Tagged Questions

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

learn more… | top users | synonyms

0
votes
0answers
5 views

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

Can the error of the sample mean, i.e., $|\bar{X}-E[X]|$, be bounded using Chebyshev inequality (or something similar)? $X$ is a discrete random variable with an unknown distribution. I am not ...
0
votes
0answers
6 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
33 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
106 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
24 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 ...
-1
votes
0answers
10 views

Estimator for a function depending on a random variable [duplicate]

Let $X$ be random variable in $m$ dimensional space. The distance between each pair of vectors $x_i^m,x_j^m$ is $D_{i,j}^m =d(x_i^m,x_j^m)$. Correlation Sum, $C(r)$ represents the probability of the ...
3
votes
1answer
62 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
42 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
14 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
votes
0answers
10 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
26 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
36 views
5
votes
1answer
158 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
29 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
51 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
35 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
22 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 \\ ...
0
votes
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
49 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
34 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 ...
0
votes
1answer
49 views

OLS versus ML estimation of VECM

A vector error correction (VECM) model has an equivalent vector autoregression (VAR) representation. (VECM) $\;\;\;\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta ...
1
vote
0answers
20 views

Learning the distribution of a phenomenon's occurrence with partial observation

I recently came across this problem: There is a phenomenon, which occurs exactly on one of a set of M (finite) places at every time step t= 1,2,... At each time step t, the place of occurrence is ...
5
votes
2answers
77 views

What's the difference between estimating equations and method of moments estimators?

From my understanding, both are estimators that are based on first providing an unbiased statistic $T(X)$ and obtaining the root to the equation: $$c(X) \left( T(X) - E(T(X)) \right) = 0$$ Secondly ...
4
votes
3answers
44 views

Cointegration - Why can't I estimate a VAR on the differences?

When talking about variables that are I(1) (the first difference is stationary), Lutkepohl book says: "...in general, a VAR process with cointegrated variables does not admit a pure VAR representation ...
1
vote
1answer
28 views

Prove that System FGLS is Consistent

In the Systems of Equations framework, such as Seemingly Unrelated Regression (SUR), suppose we have $g=1,\ldots,G$ equations. Let $\mathbf{X}_i$ be a $G \times K$ matrix, $\mathbf{y}_i$ be $G \times ...
0
votes
0answers
9 views

Price index - Bill method and Rate method

Can someone explaing the difference between these two methods for calculating price indices, i.e. how is index calculated? Or more precisely, what exactly is determined from the Bill method and the ...
7
votes
4answers
533 views

How to sample when you don't know the distribution

I'm fairly new to statistics (a handful of beginner-level Uni courses) and was wondering about sampling from unknown distributions. Specifically, if you have no idea about the underlying distribution, ...
0
votes
1answer
39 views

Parameter estimation in log linear models

Can anyone explain to me how parameter estimation is computed in log linear models? I followed this paper which is quite good, however I'm a bit confused in the parameter estimation part which is ...
4
votes
0answers
28 views

asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
2
votes
0answers
20 views

Instagram representative sample

I am doing a survey to study both sellers and consumers behaviour on Instagram in Kuwait. My question is how to decide on the sample size? What sampling design should I use since there is no sampling ...
3
votes
2answers
59 views

Estimator for $E[X]^2$

I'm trying to understand the theory of estimators. As I understand it now, if you have an r.v. $X$ and take $n$ i.i.d. samples then an estimator for $E[X^{2}]$ would be $\overline{X^{2}}$ since ...
1
vote
1answer
50 views

Definition of Scale median

Lehmann, in Theory of Point Estimation p.212 (and also on p.169), defines scale median as the solution to: $${E(X)I(X\le c)} = {E(X)I(X\ge c)}$$ given $X$ is a positive random variable, and ...
1
vote
1answer
37 views

Pointwise probability limit

Suppose we have a joint distribution of a random sample $(y_n,x_n)\in R^2$: $$ y_n = \beta_0 x_n + \epsilon_n $$ and the estimator $$ \hat \beta = argmin_\beta \frac{1}{N} \sum_{n=1}^N(y_n - \beta ...
0
votes
0answers
16 views

Getting estimates of hospital specific stroke admissions data

I am analysing a small data set on stroke process of care gathered from public sources but don't have access to hospital specific emergency stroke admissions over a period of time data except for ...
0
votes
0answers
24 views

How to make non-parametric distribution estimation with known, limited number of points of the CDF?

Is there any method to make non-parametric estimation of a cumulative distribution function (CDF) which actual points (not a sample) can be calculated numerically? I have a numerical method which can ...
0
votes
0answers
23 views

L1 distance between empirical and true distribution for discrete distributions

I have a distribution over the discrete set $\mathcal{A} = \{1, \ldots, d\}$ where the pmf is $p(.)$. That is, $p(i)$ is the probability of obtaining $i$ from $\mathcal{A}$. Given a dataset with $n$ ...
0
votes
1answer
13 views

Combining two estimates of p in a binomial estimation

I have an estimation problem for a binomial data. I got a sample and from that I can get an estimation. But I also have a kind of prior information about the p. But mind it, this prior is just a ...
1
vote
2answers
34 views

Can you compare probabilities of an epidemic by knowing R0 values?

In comparing two diseases with different basic reproduction numbers (R0), is it possible to use the R0 values to calculate the probability of an epidemic spread through a population? For example, if ...
2
votes
2answers
138 views

Gaussian noise model derivation

I have the following linear regression model, $y = f(x;w) + n$, where $y$ is the vector of true labels, $x$ is the observed data, $f(x;w) = w^Tx$, and $n$ ~ $N(0, \sigma^2)$ is the noise. Why then ...
0
votes
1answer
23 views

Finding $p(\tilde{y}|x)$ given measurement model and error distribution

Given two measurements of a variable $x$: $\tilde{y_1}=x+e_1$ $\tilde{y_2}=x+e_2$ where $e_1,e_2$ are zero-mean random variables following a bivariate normal distribution, with a known joint ...
0
votes
1answer
24 views

Maximum Likelihood solution of a zero-covariance process

Let the measurement model be: $\tilde{y}=Hx+v$ $\tilde{y}=H\hat{x}+e$ where $H$ is the basis matrix, $v$ is a constant vector equal to, say, $a$, $x$ is the measurement variable and $e$ is a ...
0
votes
0answers
14 views

Need help with importance sampling over HUGE sample space

My underlying problem is fairly simple, but the sheer size is what is causing issues. I would like to use importance sampling, but am unsure about its implementation. Problem statement: We have $N$ ...
0
votes
0answers
24 views

Recursive Bayesian Estimation, $p(C_k|\mathbf{x})$ as (discrete) likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|\mathbf{x})$, from for example a logistic regression, to ...
1
vote
1answer
60 views

Estimation based on observing sum of two variables

Let $X_1,\dots,X_n$ are i.i.d normal $N(\mu,\sigma^2).$ Suppose that we only observe $$ X_1+X_2,\dots,X_1+X_n,\dots,X_{n-1}+X_n, $$ i.e, $X_i+X_j$ for all $i<j.$ I wish to find the best estimator ...
6
votes
1answer
45 views

James-Stein Estimator with unequal variances

Every statement I find of the James-Stein estimator assumes that the random variables being estimated have the same (and unit) variance. But all of these examples also mention that the JS estimator ...
0
votes
0answers
63 views

Distributions where higher variance in sample estimates leads to smaller loss when estimating mean

Let's say I have an unknown distribution a. I do not know the shape, mean or variance of this distribution but I have access to another distribution ...
0
votes
0answers
67 views

Gaussian Process Regression/ Classification

How do we estimate parameters of the model while performing Gaussian Process Regression or Classification? While performing regression, we estimate parameters such that the model is the best fit to ...
4
votes
0answers
37 views

bias of an estimator when using stopping rules

Consider the setting where $X_1,X_2,...$ are i.i.d. real-valued random variables with $\mathbb{E}[X_i] = \theta$ and let the random variable $\tau$ be an associated stopping time. I'm wondering what ...
3
votes
2answers
98 views

Growing number of Gaussians in a mixture

Let I have a Gaussian mixture consisting of $n$ Gaussians that is already fitted (e.g. using EM algorithm) with respect to a given data set. Now I want to add one more Gaussian to make the mixture ...
0
votes
2answers
61 views

Maximum Likelihood estimator from sample distribution $N(0,\sigma^2x_i^2)$

Let independent random variable $Y_1,...,Y_n$ have respective distributions $N(0,\sigma^2x_i^2)$, where $i=1,2,...,n$ are known constants such that $x_i\neq 0$ for all $i=1,2,...,n$. Find the maximum ...