Questions tagged [estimation]
This tag is too general; please provide a more specific tag. For questions about the properties of specific estimators, use [estimators] tag instead.
1
vote
0answers
24 views
Finding the limiting expected length of a confidence interval
I have a query regarding the limiting length of a confidence interval, related to this question.
Suppose I have a sample $X_1,X_2,\ldots,X_n$ from the distribution
$$f_{\theta}(x)=\frac{1}{\...
2
votes
2answers
256 views
Why is this estimator biased?
$X_{1},X_{2},..,X_{n}$ are iid $\sim Poisson(\mu)$
than the MLE for $\theta=e^{-\mu}$ is $\hat \theta =e^{-\bar x}$
Why is this considered to be biased for $\theta$?
Is $E[\hat \theta]$ not $\theta$...
0
votes
0answers
21 views
Estimate prediction error
I actually have a a dataset of 1100 observations containing 99 features and one targeted value (Y). The idea is that my dataset is split in two: 100 observations where the targeted value is know and ...
0
votes
1answer
22 views
check understanding on unbiased and consistent estimator
I'm trying to understand the expected value of an estimate. Here's my understanding.
The expected value of the estimate $\bar x$ of the parameter $\mu$ is what the mean of xbar tends to as we ...
0
votes
0answers
31 views
minimizing variance [duplicate]
We are constructing an estimator out of two independent samples
Sample 1: $\hat\mu_1$ $var(\hat\mu_1 )=\sigma_1^2$ and sample 2: $\hat\mu_2$, $var(\hat\mu_2)=\sigma_2^2$
$$\hat\mu_3=a*\hat\mu_1+b\...
0
votes
0answers
18 views
How to prove that conditional distribution for Y given OLS for simple linear regression do not depends on original parameters? [duplicate]
How to prove that for a simple linear regression model: $$y_i=\beta_0+\beta_1 x_i+\varepsilon_i,$$ the conditional distribution
$$Y|\hat{\beta}_0,\hat{\beta}_1$$
do not depends on $\beta_0$ and $\...
3
votes
0answers
33 views
UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution
I have a random variable X with pmf
$$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$
and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$.
Here is my attempt so ...
1
vote
1answer
24 views
What's the point in using identity matrix as weighting matrix in GMM?
What is the point of using the identity matrix as weighting matrix in GMM?
GMM is the minimizer of the distance $g_n(\delta)'\hat{W}g_n({\delta})$, where $g_n = \frac{1}{n}\sum_ix_i\epsilon_i$. If we ...
0
votes
0answers
18 views
Cardinality estimation using ordered statistics
In cardinality problem (count-distinct problem) the goal is to estimate the number of unique elements in a set. HyperLogLog is one such algorithm [ref] Another approach is using order statistics, such ...
1
vote
1answer
53 views
Distribution of Maximum Likelihood Estimator
Why is the Maximum Likelihood Estimator Normally distributed? I can't figure out why it is true for large n in general. My attempt (for single parameter)
Let $L(\theta)$ be the maximum likelihood ...
1
vote
0answers
33 views
Finding UMVUE for $g(\theta)$ that satisfy $g(0)=0$ in discrete uniform $f(x\mid\theta)=\frac{1}{\theta} I_{1,…,\theta}(x)$
Let $x_1, \ldots x_n, \overset{\text{i.i.d}}{\sim}f(x\mid \theta)=\frac{1}{\theta} I_{1,...,\theta}(x)$.
I know that $T=X_{(n)}$ is complete and sufficient statistic for $\theta$ and
$$f_T(t\mid\...
1
vote
1answer
42 views
order of the time series model
I am kind of new to time series modeling.
I am trying to fit a model for a time series variable. I am trying to fit a ARMA model.
I am using R to do the analysis. When i estimate the model using both ...
0
votes
1answer
25 views
Finding MAP estimate
I think after all the reading I've done I still don't fully understand MAP estimation. I came across a problem that's leaving me dumbfounded.
Suppose $A$ ~ $N(0,\sigma^2_1) $ and $\epsilon$ ~ $N(0,\...
1
vote
2answers
48 views
Checking if a minimal sufficient statistic is complete
Let $X_1, \cdots, X_n$ be iid from a uniform distribution
$U[-\theta, 2\theta]$ with $\theta \in
\mathbb{R}^+$ unknown. Check if the minimal sufficient statistic of $\theta$ is complete.
I found ...
5
votes
1answer
130 views
Definition of Statistic
I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the ...
2
votes
1answer
31 views
Estimator based on inequality data
$X_i \sim N(\mu, \sigma^2)$ (iid), $i = 1,2,...,N$, I want to estimate $\theta = (\mu, \sigma^2)$. Problem is, I don't observe $x_i$. For each $i$, I only observe $(a_i, b_i)$, and I know that $a_i &...
0
votes
0answers
8 views
How to estimate a polling result from a given sample?
I need to estimate people's idea on some topic and I only got information from a specific group of people. The procedure that generate the group could be biased. e.g. 47% of the sample support the ...
0
votes
1answer
41 views
Approximate the data to a single curve
The question might be simple, but I am not able to find the answer. Hence I am asking here. I did search google but didn't get an answer.
I have a continuous stream of data coming from an API in the ...
0
votes
1answer
58 views
Law of Iterated Expectations Example
Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
0
votes
0answers
30 views
differentially private release of histograms (non-negative valued queries)
Two practical questions arise when releasing differentially private histograms/counts via addition of Laplace/Gaussian noise:
1) Is the result of noise addition truncated/rounded (since we know that ...
0
votes
0answers
9 views
MLE of an EGARCH(1,1) Model with Gaussian Innovations
I want to estimate parameters of an exponential GARCH(1,1); namely EGARCH(1,1) model using optimization tools at R. However, I don't want to use a ready package like rugarch or an another package. I ...
0
votes
0answers
19 views
Is the posterior distribution for the model described in this question Gaussian?
I was in the middle of writing a long answer to
Uncertainty estimation in high-dimensional inference problems without sampling?
but I was suddenly struck by doubt: since the model assumes a Gaussian ...
0
votes
0answers
20 views
Past observations & Error terms in GARCH and ARMA models
I am a bit confused concerning some of the "underlying concepts" of ARMA & GARCH models. I know that ARMA models are meant to forecast the conditional mean of a process, while GARCH models are ...
0
votes
1answer
28 views
Significance levels of the estimated logistic regression coefficients for artificially generated data sets
I'm trying to simulate 2 data sets for testing some variable selection methods for logistic regression models. The initial step is to fit the logistic regression for all candidate predictors. But ...
0
votes
1answer
34 views
Estimation of a population number
Suppose that we know the population number $n$ of a country in $2014$ for people aged between $0$ and $19$ years what would be a good simple way to estimate the population number between $1$ and $18$ ...
0
votes
0answers
13 views
Updating the kalman filter and RTS with multiple measurements at each time step
I am currently working with kalman filter in area of target tracking. The sensor we are using gets me multiple measurements at each time step. Number of measurements is not fixed, meaning that in one ...
2
votes
1answer
64 views
variance of nonparametric estimator of mean
I'm having some trouble with understanding how to calculate the variance of a non-parametric estimator.
The example comes from Wasserman's "All of statistics book"
Let $X_1, \ldots,X_n \sim \text{...
1
vote
0answers
27 views
Manuel estimation of a Garch(1,1) parameters using MLE vs rugarch package in R
I want to estimate parameters of a GARCH(1,1) model using rugarch package in R and manually(using maximum likelihood).
Firstly, I import and transfrom the data as below(Amazon return data)
...
1
vote
1answer
70 views
Understanding the details of Expectation Maximization(EM) for estimating the parameters?
When using the Expectation Maximization(EM) for estimating the parameters, every time I came across a different problem I see a totally different representation of the likelihood/Expectation function ...
1
vote
0answers
59 views
Win probability of a game character vs. two other characters
Clarification about original problem:
Dota 2 is played in matches between two teams of five players—known as the Radiant and Dire—, with each team occupying and defending their own separate base ...
0
votes
0answers
8 views
Can I improve an estimate of a coin-flip probability from a single trial using an imperfect oracle?
I have the following generative model:
I have a unknown random variable $S\in[0,1]$ and samples $s_i \sim S$. I do not observe $s_i$ directly, but instead an imperfect oracle $q_i$, which might or ...
0
votes
1answer
22 views
How to use derivatives of a function to better estimate its variance over the domain?
How to use derivatives of a function to better estimate its variance over the domain?
I have a scalar smooth function $f(x)$ and a multivariate random variable $x$ with known distribution (e.g. ...
0
votes
0answers
26 views
Simulating from an Epanechnikov kernel density estimate in MATLAB / exact form of the Epanechnikov kernel in MATLAB?
It's my first time posting, so apologies if I'm breaking any etiquette. I've used MATLAB's ksdensity function to estimate a density using the Epanechnikov kernel and would now like to make repeated ...
3
votes
1answer
87 views
Interpreting BLUPs or VarCorr estimates in mixed models?
I am referring to the question. When estimating random effect (RE) variance or correlation, the estimations are different in VarCorr(mod) function and when ...
1
vote
0answers
25 views
What is the difference between model prediction and model estimation? [duplicate]
Could you please explain for me what's the difference between model prediction and model estimation ? What is the difference between a prediction interval and an estimation interval?
0
votes
0answers
16 views
Good turing smoothing for unigram LM
I was wondering if it is at all possible to use good turing smoothing for unigram language model?
I know that this smoothing technique helps distribute the weights from most occurring words to less ...
1
vote
0answers
33 views
Need question answered [closed]
With a universe of 143, 30 of those were queried/sampled at random, giving answers of (dividing into groups of) A, B, C,and D. Question: If all 143 were sampled, what is the probability that the ...
0
votes
1answer
23 views
Integrating out an extra parameter in Maximum Liklihood estimation
In estimation theory I have seen maximum likelihood being used assuming additive Gaussian AWGN where the signal is a function of multiple parameters(like frequency, time delay, phase, bit).
Sometimes ...
0
votes
0answers
26 views
Showing that multiple linear regression breaks down with less variables than parameters to estimate
I consider a multiple linear regression model with only two explanatory variables.
$\ y=\beta_0+\beta_1x_1+\beta_2x_2+\epsilon$
fulfilling the CLM properties.
From the moment restrictions
$\frac{1}...
0
votes
0answers
22 views
Bonferroni confidence region for shifted Laplace parameters
Consider the shifted Laplace distribution with the density:
$$f(y)=\frac{\theta}{2}e^{-\theta|y-\mu|}\quad, \quad y\in \mathbb R$$
Using the Bonferroni method, construct a $100(1-\alpha)\%$ ...
5
votes
1answer
52 views
Is MLE intrinsically connected to logs?
My mathematical exploration led me the following claim:
Claim: MLE is fundamentally connected to logs (and KL divergence, which also uses logs). It’s not correct to say log shows up simply to make ...
3
votes
1answer
50 views
Consistency in uniform distribution
I know what consistency is but in options C and D both U and V are given whose covariance is quite difficult to find.
0
votes
0answers
76 views
Maximum likelihood estimators of $\theta$ in $U(2\theta-1,2\theta+1)$ distribution
I understand why (D) is one of the answers but i dont know about the rest?
0
votes
0answers
56 views
Derive the Likelihood Ratio Test for multivariate normal and specific covariance matrix
Let $X_1,\ldots, X_n$ be i.i.d. $N(µ, C)$ random $p$-vectors.
Derive the Likelihood Ratio Test for
$H_0: C = σ^2(1 − ρ)I_p + ρ1_p1_p^T$, where
$1_p =
\begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{...
5
votes
1answer
49 views
Can MCMC algorithm estimate partition function (normalizing constant)?
Importance Sampling can estimated the normalizing constant by averaging the weights (the ratio of unnoramlized distribution and importance distribution). Is there anyway that MCMC algorithm can ...
1
vote
0answers
59 views
Estimation of parameter $\widehat\beta$ in the linear model [closed]
Consider the simple linear model $Y=X\beta+\varepsilon$ where $\varepsilon\sim N_n(0,\sigma^2I).$
It known that $\widehat\beta=(X^tX)^{-1}X^tY$.
Also,
$$\pi(\beta\mid Y)\propto\Gamma\left[\frac{1}{...
3
votes
1answer
59 views
Recommended Mutual Information Estimator for Continuous Variable
The mutual information seems to be quite an interesting measure of the relationship between variables. As such I wanted to apply it to investigate the relationship of two continuous variables $X$ and $...
2
votes
0answers
141 views
MLE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{2(\theta-x)}{\theta^2}\mathbf1_{0<x<\theta}$
Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables with pdf
$$f(x\mid\theta)=\begin{cases}\frac{2(\theta-x)}{\theta^2}&,\text{ if }0<x<\theta \\ 0 &,\text{ otherwise }\end{cases}$$
...
1
vote
1answer
30 views
Number of parameters in Bayesian Classifier
Problem
Assume we have a Bayesian classifier with the three following features to determine whether a software user is a student, an ...
2
votes
1answer
51 views
Difference between empirical distribution and the data-generating distribution? [closed]
I understand that an empirical distribution is basically sampling from the sample set with replacement. However I am not quite sure how $ \hat{p}_{data} $ and $ p_{data}$ in Maximum likelihood ...
