLE for $\tau(\theta)$ is asymptotically efficient ...
0
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1answer
33 views

Efficient estimators and CRLB

An estimator is efficient if it reaches the Cramér-Rao Lower Bound and since it is efficient, it is also the UMVU estimator of the parametric function $\tau(\theta)$. But Cramér-Rao inequality and the ...
2
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1answer
141 views

Are complete statistics always sufficient?

I know that a complete sufficient statistic $T$ is such that 1) $T$ is sufficient for $\theta$, unknown parameter and 2) $T$ is complete. So, is it always the case? If the answer is not, what ...
2
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2answers
78 views

Cramér-Rao inequality and MLEs

I know that if it exists, a regular, unbiased estimator $T$ for $\tau(\theta)$ attains the Cramér-Rao Lower Bound (next, CRLB) if and only if I can decompose the score function as follows: $S(\theta)=\...
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63 views

Fisher information for uniform distribution [closed]

If I want to compute the CRLB for iid uniform on $[0,\theta]$. I need in the denominator this expression: $E_\theta\left[\left(\frac{\partial \log f(X)}{\partial \theta}\right)^2\right]=nE_\theta\left[...
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32 views

Estimation of the mean when data points are not independent

When we estimate the mean with 95% confidence, this formula is used: $\hat{\mu}=\overline{\mu}\pm1.95\times\frac{\overline{\sigma}}{\sqrt{N}}$, where $\overline{\mu}$ is the sample mean, $\overline{\...
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18 views

Point configuration given distribution of distances

Does anybody know if there is a method/algorithm to find coordinates for $N$ points, given a probability density function from which the $N(N-1)/2$ different pairwise distances between them can be ...
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39 views

Limits of integration of Pitman Estimator for Laplace (Double exponential) distribution

I am struggling, in general and specifically, with trying to determine the limits of integration for the Pitman Estimator to find a Minimum Risk Equivariant (MRE) estimator of a location parameter ...
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3answers
2k views

Is p-value a point estimate?

Since one can calculate confidence intervals for p-values and since the opposite of interval estimation is point estimation: Is p-value a point estimate?
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1answer
2k views

How to derive the likelihood function for binomial distribution for parameter estimation?

According to Miller and Freund's Probability and Statistics for Engineers, 8ed (pp.217-218), the likelihood function to be maximised for binomial distribution (Bernoulli trials) is given as $L(p) = \...
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1answer
61 views

complete sufficient statistic exercise

I have to find complete sufficient statistic of the following pdf $$f(x|\theta)=\frac{\theta}{(1+x)^{(1+\theta)}},\quad 0<x<\infty,\theta>0.$$ My Attempt: The joint density $$f(\mathbf x|\...
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32 views

estimate ranking and winning probability of N players after observing the each pair of players' matching results

Question: N people are ranked from the highest to the lowest with no ties, but we do not know the precise rankings. When two people have a match, the one with higher rank wins with a probability p > 1/...
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14 views

Using Horvitz-Thompson for estimation from a simple random sample with unknown membership probability

I learnt about the Horvitz-Thompson estimator yesterday, and am trying to apply it to the degenerate case where $p$ is uniform for each, but I seem to have run into a bit of a confusing situation. ...
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1answer
123 views

Estimating sample mean from a biased sample (whose generative process is known)

I'm working on a problem where I'm trying to estimate some property of a dataset from a small non-uniform sample. (Let's just take the population mean because it's simple.) Formally, assume I have ...
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1answer
183 views

point estimation--an interview

I've encountered the following interview question: ...
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32 views

Finding the Estimated Standard Error of a 10%/90% data division estimator

I am learning about estimated standard error of an estimator, which is defined as the square root of the variance of the estimator: $\sqrt{V(\Theta)}$ Given the data, assumed to be from a normal ...
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54 views

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 $$f(x;\...
3
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1answer
115 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 estimator ...
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1answer
35 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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4answers
107 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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1answer
337 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 ...
3
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1answer
58 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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0answers
37 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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1answer
158 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, how ...
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1answer
131 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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1answer
29 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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70 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
59 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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2answers
398 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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0answers
57 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 $\...
6
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1answer
150 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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167 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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0answers
147 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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38 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
233 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 ...