"A rule, method, or criterion for arriving at an estimate of the value of a parameter."

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15
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1answer
182 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 ...
0
votes
0answers
9 views

formal tests for ordinal probit model

What are the basic post estimation tests for an ordinal probit model? How do we check parallel line assumption for ordinal probit model? the omodel comand of stat only works for probit model, i have ...
0
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0answers
46 views

Estimator of The Mean of the Ratio of Uniformly Distributed Variables

Given two random variables, $ X \sim U \left[ {\mu}_{x} - \frac{{l}_{x}}{2} > 0, {\mu}_{x} + \frac{{l}_{x}}{2} \right] $ and $ Y \sim U \left[ {\mu}_{y} - \frac{{l}_{y}}{2} > 0, {\mu}_{y} + ...
0
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0answers
6 views

Worried about hidden costs of using this loss function to fit weights

I have the following model: $$\frac{Y}{T} = f(X \beta)$$ where beta is a vector of weights, and Y and T - Y are greater than 0. I want to fit the $\beta$ vector using the loss function $$ ...
1
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0answers
108 views

Summary of estimator properties (consistency, bias, sufficiency, etc.)

I've read about various properties of estimators, but I'm wondering if there's some source with a summary (maybe a list, table, or graphic) of the properties for different kinds of estimators. ...
1
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0answers
18 views

Derivation of Variance for formula of Cohen's d statistic

Cohen’s d is one of the most common ways we measure the size of an effecthttp://en.wikiversity.org/wiki/Cohen%27s_d. Cohen’s d simply a measures the distance ...
2
votes
1answer
57 views

Maximum likelihood of constrained distribution

A random variable $X$ is represented by the following CDF: $F(x)=(1+\frac{1}{x^2})^{-\beta}$ , $x\in(0, \infty), \beta>0$ Question: How do you get the MLE of $P(X>1)$ for the distribution? I ...
2
votes
1answer
50 views

Density estimation and histograms

This is an excerpt from BW Silverman's Density Estimation for Statistics and Data Analysis: The oldest and most widely used density estimator is the histogram. Given an origin $x_0$ and a bin ...
0
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0answers
36 views

Practical application of Cramer-Rao lower bound to calculate the variance of estimator

I would like to use the Cramer-Rao lower bound to help me estimate the variance of my maximum likelihood estimator, across a range of different samples of data. My question is, how do I do this ...
1
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1answer
35 views

Confidence interval for a function of the MLE

I am studying an old assignment in which I have calculated the MLE for a sample from an exponential distribution. It then gives the formula for the median of an exponential distribution ...
2
votes
1answer
51 views

Why are inf and sup used in the definition of minimax estimators?

An estimator $\hat{\delta}$ is minimax iff $$\sup_\theta R(\theta,\hat{\delta})=\inf_\delta\sup_\theta R(\theta,\delta)$$ or in english iff out of all estimators it has the least maximum risk. For ...
1
vote
1answer
68 views

Going from derived estimators to their implementation in software

Estimation and Inference in Econometrics by Davidson and MacKinnon (1993 edition, the older one) on page 552, ch 16.3 'Covariance Matrix Estimation' states: "Consequently, the matrix \begin{eqnarray} ...
0
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0answers
21 views

Efficient implementation of this estimator when software runs out of memory

Further to the excellent discussion and answers on projection matrices here, I am wondering if there are perhaps more gains to be made when implementing this estimator \begin{eqnarray} (X' P X - ...
4
votes
2answers
125 views

Speeding up hat matrices like $X(X'X)^{-1}X'$ (projection matrices) and other aspects of custom-built estimator when software runs out of memory

Is there a way to speed up $Z(Z'Z)^{-1}Z'$ type matrices? I am implementing the expression below directly using a matrix language and my program frequently crashes while if I run OLS on them using a ...
0
votes
1answer
67 views

Normal method of moments derivation explanation of Algebra step

In deriving normal estimators using method of moments, why does the below equality hold? $$ \frac{1}{n} \sum X_i^2 - \bar{X}^2 = \frac{1}{n} \sum (X_i - \bar{X})^2 $$ This is from Example 7.2.1 from ...
-2
votes
1answer
40 views

What is the Mean Squared Error for this estimator?

I understand that to find the MSE, i must find the variance and bias and add them together. I've had trouble calculating either of these so a breakdown would be immensely helpful. The estimator is: ...
0
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0answers
10 views

Dependence of PDF of LLR of symbols

I have a system model with $y=hs+\sum_i^n gx+n$ where h is rayleigh fading desired channel, g is interfering channel x is interfering symbols. $\hat{s}=w*y$ where w is MMSE filter. On what factor pdf ...
0
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0answers
18 views

Finding a consistent estimator mathematically

This is my first post on this website so hopefully everything will go smoothly. Let me first ask the question, then go over my problem. Q: Suppose that we are given $({X_{1i},X_{2i}})$ which is a ...
-1
votes
1answer
28 views

Estimator $\gamma = \sum a_i\times x_i$ , where $X_i \sim \exp(t_i \theta )$ Show $\gamma$ is unbiased if $\sum a_i/t_i = 1$

I'm getting really confused with the estimators in this question! $X_i \sim \exp(t_i \theta x)$ where $t_i$ are positive constants. The MLE for $\theta = \frac n{\sum t_i x_i}$ And $\phi = ...
3
votes
1answer
55 views

Does MCD estimator suffers from swamping effect?

If there are multiple outliers in the data set, the Mahalanobis distance suffers from masking and swamping effects. In order to rectify this problem, robust estimation of location and scale, such as ...
3
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0answers
18 views

Fixed parameter estimates of parent factors in a nested design

Summary: What is happening with parameter estimates of factors that are the 'parents' of nested factors? Data: My analysis involves testing the effect of different parameter settings for automatic ...
3
votes
2answers
117 views

Self-study: Finding the maximum likelihood estimates of the parameters of a density function

Consider a random sample $x_1,x_2,...,x_n$ from a newly-generated distribution, whose probability density function is given below \begin{equation} f(x|\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( ...
3
votes
1answer
34 views

How to prove that the permutation of the points are the minimal sufficient statistics for Cauchy distribution?

I see this everywhere that the permutation of the samples $X_{(1)}, ..., X_{(n)}$ is the minimal sufficient statistic for the Cauchy distribution [1]. It is clear that it is a sufficient statistic,but ...
0
votes
1answer
88 views

Horvitz-Thompson estimator for two-stage cluster sampling

So I want to apply the Horvitz-Thompson (H-T) estimator to two-stage cluster sampling. The H-T estimator is defined as: $$\sum\frac{Y_i}{\pi_i}$$ where $\pi_i$ is the probability of including the ...
0
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0answers
12 views

Multivariate Generalization of (Fisher) Efficiency Ratio

Fisher defined efficiency as the ratio of variance of the most efficient estimator to the variance of the estimator in question. The Cramer-Rao bound allows easy computation of the variance of the ...
0
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0answers
16 views

Derivation of description length

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with $N$ data points, our estimation error for $\hat ...
2
votes
0answers
17 views

linearization of an estiamtor

Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional ...
2
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0answers
39 views

Unbiased estimator of the variance [duplicate]

I am going through the book The Elements of Statistical Learning and I'm finding it extremely terse. I have a background in probability but not statistics so perhaps that is why. Anyway, in Chapter 3, ...
0
votes
1answer
47 views

Posterior mode estimator unchanged under coordinate transformation?

I'm looking at a data set where the posterior mode has noticeably less "bias" than the posterior mean and posterior median, and somewhat less error. However, the posterior mode is not invariant under ...
0
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0answers
20 views

Source of least square estimator of Poisson parameter

I have $X \sim \text{Poisson}(\theta)$ I need to see how least square estimator of $\theta$ is obtained. Is there anything online showing that?
1
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0answers
19 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
1
vote
1answer
79 views

What are the implications of estimating a covariance matrix from a correlated sample?

Given a sample of $n$ independent observations $x_1,...,x_n$ (where $x_i$ are $p$-dimensional column vectors), the $p \times p$ sample covariance matrix is defined as ...
1
vote
1answer
42 views

Pointwise probability limit

Suppose we have a joint distribution of a random sample $(y_n,x_n)\in R^2$: $$ y_n = \beta_0 x_n + \epsilon_n $$ and the estimator $$ \hat \beta = argmin_\beta \frac{1}{N} \sum_{n=1}^N(y_n - \beta ...
0
votes
0answers
52 views

Finite variance of harmonic mean estimator when samples are bounded

Harmonic mean estimator is notorious for the possibility of having infinite variance. Now I want to show that it has finite variance when samples are bounded. I am wondering whether my following ...
0
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0answers
39 views

Generalized minimum chi-square estimators?

I need to implement the generalized minimum chi-square estimators (alternative to L-moments method and maximum likelihood estimate (MLE)) for estimate the parameters of the gamma distribution. My ...
1
vote
1answer
37 views

Deriving the formula for the mean of a stratified sample

The formula for the mean of a stratified sample $\bar Y_s$ is: $$\bar Y_s = \frac 1 N \sum_i N_i \bar Y_i$$ where $N$ is the sample size for all strata, and $N_i$ and $Y_i$ are the sample size and ...
0
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0answers
37 views

Help in calculating OLS estimators under certain constraints/modifications

I am new to these forums and to econometrics as a whole. I was hoping someone would be able to give me a nudge in the right direction with this problem. I've done extensive research both online and in ...
0
votes
1answer
277 views

How can you prove that the naive estimator is less efficient than the OLS estimator

The "naive estimator" is an estimate of the slope obtained by joining the first and last observations and dividing the increase in the height by the horizontal distance between them. Given that the ...
2
votes
2answers
95 views

Obtaining an estimator via Rao-Blackwell theorem

Let $X_1,...,X_n$ be iid with pdf $$f(x|\theta) = exp(\theta -x) I(x)_{(\theta, \infty)}$$ It is asked to find an unbiased estimator for$ \theta $ that is function of a sufficient statistical for ...
0
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0answers
16 views

What are the canonical data sets used for testing robust linear fitting?

The UCI database (link) is one of the repositories for canonical data. It has ~295 data sets for use. There are many others. (link) While data can be useful not all data is relevant for all ...
1
vote
1answer
1k views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
1
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0answers
26 views

Poisson counting process parameter

Two quick questions: What's the maximum likelihood estimator of the parameter of an homogeneous Poisson counting process? To estimate $\lambda$ I'm currently using number of events/total time, ...
1
vote
2answers
77 views

Limiting joint distribution of estimators; Functional Statistics; Influence curves;

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
1
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1answer
48 views

Conceptual questions on Entropy and estimation

Learning Informative Statistics: A Nonparametric Approach paper presents an approach to parameter estimation by entropy minimization. There are other related works "Minimum-entropy estimation in ...
0
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0answers
50 views

What is the plim estimator of $\beta$, What would we need to obtain a consistent estimator of $\beta$?

Suppose that the true model is $$Y_i = βZ_i + u_i$$ However, the researcher/analyst can only observe $X = Z + w$, where $w$ is a measurement error with zero mean and constant variance $\sigma^2_w$. ...
5
votes
2answers
228 views

Drawing numbered balls from an urn

PROBLEM There is an urn with a set of balls where each ball is labeled with a different integer. The numbers on the balls are known and are not a range of integers. For example the set of balls could ...
3
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0answers
132 views

Better estimator of expected sum than mean

I am trying to find the optimal estimator for the maximal expected $\Sigma X_i$ where $X_i$ is sampled from an unknown distribution which is chosen to be maximal. To clarify and simplify, there are ...
0
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0answers
18 views

Maximum Likelihood estimators in reation to linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
2
votes
1answer
356 views

How to calculate the scale parameter of a Cauchy random variable

Let $(X_n)$ be iid random variables and suppose they have mean 0 and follow Cauchy distribution. I know I can set the location parameter to 0. My question is how to find the corresponding scale ...
1
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0answers
46 views

Acronyms to use for Bayesian posterior predictive distribution estimators

I am considering writing an article that discusses the Bayesian MMSE and MAP of the posterior predictive distribution. I was wondering if there are acronyms that have been used so that instead of ...