Questions tagged [estimators]

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

Filter by
Sorted by
Tagged with
4
votes
1answer
60 views

Prove that sample covariance matrix is positive definite [duplicate]

Consider the $p \times p$ sample covariance matrix: $$\mathbf{S} = \frac{1}{n-1} \cdot \mathbf{Y}_\mathbf{c}^\text{T} \mathbf{Y}_\mathbf{c} \quad \quad \quad \mathbf{Y}_\mathbf{c} = \mathbf{C} \mathbf{...
0
votes
1answer
60 views

Expectation of sample variance $E(s^2)=\sigma^2$

Let $$s^2=\frac{\left(\sum_{i=1}^n y_t^2\right)-n \bar y^2}{n-1}$$ be an estimator of $\sigma^2$. Let $$E(yAy^T)=Tr(A\Sigma)+\mu^TA\mu$$ be an identity, where $y$ is a random vector, $A$ is a suitable ...
0
votes
1answer
21 views

Copula from small samples

Which copula estimation approach performs better when the empirical data to be modeled has a small sample size? Parametric copulas (Gaussian, t-, Gumbel, Clayton, etc), or Non-parametric (empirical) ...
1
vote
1answer
22 views

Can't Follow the Algebra in a Estimator MSE Comparison

Little bit of background - working through some maths and stats autodidactically. I simply can not follow the algebra of the following worked example comparing the MSE of two estimators. I can not ...
0
votes
0answers
16 views

Are these two linear estimators equivalent?

Assume we have data $y_i,i=1,\cdots, m$ at points $(x_{1,1},x_{1,2})$ and $y_i,i=m+1,\cdots, n$ at points $(x_{2,1},x_{2,2})$ from model $y=k_1x_1 + k_2x_2+\epsilon$. We want estimate the expected ...
2
votes
1answer
27 views

SEM: can GLS estimation be used under severe nonnormality?

In Randall E. Schumacher's and Richard E. Lomax's book "Beginer guide to SEM", the writers keep saying that if non-normality is sensed, where you can't use ML, you can go for GLS/ADF/WLS. ...
4
votes
1answer
33 views

In SEM, what is the difference between ADF and WLS estimation methods?

In some books both WLS and ADF are considered different methods. In other books, they acknowledge that ADF = WLS, so they are used interchangeably throughout the book. How solve this confusion?
1
vote
0answers
31 views

Proving equal OLS estimators

I am trying to prove that equals in the following two equations given that the vector are the same in both: The only two assumptions for both equations given are the homoskedastic assumption () ...
1
vote
1answer
38 views

Cokriging and collocated cokriging data requirements

In this wiki article and elsewhere in educational materials/papers, I have seen people refer to the idea that secondary data, if used (appropriately) in cokriging or collocated cokriging, is usually ...
0
votes
0answers
16 views

What's the distribution of maximum likelihood estimate on linear regression parameters with small sample size?

$y = k_1x_1 +k_2x_2 + \sigma\epsilon$ $\epsilon \sim D$, wehre $D$ is a known distribution e.g. $N(0,1)$. With $n\rightarrow \infty$, $\frac{1}{\sqrt{n}}((k_1,k_2,\sigma) - (\bar{k_1},\bar{k_2},\bar{\...
1
vote
0answers
33 views

Using variance of sample to calculate unbiased estimate of population variance

I am trying to follow through the survey sampling chapter of Rice's statistics book. Denote the sample values by $X_1, X_2, \ldots, X_n$ and the population values by $x_1, x_2, \ldots, x_N$ (so the ...
1
vote
1answer
55 views

Where to find a citation for the n/(n+1) sample variance correction?

In my Master's thesis project I could not show my (normally-distributed) samples to have a common population variance (through Levene's test or otherwise), so I could not use the n/(n-1) Bessel's ...
3
votes
0answers
71 views

Are some statistical moments harder to estimate than others? [closed]

Statistical moments include the mean, variance, skewness, kurtosis, etc. Since we hardly ever have population data and work with finite sample datasets, we often have to rely on estimators of these in ...
1
vote
2answers
66 views

How to include the observed values, not just their probabilities, in information entropy?

Shannon entropy measures the unpredictability in a random variable's outcome as the weighted average of the probabilities of that variable's outcomes or observed values. However, it discards the ...
1
vote
1answer
25 views

Estimating survival functions for interval censored data

I have a longitudinal dataset of about 350,000 individuals who had diagnostic measurements taken within a specific time period. The number of measurements per individual varies and the measurement ...
1
vote
1answer
29 views

What is the distinction between bias in prediction and parameter estimation?

I am trying to understand the distinction between bias in prediction and parameter estimation. This example in Gelman, Bayesian Data Analysis, 2nd ed. 2004 pp. 255-256 is very confusing to me. Why do ...
0
votes
0answers
23 views

How important is triangle inequality for statistical estimators?

(Pearson's) correlation is a measure of co-dependence that does not fulfill certain axioms such non-negativity and triangle inequality. In layman's terms, how would you describe what triangle ...
0
votes
0answers
5 views

How to compare estimators for consistency between in-sample and out-of-sample fits?

What general procedures are out there for quantifying how well an estimator (such as for the mean, standard deviation or correlation) of a continuous random variable gives a consistent picture of its ...
1
vote
1answer
58 views

Most Efficient Estimator and Uniformly minimum variance unbiased estimator

I am studying Estimation theory from "Introduction to theory of statistics" by "Mood and Graybill". After completing I thought I understood UMVUE (uniformly minimum variance ...
2
votes
2answers
43 views

“… because sample mean gets different values from sample to sample and it is a random variable with mean $\mu$ and variance $\frac{\sigma^2}{n}$.”

This answer by user "sevenkul" says the following: The sample mean $\overline{X}$ also deviates from $\mu$ with variance $\frac{\sigma^2}{n}$ because sample mean gets different values from ...
2
votes
1answer
31 views

Citation: Sample mean as consistent and unbiased estimator of the expected value

A reviewer asked for a citation that the sample mean is a consistent and unbiased estimator of the expected value and therefore converges towards the expected value. I know I can easily do the ...
1
vote
0answers
18 views

Can you use a test statistic like Anderson–Darling for parameter estimation?

I was looking at the Wikipedia article for maximum spacing estimation and this got me thinking. The idea behind this method is that if you know the true distribution of the data, then its CDF should ...
0
votes
0answers
34 views

Taking Expectation Over Inverse Sum of Indicator Functions?

I'm working with a zero inflated Poisson distribution that has the following pmf: $$f(y|w,\lambda)=wI[y=0]+(1-w)\frac{e^{-\lambda}\lambda^{y}}{y!}$$ I would like to find the expectation of the ...
2
votes
1answer
12 views

Variance Estimator Change if we know Population Mean? (Normal dist. example)

For a normal distribution $N(\mu, \sigma^2)$ a commonly used unbiased and consistent estimator of variance is $$\hat \sigma^2=\frac{\sum_ix_i^2 + n(\bar x)^2}{n-1}=\frac{\sum_i(x_i-\bar x)^2}{n-1}$$ ...
3
votes
1answer
46 views

Unbiasedness of an estimator

I want to show that the following estimator is an unbiased estimator of E(X): $\frac{2}{N}\sum_{i=1}^{N/2}(X_i + \mu) + \frac{2}{N}\sum_{i=N/2+1}^{N}(X_i - \mu)$ In other words, I want to show that ...
0
votes
0answers
12 views

Bias/variance of an estimator versus bias/variance of a model

I have come across two different but similar definitions of bias and variance. In the book Deep Learning book, in section 5.4, bias and variance are defined for an estimator, where bias is the ...
0
votes
0answers
26 views

Two-dimensional Multinomial distribution and an estimator under the assumption of independence

We have two-dimensional multinomial distribution $Mult(n, p)$, where $p = (p(x, y))_{x \in \mathcal{X}, y \in \mathcal{Y}}$ is a matrix containing probabilities of outcomes of $(X, Y)$: $p(x, y) = P(X=...
0
votes
1answer
51 views

Unbiased estimatior for $\bar{x} $ from a Random Sample with unequal selection probability

I have the following population: Where the left column is the age of our individuals and the right column is their weight (in kg). The exercise tells us that we use Random Sampling with no ...
1
vote
0answers
15 views

Negatively correlated estimators for the AR-1 process

I have the following question. Assume we have a stochastic process \begin{equation} y_t = \gamma + \phi y_{t-1} + \epsilon_t, \ \epsilon_t \sim \mathcal{N}(0, \sigma^2), \end{equation} where $|\phi| &...
0
votes
0answers
37 views

Does multicollinearity produce wrong beta estimates?

This is the first time for me to ask a question here. I'm sorry that if I break any rule here. I have encountered a problem about the consequence of multicollinearity. During reading the explanation ...
0
votes
0answers
11 views

What are the available tools (results) that can be used to pin down the rate of convergence of an estimator besides CLT?

What are the available tools (results) that can be used to pin down the rate of convergence of an estimator besides CLT? It would be great if you could illustrate how to use the tools (results) with ...
0
votes
0answers
12 views

Variance estimator using mixture of scaled and unscaled data

Given two datasets: $X_1, \dots, X_n \sim N(1, \sigma^2)$ and $X_{n+1}, \dots, X_N \sim N(1, 2\sigma^2)$ My proposed estimator for $\sigma^2$ is simply a scaled combination of both classical ...
0
votes
0answers
24 views

How to estimate differences between two normal means when variances are tested to be unequal

Let $X_1 X_2...X_n, Y_1 Y_2...Y_n$ be $\sim N(\mu_x$, $\sigma_x^2$), $\sim N(\mu_y, \sigma_y^2$). I want to estimate ($\mu_x - \mu_y$) and have used F-test to see that $\sigma_x^2$ $\neq$ $\sigma_y^2$ ...
0
votes
1answer
23 views

Standard Error as an estimator

The Standard Error is said to estimate the true standard deviation of the sampling distribution of the difference in the sample means. But we also studied that estimators are calculated only for ...
0
votes
2answers
65 views

Proving the consistency of this OLS estimator for $\hat\beta_1$?

So in this particular linear regression model we are given that $\beta_0=0$. The goal is to find the estimator, $\hat\beta_1$, and show that it is consistent. I managed to find $\hat\beta_1$ as ...
0
votes
0answers
10 views

R: Any implementation of Weighted-average least squares (WALS) estimators?

I just stumbled upon Weighted-average least squares (WALS) estimators. One of the authors refer to a Matlab and Stata package, but is anyone aware of an implementation/example/package in R? Thanks in ...
0
votes
0answers
44 views

Explain the descriptive statistics notion of population (distribution) to a measure theorist

My understanding of probability theory is a rigorous measure-theoretic one and I know almost nothing about descriptive statistics. However, I need to understand some of the commonly used notions in ...
1
vote
0answers
41 views

Techniques for finding UMVUEs

I'm learning about the different techniques available to find the UMVUE such as Rao-Blackwell and Lehmann-Scheffé theorems. My question is how to know when is better to use one method from the other ...
1
vote
0answers
34 views

How to construct the likelihood function of compound Poisson process?

Since in the compound Poisson process (CPP), the jumps occur according to the Poisson process with intensity $\lambda(t)$. The jumps size is iid random variables and itself independent of the Poisson ...
0
votes
0answers
13 views

ADAM First and Second Moments

I'm trying to understand why ADAM uses the gradient and the gradient squared respectively as estimators the first and second moments. I was assuming that the random variable we were estimating was the ...
2
votes
0answers
19 views

Adjusting for non-response bias in a survey

Say I have a target population of 10,000 for a survey, and only 2,000 respond, because they likely feel strongly about the survey topic (either happy or angry). I have clear, abundant non-response ...
0
votes
0answers
20 views

Estimate binomials given monotonic probabilities

I am given $N$ ordered coins and for each coin $i=1,..,N$ some trials $X_i \sim Bin(n_i, p_i)$. The coins are ordered in the sense that I know a priori that $0\leq p_1\leq p_2 \leq ... \leq p_N \leq 1$...
0
votes
1answer
49 views

Variance of an integer-valued parameter estimator for Poisson distribution

Supposing I have a Poisson distributed random variable $X \sim \text{Poiss}(\lambda)$ with a parameter $\lambda$ that could take integer values only. Let $x$ be a single observation of a random ...
0
votes
0answers
19 views

stabilizing method for normal distribution $N(\mu, \mu^p)$

Consider a sequence of n i.i.d. sample $X_n$ from normal distribution $N(\mu, \mu^p)$, $p$ interger, I am curious about the stabilizing method for the estimator $\hat{\mu}=\bar{X}$. Apparently, by CLT ...
0
votes
0answers
14 views

Consistent estimator of E(Var(Y|X)) for continuous random variables

Consider a set of $n$ i.i.d. observations $\{(x_i,y_i)\}_{i=1}^n$ of a pair of random variables $(X,Y)$ with finite mean and variance. I am interested in estimating the unexplained variance $\mathbb{E}...
0
votes
3answers
67 views

Show that the two estimators are unbiased for $\theta$ [closed]

$X_1$ and $X_2$, one accurate than the other, are subject to the standard deviations, $\sigma$ and 1.25$\sigma$ respectively. $X_1$ occurred 6 independent times, giving a mean of $\bar{x}_1$ while $...
0
votes
0answers
17 views

What determines the precision of my estimator?

I suppose I want to run the following regression: $$y_{ist} = \beta_0 + \beta_1 \tau_{st} + \beta_2 T_t + \beta_3 \tau_{st} T_t + \epsilon_{ist} $$ $\tau_{st}$ is my continuous treatment variable. ...
6
votes
2answers
254 views

Variance and asymptotic normality of $\frac{1}{n-1}\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2$, where $X \sim \mathcal{N}(0,1)$

Consider a length $n$ vector $\mathbf{x}$ containing $n$ i.i.d. observations $\{x_i\}_{i=1}^n$ of a standard normal random variable $X$. Let $\mathbf{z}$ be a length $n-1$ vector whose entries are $...
0
votes
0answers
16 views

With a survey sample that includes probability weights, is taking the mean without using weights a biased/inconsistent estimator?

If you are estimating the population mean. or would it be sampling bias if one were to take an unweighted mean?
1
vote
0answers
38 views

Nonparametric hypothesis testing to infer that the population variance belongs to a certain interval?

Let $n \in \mathbb{N}$ be either or small or large. Let $\{x_1 \dots x_n\} \subset \mathbb{R}$ be a random sample generated by a random vector $X \in \mathbb{R}.$ The test(s) I'm aiming for are about ...

1
2 3 4 5
12