Any statistical process which seeks to approximate an unknown value, such as a distribution, a point estimate (e.g. mean), or confidence interval.

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2
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2answers
60 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( ...
0
votes
0answers
34 views

Hayashi yoshida estimator for correlation not coming between -1 to 1

I took two time series data with 141 data points in total with time stamps. i found out actual correlation between them which is about 0.97. Now i find the Hayashi Yoshida estimator for correlation. ...
12
votes
2answers
634 views

What is the logic behind method of moments?

Why in "Method of Moments", we equate sample moments to population moments for finding point estimator? Where is the logic behind this?
1
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0answers
44 views

Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient

Consider when $X_1, ..., X_n \sim $ Bernoulli($p$). We want to estimate $p(1-p)$. Suppose $\hat{p}=\frac{1}{n}\sum_{i=1}^nX_i$. Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient ...
2
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0answers
10 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 ...
1
vote
1answer
37 views

Correctness of a proof for Hodges' estimator

We know the following is Hodges' estimator: $$ \delta_n = \begin{cases} \bar{X}_n & |X_n| \geq n^{-1/4} \\ a\bar{X}_n & |X_n| < n^{-1/4} \\ \end{cases} $$ where $X_1, ..., X_n \sim ...
0
votes
1answer
20 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
14 views

the total number of over lapped sets [closed]

I have a number of files that have specific features like this : feature1: the number of files that have it are 100 feature2: the number of files that have it are 50 feature3: the number of files ...
2
votes
2answers
55 views

estimating the value of a property (real estate) using the hedonic regression

I'm trying to estimate the value of a property depending on the property characteristics. I did some research and I found out, that it would be better to use the Hedonic Model/Regression instead of ...
0
votes
1answer
32 views

Predicted lm() means of log-transformed and untransformed data not equal

Why is the back-transformation of the predicted values so different from the observed when the observed are log-transformed? Sample data: ...
5
votes
1answer
63 views

Error bars on log of big numbers

I am calculating a quantity of the following form: $\mu = \log( \frac{1}{n} \sum_{i=1}^{n} e^{\phi(X_i)} )$ via MC. $X_i$ are iid and I can sample them. I want to give error bars\ confidence ...
1
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0answers
23 views

Unbiased Estimator of Product

Suppose there are stationary times series $\{A_i\}_{i=1}^{T},\{B_i\}_{i=1}^{T},\{C_i\}_{i=1}^{T},\{D_i\}_{i=1}^{T}$, which may not necessarily be independent processes. We know that ...
5
votes
1answer
40 views

Show the shortest confidence interval of a normal distribution

I'm having trouble formally showing a problem I have been given. It goes as so: Show that among all $(1-\alpha)*100$% confidence intervals of the form ...
3
votes
1answer
62 views

Unbiased estimator and sufficient statistic from discrete uniform distribution

$z_1,...z_n$ is a sample from a discrete $\{1,...,N\}$ uniform distribution. I have two questions: 1; I want to find an unbiased estimator for N, with the help of $z_1$ 2; I want to find a 1 ...
0
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0answers
17 views

Confidence intervals for maximum likelihood estimator with constraints

Let us suppose I have a maximum likelihood estimator for a multivariate parameter $\vec{\theta}$. The parameter is subject to the following constraints: $\theta_i \in [0,1]$ $\sum_i \theta_i = 1$ ...
0
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0answers
14 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 ...
6
votes
1answer
147 views

Beginner learning resources : Pdf and likelihood function for non-Gaussian time series model

I am struggling with exercise problems related to blind system identification where the knowledge about the source input is assumed to be known using maximum likelihood estimation of univariate time ...
0
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0answers
37 views

Density function for non-gaussian case

Consider a linear time series model - Autoregressive model ($y$) of dimension $n \times 1$ where $n$ is the number of data samples and only one variable AR model of order $p >1$ is excited by ...
1
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0answers
13 views

Define latency by enter/exit counts?

Is it possible to estimate latency in following case? Assume that one has a black box and objects that go through it. There exists system that logging facts of entering and leaving the box. And all ...
0
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0answers
12 views

Parameter estimation using lmom package

As a part of my risk management job, I need to try to fit various distributions to loss data. I have been using the lmom package in R for estimating parameters for ...
0
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0answers
7 views

Classical Approach to State Estimation?

Parameter estimation methods can be either classical (e.g. least squares, maximum likelihood) or Bayesian (e.g. maximum a posterior). But all state estimation methods I can think of (e.g. Kalman ...
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0answers
3 views

How do you estimate $\alpha$ parameter of a latent dirichlet allocation model?

Blei has shown that it is possible to estimate $\alpha$ in a LDA model, but I have yet to find a library (any library; C, C++, Java, ...) to do so. Usually, implementations (including Blei's) treat ...
3
votes
1answer
39 views

Convergence in probability for two statistics of Laplace random variables

Suppose $X_i$ are iid random variables. $X_i \sim \mathrm{Laplace}(\lambda)$. Also define: $$ U_n = \frac{1}{n-2}\sum_{i=1}^n{|X_i|} $$ $$ V_n = \sqrt{\frac{1}{n-1} \sum_{i=1}^n{X_i^2} } $$ Given ...
1
vote
1answer
35 views

Clarification on a paper regarding estimating N from a Binomial Distribution

I was wondering if someone could clarify the following for me. In the paper "Inference for the binomial $N$ parameter" by Adrian Raftery, his first example outlines the posterior of $N$ given $x$ as ...
1
vote
1answer
33 views

Estimation of VECM via ML and OLS

Take a vector error correction (VECM) model: $$\;\;\;\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-(p-1)}+\varepsilon_t$$ where $\Pi=\alpha \beta'$ and each row of ...
2
votes
2answers
60 views

Computational Statistics question

I've got a tricky computational statistics problem and I was wondering if anyone could help me solve it. Okay, so in your left pocket is a penny and in your right pocket is a dime. On a fair toss, ...
0
votes
0answers
46 views

Best linear unbiased estimator

I have a sample of N stocks. I have the following information: For each stock i, I have an estimate of variance (of returns) $\hat{\sigma}^2_{i}$. I also have a fitted variance, denoted by ...
0
votes
0answers
10 views

Estimating a Poisson process when arrivals occur out of order?

TL;DR: Given a Poisson process with constant intensity $\lambda$, but you are hearing about arrivals out-of-order, how would you (sequentially/iteratively) estimate $\lambda$? What if the problem is ...
2
votes
0answers
39 views

Does Hoeffding's inequality apply to sampling from finite populations?

Based on Hoeffding's theorem, one could easily find the minimum number of samples required for the inequality $\Pr \left(|\bar{X} - \mathrm{E} [\bar{X}]| \geq t \right) \leq \delta$ to hold as ...
0
votes
1answer
19 views

Fit stochastic differential equation to data

Could I have some review of the method I used to fit following SDE: dX = f(t) dt + s X dW Fitting method: Calculated sample for sdW from our data as: sdWt = ...
0
votes
1answer
21 views

Estimate effect on mean of dependent variable of an increase in the independent variable in a linear regression

Suppose I have the linear regression equation: Y = B0 + B1(x) How do I find the estimated effect on mean Y of an additional 50 to x? I believe this is the multiplicative effect.
1
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0answers
63 views

Can Chebyshev inequality be used to bound the error of the sample mean?

Can the probability of error of the sample mean, i.e., $\Pr(|\bar{X}-E[X]| \geq \epsilon)$, be bounded using Chebyshev inequality (or something similar)? $X$ is a discrete random variable with an ...
0
votes
0answers
19 views

Basic questions about stochastic gradient descent / Robbins and Monro algorithm

I have a LOT of time series observations and I would like to estimate a simple AR(1) model $$ y_t =c+ \phi y_{t-1}+ \varepsilon_t \qquad \varepsilon_t \sim \text{N}(0, \sigma^{2}) $$ with parameters ...
1
vote
1answer
41 views

proving the asymptotic distribution of the mean

Let ${X_t} = \mu + \sum\limits_{j = - \infty }^{ + \infty } {{\psi _j}{\varepsilon _{t - j}}}$ with $\varepsilon$ is a white noise iid with variance $\sigma^2$ , $\sum\limits_{j = - \infty }^{ + ...
5
votes
3answers
115 views

Fast density estimation

Suppose you are trying to estimate the pdf of a random variable $X$, for which there are tons of i.i.d. samples $\{X_i\}_{i=1}^{n}$ (i.e. $n$ is very large, think thousands - millions). One option is ...
1
vote
1answer
30 views

Estimate of Coefficient Variance in multiple regression

I'm trying to compute an estimate for the variance of the estimated coefficients in a non-linear regression using the formula described in link. I can't figure out how to build $F_{ij}$ Let's ...
3
votes
1answer
79 views

Robust estimates of the covariance matrix in the big data space

I am trying to compute the robust estimates of the covariance matrix (and also the mean) in the big data space. I am aware of FastMVE and FastMCD (Minimum Covariance Determinant and Minimum Volume ...
2
votes
1answer
44 views

What's the statistical method where you add a certain number to each sample to make the distribution slightly more uniform?

Please forgive my lack of knowledge - it's been a while since I've taken classes in statistics, and even then, it was not my strong point. I'm trying to recall a method used to upweight all values in ...
0
votes
0answers
20 views

How to estimate Extreme value distribution parameter

Assume that I have non-negative Gamma random variables $\{X_i,i=1\dots n\}$ and I want to find $M_n=\max\{ X_i\}$. I want to apply generalized extreme value distribution (GEV), but how to find $\mu$, ...
2
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0answers
17 views

Evaluating survival models in the presence of covariate-dependent censoring

I have a censored survival analysis problem with the following characteristics: Failure times are discretized The censorship distribution depends on certain covariates I don't have a ...
1
vote
1answer
38 views

Is there any method to quantify parameter estimation uncertainty of method of moments fitting technique?

If I want to fit a distribution (let's say we can be certain about the type) to observations using maximum-likelihood method, I have many options to express the parameter estimation uncertainty due to ...
5
votes
3answers
41 views
6
votes
1answer
191 views

Estimating parameters for a binomial

First of all I'd like to precise that I'm not an expert of the subject. Suppose to have two random variables $X$ and $Y$ that are binomial, respectively $X\sim B(n_1,p)$ and $Y\sim B(n_2,p),$ note ...
0
votes
1answer
36 views

Point estimation MLE and MME

Consider the family of probability mass functions given by f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,.... and indexed by parameter k E Z. For a random sample of size n, derive with justification: a) ...
2
votes
1answer
55 views

Conceptual question on estimation : How to calculate the variance of estimation error

EDIT/ UPDATE: I have understood CRLB & why we need it. But my problem is something else. In book ...
2
votes
1answer
36 views

Average of Dependent Variables

Suppose $X_1, \ldots, X_n$ are dependent varibles with identical marginal distribution. Denote the common population mean as $\mu_0$. In this case, is $\frac{1}{n} \sum X_i$ a reasonably good ...
0
votes
1answer
28 views

Inverse Gamma Prior with Scale Parameter set to 1

\begin{align*} X_{ij} \mid \mu_i , \sigma^2 & \sim N(\mu_i, \sigma^2) \nonumber \\ \mu_i & \sim N(\mu_0, \tau^2) \nonumber\\ % S_i^2 \mid \sigma^2 & \iid \chi_{n-1}^2/(n-1) \nonumber \\ ...
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0answers
15 views

Transforming frequency data into a rating system

I'm working on a project for fun using data (items from a persistent video game) I've gathered from the web. At the moment, the data consists of around 180,000 rows which will probably grow quite ...
3
votes
2answers
54 views

Fitting parametric CDF to ecdf

There is a random variable $X$, but the only data I observed is actually its empirical distribution function (at a suitably fine grid). That is, I only observe $\hat{F}(x)$:=$\#\{x\leq u\}\over N $ ...
0
votes
1answer
36 views

On approximating the MSE of an estimator

I'm trying to approximate the MSE of an estimator through simulation, in particular estimators of the form $$ \hat{\theta} = \sum_{i=1}^N w_i X_i $$ Where $X = \{X_1,...,X_N\}$ are i.i.d. samples ...