# Questions tagged [estimators]

A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].

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### Sum of asymptotically independent random variables - Convergence

Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$. If $X_i\sim Ber(\pi_i)$, I ...
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### Must maximum likelihood method be applied on a simple random sample or on a realisation?

I guess my trouble is not a big one but here it is: when one applies maximum likelihood, he considers the realization $(x_1, \dots, x_n)$ of a simple random sample (SRS), leading to ML Estimates. But ...
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### Asymptotic unbiasedness + asymptotic zero variance = consistency?

Here, Ben shows that an unbiased estimator $\hat\theta$ of a parameter $\theta$ that has an asymptotic variance of zero converges in probability to $\theta$. That is, $\hat\theta$ is a consistent ...
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### Is Coefficient of Variation a valid measure of relative efficiency?

I'm wondering if it is always valid to use Coefficient of Variation (CV) to determine relative efficiency of parameter estimators, and to compute statistically equivalent sample sizes based on that ...
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### Using Rao-Blackwell to improve the estimator of P(X/Y < t)

X and Y are independent N (0, 1) random variables, we want to approximate P (X/Y ≤ t), for a fixed number t. The first part of the problem was to describe a naive Monte Carlo estimate. I described ...
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### What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves? [duplicate]

!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
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### Terminology clarification about sample moments

According to MathWorld (link): "The sample raw moments are unbiased estimators of the population raw moments". While in Wikipedia (link) it is said: ...the $k$-th raw moment of a population ...
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### Why can we get better asymptotic global estimators even for IID random variables?

Let $X_1,...,X_N$ be IID random variables sampled from a parametrised distribution $p_\theta$, and suppose my goal is to retrieve $\theta$ from these samples. We know that the MLE provides an ...
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### Standard practice to show Biased CRBs

I have a problem with four-parameter estimation. I have derived the variances for the estimated parameters using Monte Carlo simulations (numerical ones) and theoretical ones using the inverse of the ...
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### Intuition behind between-group covariance matrix from MANOVA?

Suppose that we have samples from $m$ different $p$-dimensional normal multivariate distributions, where they share a common covariance matrix $\Sigma$ but the mean vectors may be different for each ...
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### Why does not this underlying hypergeometric distribution lead to unbiased estimators?

This example is take from Lippman's "Elements of probability and statistics". Let N be the number of fish in a lake the warden wants to estimate. He catches 100 fish, tags them and releases ...
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### Is an estimator that always have a value of zero is a linear estimator?

Consider a simple linear regression model: $$Y=\beta_0+\beta_1 X +u$$ Here, we can consider an estimator that does not use any data: $$\hat{\beta}_1=0$$ That is, regardless of the observed data, the ...
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### What is a measure of hardness-of-approximation by samples?

Suppose there is a large vector $\mathbf{x}$ of real numbers, and I want to estimate a certain aggregate function $f(\mathbf{x})$ by taking a small sample of the population $\mathbf{x}$. I would like ...
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### Optimality criterion for mean estimators

Assume a sample size of $n>5$, a given variance $\sigma^2 > 0$ and a $\delta \in (2e^{-n/4}, 1/2)$. Proof that there exists a distribution with variance $\sigma^2$ such that for any mean ...
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Let $\hat{\theta}$ be a biased estimator whose bias depends on the true value $\theta_0$, such that $E[\hat\theta|\theta_0]= f(\theta_0)\neq \theta_0$. Let $t_{sample}$ be a sample realization of $\... • 3,083 1 vote 0 answers 42 views ### Showing that the estimator of the log posterior used in stochastic gradient MCMC is unbiased In the SGLD paper as well as in this paper it is claimed (paraphrasing) that the following estimator: $$\widetilde{U}(\theta) = -\dfrac{|\mathcal{S}|}{|\widetilde{\mathcal{S}}|} \sum_{{x}\in \... • 33 3 votes 1 answer 107 views ### Confidence interval on ratio of estimates for exponential random variables Given exponential random variable X, the MLE for the scale parameter is \hat{\beta_x} = \bar{x}, and the confidence interval for that estimate is:$$\frac{2n\bar{x}}{\chi^2_{\frac{\alpha}{2},2n}} &... • 1,148 0 votes 1 answer 80 views ### What is the variance decomposition method? For$i = 1, \ldots, m$and$j = 1, \ldots , n$we have observations$x_{ij}$. We can assume that $$x_{ij} = y_{i} + z_{ij}, \qquad y_{i} \sim \mathcal{N}(\mu_{y},\sigma_{y}^{2}), \quad z_{ij} \sim \... • 101 2 votes 1 answer 78 views ### Maximum Likelihood Estimation for a Unique Probability Density Function In the context of estimating parameters for a uniquely distributed set of independent and identically distributed random variables, I am examining the following probability density function f(x|\... • 425 0 votes 0 answers 21 views ### How to show that the influence function of minimum density power divergence estimator with positive tuning parameter is bounded? In the linked paper, in the influence function section, the term {u_{\theta}(y)}{f_{\theta}(y)}^\alpha is directly called bounded which i do not get the explanation of? Here \alpha > 0 is the ... 2 votes 1 answer 91 views ### Mathematical Step for consistency Let me state my problem from the beginning: Let i be an index representing countries (i = {1,2,\ldots,N }), and t represent time, denoted as available data for country i (t = {1,2,\ldots,T_i }... • 279 0 votes 1 answer 76 views ### Unable to estimate AR(p) coefficients and \sigma^2 I am currently trying to solve this problem pertaining to the Yule-Walker equations: Let \{X_t\}_{t\in Z} be a causal autoregressive process given by$$X_t = \varphi X_{t−2} +W_t$$with \{W_t\}_{t\... 0 votes 0 answers 26 views ### One off the advantages of the bootstrap is that i don't have to worry about having a good estimator? I'm learning about the theory of estimators and saw that sometimes the analytical formula of the estimator has to be diferent off the formula for the parameter, for example the standart deviation, and ... 1 vote 0 answers 25 views ### Large samples property of bayes procedures I was reading through Wasserman's All of Statistics and I came across this property in the Bayesian statistics chapter: I think I don't really get what is supposed to be the intuition behind it, and ... 3 votes 1 answer 121 views ### Is the sample mean an unbiased estimator of population mean in the presence of autocorrelation? I've seen previous questions here that the sample mean can be considered an unbiased estimator of the population mean. e.g.1, 2. While the examples seem to refer to independent sample points, it seems ... • 539 1 vote 1 answer 66 views ### How to estimate how heavy a tail is? Suppose I have data coming from a single variate distribution. I want to estimate how heavy the tail of the distribution is. For example, if the data comes from the Zipf distribution, I would want the ... • 145 1 vote 1 answer 108 views ### Difference between consistent and unbiased estimator [duplicate] I have a problem where I have to think of an example to explain a practical example of consistency and unbiased. The example I thought of is the sample mean. Consistency is when the estimator (sample ... 0 votes 0 answers 26 views ### How to derive the (partial) maximum likelihood estimator for a simple autoregressive model I am trying to derive two maximum likelihood estimators which I have seen in a statistics book, but I am unable to derive them and would really like some help. It goes like this: Consider the simple ... 1 vote 1 answer 170 views ### Is convergence in probability implied by consistency of an estimator? Every definition of consistency I see mentions something convergence in probability-like in its explanation. From Wikipedia's definition of consistent estimators: having the property that as the ... 0 votes 0 answers 9 views ### Variations of Correlation Coefficient of Simple Linear Regression with Estimators [duplicate] Suppose we are using an Ordinary Least Squares (OLS) estimator of \alpha_{0} and \alpha_{1} for the simple linear regression below:$$ H_{i} = \alpha_{0} + \alpha_{1}X_{i} + \epsilon_{i} $$How ... 4 votes 1 answer 182 views ### Is it more correct to say "bias of the standard error of the estimator" or "bias of the standard error of the estimate" I understand an estimator is a "rule" (e.g., a function, say g) that produces an estimate (\hat\theta) of an estimand (say, a population parameter, \theta). My question: is it ... 1 vote 1 answer 67 views ### Does increasing number of observations lead to the decreasing of Mean Square Error of consistent estimators? I know that not all weakly consistent estimators exhibit MSE-consistency : https://stats.stackexchange.com/a/610835/397467. Anyway, does increasing the sample size leads to a reduction in their mean ... • 11 0 votes 0 answers 44 views ### How to compute the FGLS estimator for simulated data in Matlab (or any other language really)? first time posting here so please let me know if I can improve my question. As an exercise, I have to simulate 1000 iterations of a sample of 500 observations from a linear model:$$ y_i = \beta_0 + \... 3 votes 1 answer 51 views ### Properties of statistical estimators when data is a collection of estimates Assume I have a statistical estimator$\theta$that has nice properties (say, unbiased and consistent) when the data$Y=\{y_1,y_2,\dots,y_n\}$is i.i.d. (possibly with additional assumptions). But now,... • 213 3 votes 1 answer 132 views ### How does Huber compute the$\operatorname{var}(s_n)/E[s_n]^2$and$\operatorname{var}(d_n)/E[d_n]^2$? (N.B. I am cross posting this question from math stackexchange since after x days I have still not received any responses.) How does Huber in book 'Robust statistical procedures' in chapter 1 compute ... • 31 0 votes 0 answers 26 views ### How to know errorbars on accuracy to nonlinear fit:$A(1-\cos(x+\phi))\$ with poisson noise?
I am trying to write some code that accurately estimates the parameters for the following function: $$Y = A(1-V \cos(X+\phi))$$, where this output data is poisson distributed. To do this, I first ...