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

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8 views

Is M-estimation valid only for regression models?

Is M-estimation valid only for regression models or does it's working hold good for robust estimation of parameters in other statistical models? I understand that M-estimators are asymptotically ...
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3answers
30 views

Square of the Sample Mean as estimator of the variance

Suppose we have the following random variables $X_1$, $X_2$,....$X_n$,.., that are $iid$ but we dont know what distribution they follow. I know that the sample mean $\bar{X}$ is an unbiased ...
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0answers
14 views

Reporting Data with Estimates and SE

What do you think? When reporting data in an article, a usual way is to show the mean values +/- the standard errors in a table. For Example, when you measure Abundance of pigeons in parks of five ...
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1answer
28 views

Variance of estimator(exponential distribution)

I have exponential distributed data $Exp(\lambda)$ with sample n = 50. Also, The sample mean = 2.17. I need to find the estimator of parameter $\lambda$ by the method of moments and to build 95% ...
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16 views

Are point estimators and set estimators the same thing?

My professor defined a point estimator as A mathematical rule that maps a random sample $\lbrace x_i \rbrace_{i=1}^n$ into a 'best guess' at the parameter $\theta$ I am confused about what ...
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33 views

What are the bias and variance of a model returning the observed mean for a training set?

It seems to me that bias = variance = 0 but MSE > 0, possibly very high, so clearly my intuition, and math, are wrong. For a training set $T$ and a regression problem let $M(T) = \text{Ave}(y(T))$. ...
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2answers
888 views

When is a biased estimator preferable to unbiased one?

It's obvious many times why one prefers an unbiased estimator. But, are there any circumstances under which we might actually prefer a biased estimator over an unbiased one?
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1answer
52 views

Identification from minimum value of truncated distribution

Suppose that a given population is endowed with a pair of characteristics $T$ and $K$. Let's think of these characteristics as random variables $$(T,K) \sim \operatorname{BiNormal}((\mu_T, \mu_K), ...
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12 views

Cancelling roots in ARMA(1,1) with external regressors

I am trying to find out what cancelling roots would imply for the estimators of my external regressors in my ARMA(1,1) model. Unfortunately however I'm stuck in my final step since I'm insecure about ...
4
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1answer
198 views

root-n consistent estimator, but root-n doesn't converge?

I've heard the term "root-n" consistent estimator' used many times. From the resources I've been instructed by, I thought that a "root-n" consistent estimator meant that: the estimator converges on ...
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0answers
10 views

(Self-study) Optimal Estimators that minimizes MSE

Show c||x-(x_bar)(1)||^2 minimizes MSE among all such estimators for c = 1/(n+1) I understood everything about Lehman-Scheffe, Rao-Blackwell, UMVU and such, but I have no idea how to get started on ...
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0answers
11 views

Asymptotics of the estimator y'y/y'x in a linear model

I am trying to learn to understand how to derive asymptotic distributions. In an exercise, I am trying to analyze the asymptotic behaviour of the estimator $\hat{\beta} = \frac{y'y}{y'x}$, where $y = ...
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2answers
133 views

Is bias a property of the estimator, or of particular estimates?

As an example, I often encounter students who know that Observed $R^2$ is a biased estimator of Population $R^2$. Then, when writing up their reports, they say things like: "I calculated Observed ...
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0answers
16 views

What is a good estimator for the reciprocal of covariance?

Let $X,Y$ be random variables with unknown but nonnegative covariance. What is a good estimator for $1/\operatorname{Cov}[X,Y]$? Specifically, how does one deal with negative sample covariance when it ...
1
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0answers
7 views

The estimator of variance of linear regression targets

In Section 3.2 of the book The elements of Statistical Learning (2ed), I read the text: Typically one estimates the variance $\sigma^2$ by $$ \hat{\sigma}^2 = ...
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0answers
21 views

Linear regression: Estimator for OLS and Minimum Absolute Deviations

I've received conflicting information in 2 different statistics classes, and I want to better understand the problem before I ask either of them for clarification Prof 1: We use OLS for linear ...
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0answers
19 views

Procedure for calculating a sampling distribution

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
3
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63 views

Paired comparison of instruments using different measurement samples

I have instruments A, B, C and D - I'm in search of the best one. The problem: For illustrative purposes, let's use an example of evaluating the best among the instruments measuring difference in ...
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0answers
9 views

help me understand the proof in the paper “restricted ridge estimation”

I'm reading the paper "restricted ridge estimation" by Grob(2003). I can not understand the proof of theorem 1 in this paper. I don't know how this estimator $\hat{\beta}_{r}(k) = ...
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1answer
89 views

Comparing spread (dispersion) between samples

EDIT: This question has been heavily paraphrased and re-asked in a broader, but better way here: Paired comparison of instruments using different measurement samples This is going to be a long one, ...
0
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1answer
21 views

Most likely event in a multinomial distribution setting

I'm looking at the following scenario: $k$ categories, distributed by a multinomial ($p_1,\dots,p_k$) such that $p_1 \ge \dots \ge p_k$. Draw $n$ samples. I'm interested in estimators/lower bounds ...
1
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1answer
26 views

Compare two multivariate distributions

I have a multivariate distribution (for which I know the parameters) that I simulate data from. I then fit several distributions to this simulated data using several different approaches (similar to ...
1
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1answer
33 views

Linear Regression - Conditions for unbiased estimate

When is the linear regression estimate of $\beta_1$ in the model $$ Y= X_1\beta_1 + \delta$$ unbiased, given that the $(x,y)$ pairs are generated with the following model? $$ Y= X_1\beta_1 + ...
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1answer
34 views

If $X \sim \mathcal{P}(u)$, show that $S=(-1)^X$ is the UMVUE of $e^{2u}$

If $X \sim \mathcal{P}(u)$, show that $S=(-1)^X$ is the UMVUE of $e^{2u}$. I can't figure this out, finding UMVUE always confuses me. Any help is greatly appreciated
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3answers
56 views

How to Prove Unbiased Estimator

I'm unsure of how to convince myself that $$\hat{\beta} = \frac{\sum X_i Y_i}{\sum X_i^2}$$ is an unbiased estimator when the regression model $$Y_i = \beta X_i + \epsilon_i$$ follows basic OLS ...
4
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1answer
69 views

“Consistent estimator” or “consistent estimate”?

Question: Are both expressions "consistent estimator" and "consistent estimate" meaningful? The quote below is intended to be illustrative; however, I am interested in the question above in a ...
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27 views

robust estimator when $\mu/\sigma$ is constant

I have a large set of measurements consisting each of about 100 points, (normally distributed), with up to 20% outliers. The outliers are all shifted towards positive values. From the physics, I know ...
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0answers
29 views

Minimum mean squared error linear combination of random variables

Consider the following objective function: $$ \mathbb{E}((Y-X\beta)^2)\rightarrow \min_\beta $$ where $Y$ and $X$ are (generally not independent) random variables and $\beta$ is a constant. That is, ...
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2answers
56 views

Estimating unconditional variance in time series

Consider a time series process with a well-defined, finite unconditional variance. Given a realization of the process (a time series) and a model for it, there are at least two ways of estimating the ...
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0answers
33 views

When to assume that $\Pr(X=x, Y=y) = |\mathcal{Y}|^{-1} \Pr(X=x|Y=y)$?

Background Let $\mathcal{X} = \{x_1, x_2, \ldots, x_n\}$ be a set of samples, each that corresponds to a label in the set of labels $\mathcal{Y}$. Ideally, our objective is to find the joint ...
4
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1answer
51 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
54 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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33 views

Show that weighted least squares estimator for a specific model is not consistent

Here is the background for this problem: $\qquad\qquad\qquad$ $Y_{1},...,Y_{n}$ iid $N(\mu,c^2\mu^2)$, $\,\,$ $c^2$ known. $\,$ The problem is as following: Consider the above model. Define ...
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32 views

Negative of log-likelihood on test data decreases but the parameters Mean Square Error(MSE) increases. How to justify the situation?

We develop an EM algorithm to model a problem. We generate some synthetic data of the model with parameters $\Theta$. We call the data $\text{D}$ which is decomposed it into two separate sets, ...
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1answer
29 views

Normal approximation of parameter $p$ of $Bin(n,p)$? [closed]

I've seen normal approximation applied for approximating a binomial distribution $B(n,p)$. However, if one estimates the parameter $p$, then can the parameter $p$ be "normally approximated" just as ...
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189 views

Anscombe-like datasets with the same box and whiskers plot (mean/std/median/MAD/min/max)

EDIT: As this question has been inflated, a summary: finding different meaningful and interpretable datasets with the same mixed statistics (mean, median, midrange and their associated dispersions, ...
3
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0answers
28 views

Relationship beween SEs of two point estimates

Good evening everybody! There's a very odd question from an old exam in Introductory Statistics that has been preoccupying me for a couple ofhours: What is the relationship between the Standard ...
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0answers
18 views

Sample Properties for Estimation

Newbie here. I want to estimate the nationality of a person from the nationalities of his family. I am looking for ideas/pointers for the following: How can I know if his family is a good sample ...
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0answers
12 views

Trouble understanding equations with a mix of variances, expected values and means

Let's assume we don't know the real value $\mu$ of the average of a of random variable $x$. We can find an estimator for the variance using the Bessel correction: $$\widehat{V(x)} = ...
1
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1answer
39 views

Help to understand this. Expected value of $S^\alpha$ in Gaussian distribution

Lets $X_1,\cdots,X_n$ be simple random sample from $\mathcal{N}(\mu,\sigma)$. $\overline{x}$ is sample mean. Let $$S^2=\begin{cases}\sum_{i=1}^n (x_i-\mu)^2, \mathrm{ where\ } \mu \mathrm{\ is\ ...
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0answers
23 views

Estimating multimodal (1-3 modes) signals

I am trying to estimate measured signal, which has multimodal behaviour, usually 1-3 modes (see trimodal sample frequencies below for example), but in one experimental setup it's 1, 2 or 3 all the ...
1
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1answer
189 views

Is this an unbiased estimator?

I'm working in the following problem: Let X be a sample of size = 1 from a Poisson distribution with parameter $\lambda$, and let $h(\lambda) = e^{-3\lambda}$. a.) Check if $T = (-2)^X$ is an ...
5
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1answer
50 views

why is $1/n \sum_{i} (X_i -10)^2$ unbiased

Let $\{ X_1,X_2,...,X_n \}$ be n observations randomly drawn from normal distribution with mean $10$ and unknown variance. Prove that the estimator $1/n \sum_{i} (X_i -10)^2$ is unbiased. Why is this ...
0
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0answers
47 views

How to prove the unbiased and biased estimator of autocovariance function?

These two estimators are commonly referenced as sample autocovariance functions. I'm curious how you're to show the first is an unbiased estimator, while the second is a biased one. And how would ...
4
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4answers
208 views

What is the difference between bias and inconsistency?

I am trying to learn about bias in simple linear regression. Specifically, I want to see what happens when the $cov(e,x) = 0$ assumption of the simple regression is violated. If this assumption is ...
0
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0answers
27 views

The variance of a biased estimator

This builds on an an earlier question from Math SE. I am just starting to learn about the simple regression model. In particular, I am trying to understand what happens to $\hat{\beta_1}$ when the ...
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2answers
184 views

a question on negative mean square error

for simple random sampling, i have calculated some mean square errors of many types of ratio estimator. Well, i obtained negative mean square error. Is there a mistake? negative MSE is a normal ...
30
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3answers
1k 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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0answers
18 views

Is there any paper about the distribution of difference of log-normal variable?

I am working on the problem relating to the difference of log-normal distribution. I have found several papers about this topic, however, none of them gives me the answer I want. More specifically, ...
0
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0answers
20 views

Pandey and Dubey estimator.

I am studying sampling theory Pandey and Dubey(1988) proposed the following product estimator. $\bar y_{PD} = \bar y \left( \frac{\bar x + C_x}{\bar X +C_x}\right)$ And its Mean square error is ...