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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18 views

mean of a transformation of a random variable: approximating the mean by taylor expansion

I am interested in approximating the mean $E[f(X)]$ where $X$ is a random variable with mean $\mu$, variance $\sigma^2$ and support $\{0,...,n\}$ where $$f(X):=\sum_{i=X}^n \beta(i)$$ with ...
3
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19 views

Calculating a Constrained Mean

Assume that I have ten measurements of a physical quantity that must be non-negative (e.g., mass). Because of measurement error, some of the results are nevertheless negative. Can someone suggest a ...
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1answer
36 views

Using results of regression on raw observation values to approximate results of regression on relative change between observations (Simple, Linear)

this is my first time on Stack Exchange so if I did something wrong please tell me. I have a time series dataset. There is an observation $(y,x)$ for each continuous time $t$. Let’s say for each day ...
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1answer
18 views

Does the estimated overdispersion parameter of Negative Binomial depend on mean

Negative Binomial distribution can be parameterized using mean, $\mu$, and overdispersion $\psi$, so that the variance of NB is $\mu + \frac{\mu^2}{\psi}$. We know there is no analytical solution for ...
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39 views

Maximum Likelihood Estimation with Known Parameter Distribution

Consider i.i.d observation vector ${\bf x}$ from a distribution $F$ depending on vector of parameters $\boldsymbol{\theta}$ and single parameter $\alpha$. We would like to estimate parameters ...
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1answer
13 views

On using an orthogonal series to estimate a regression function

Suppose I have a function $g\in L_2(\mathbb{R})$, and we observe variables two -vectors $(Y_i,X_i)$ such that $Y_i = g(X_i) + U_i$ for some IID error terms $U_i$. If I want to estimate $g$, I want to ...
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3answers
86 views

Can I write estimate $\pm$ its standard error?

Suppose I have an MLE estimate $\hat{\theta}$ for a parameter $\theta$, and $\hat{\sigma}$ is the sqrt of the inverse of the negative of the Hessian of the log likelihood at $\hat{\theta}$. Can I ...
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1answer
26 views

Statistical Practices Using Sparse Data: Methods for Approximating Standard Deviation

Suppose I know that for a discrete, non-negative r.v. $X$ that $X | X \geq 1$ has $\mu = 3.3$ while when $X \geq 0$ has $\mu = 2.1$. That is, the subset of the population that already has a value ...
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45 views

Poisson distribution confidence interval

Edit : this is the data I used for the first part of the problem : \begin{matrix} Rocks & 0& 1& 2& 3& 4& 5& 6 & 7\\ samples & 12& 27& 28& 19& ...
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21 views

Problems with curve fitting

I have a longitudinal dataset to which I am trying to fit the following model : $$ y_{i,j} = \frac{1}{1+a_{i} \exp(-r_{i}t_{i,j})} + \varepsilon_{i,j} \tag{1}$$ The setting : $i$ ($1 \leq i \leq ...
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23 views

Can I estimate Variance of Gamma from Negative Binomial distribution distributed data, given NB is Gamma-Poisson compound

I believe the data I have follows Negative Binomial distribution (over-dispersed Poisson). We know Negative Binomial is a Gamma-Poisson compound distribution. The variance of this Gamma distribution ...
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38 views

Package ‘fitdistrplus’

I am trying to use the package ‘fitdistrplus’ in R to fit one non standard distribution to my data set. I am trying to copy the methods the package creators used in their tutorial for specifying a ...
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1answer
79 views

Learning to write your own estimation algorithms - where to start?

I am trained as an epidemiologist. Early in my career, statistical models were a black box, and when in doubt I was advised to go see a statistician. This never sat well with me, so I became ...
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3answers
117 views

Different non-parametric methods for estimating the probability distribution of data

I have some data and was trying to fit a smooth curve to it. However, I do not want to enforce too many prior beliefs or too strong pre-conceptions (except the ones implied by the rest of my question) ...
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1answer
33 views

What do you call a parameter that is estimated from historical values?

There are several methods to estimate parameters in a model (MLE, MAP, GMM). Does the process of estimating a parameter from historical data have a name?
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2answers
47 views

Compute moment and quantiles of a stream of data

I'd like to compute the moments and quantiles of a random variable which is the output of a sensor. I don't intend to store all the values this sensor outputs (let's say it outputs one value each 15 ...
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1answer
62 views

construct the maximum likelihood estimator

Let $a_{1},a_{2},a_{3}$ be independent with a normal(0,1) distribution. Define $X_{1},X_{2},X_{3}$ by $X_{1}=a_{1}$, $X_{2}=\theta X_{1}+a_{2}$ and $X_{3}=\theta X_{2}+a_{3}$ Find the MLE for $\theta$ ...
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1answer
24 views

Literature on nonparametric density estimation

I am about to write my bachelor thesis about non-parametric density estimation, especially kernel density estimators and their application in classification. As I am quite new to looking for academic ...
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1answer
32 views

Expectation expansion query

I am trying to do a proof. Define the best Bayesian estimator by $\theta^B=E(\theta|x)$. Prove that for another estimator $\gamma$ of $\theta$, we have $MSE(\theta^B)\leq$$MSE(\gamma)$. Proof: ...
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17 views

Logistic mixed model

In the logistic mixed model ${\rm logit}(P(Y_i=1))= α + βX_i + u_i + ε_i , i=1,...,m$, when we know $u_i\sim \mathcal N(0,σu^2)$, and $ε_i\sim\mathcal N(0,σi^2)$, and if we know $σi^2$ in each area ...
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16 views

Possibilities of accelerating EM algortihm

I'm trying to use the EM to estimate some parameters. I've programmed and it delivers. The problem however is that for each run of my programme, it can take either 5 seconds, 1min, 3min or more to ...
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1answer
21 views

Can I pull a confidence interval out of a single sample by dividing it into sub-samples?

Let's say I have taken a sample $S$ of a population. I am trying to figure out the population mean. Because I have only made one sample, the best I can do is assume that $\bar S$ is the mean of the ...
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18 views

Maximum likelihood estimation (MLE) for Markov Chain initial distribution?

I am working on using MLE to estimate a Markov Chain, I have successfully estimated the transition matrix $A$, using the method provided in ...
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11 views

Long Run VAR alpha and beta significance levels

I am using a VAR with 2 variables and 4 lags. I am combining the coefficients of these variables to get an overall alpha and beta value for in the form $Y = \alpha + \beta X$. In order to get the long ...
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17 views

Estimation of a vector with a big covariance matrix

I have a Gaussian vector with a known covariance, given by a Covariogram (covariance function). Inverting this matrix (lets say of size 5000x5000 and above) is not reasonable. Is there any known ...
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12 views

Estimating distribution function for dependent observations

Let $X_1,\dots,X_n$ be identically but not necessary independent distributed with distribution function $F$. I'd like to estimate $F$ efficiently. In case, $X_1,\dots,X_n$ are i.i.d., we can estimate ...
2
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1answer
65 views

What inferential method produces the empirical CDF?

The empirical cdf is an estimate of the cdf. What kind of estimation method (such as method of moments, MLE, ...) constructs the empirical cdf? Is the empirical cdf a nonparametric estimate? Do ...
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2answers
30 views

Estimating the number of streets in $CITY?

I'm not sure if this is the right place to ask these kinds of questions. I want to estimate the number of streets there are in a certain city. Any advice? How would you begin? (To be precise, I want ...
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0answers
21 views

Variance estimation from two gaussian distributions

Assume a stochastic process with observation $r$, and two hypotheses : $X \sim (0, \sigma^2)$ and $Y \sim (0, \sigma^2 + \tau^2)$. When we observe/receive $r$ we don't know which hypothesis $X$ or ...
3
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28 views

Laplace rule of succession for a Poisson process

Suppose that I measure for an interval of time $t$ a Poisson process that has unknown average rate $\mu>0$ and I observe no event. What could be a reasonable estimate of $\mu$ from a Bayesian point ...
2
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30 views

Estimation of unknown vector's amplitude with Gaussian noise

I have the following model: y = P v + n Where y is the vector of observations, v is a unit vector and n is a Gaussian random noise whose covariance matrix is the identity matrix. P is a positive ...
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20 views

Parametric transition matrix in Markov Chains

I am trying to model a discrete-time MC with transition probabilities that depend on some function of parameters i.e $p_{ij} = f(X_0,X_1)$. Suppose we take a log-linear model where $p_{ij} = ...
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39 views

Asymptotic Distribution of beta distribution parameter

I am trying to create a t-interval for the first shape parameter of a Beta distributed variable based on a random sample. $ f(x;\theta,\beta) = ...
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42 views

Need a working algorithm to find out optimal kernel bandwidth for density estimation

I am looking for a working algorithm for find out optimal kernel bandwidth for density estimation. I need to write my own program in pascal instead of using R or Matlab. So far all algorithms I ...
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18 views

Estimation based Distance observations

Let $X_1,\dots,X_n$ are i.i.d. copies of $X$, however I don't know the value of them. I only know the distance for every pair $(i,j)$. Now, assume that we are given $Y$ and $Z$ which are also i.i.d. ...
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1answer
101 views

How to keep time invariant variables in a fixed effects model

I have data on a large Italian firm's employees over ten years and I would like to see how the gender gap in male-female earnings has changed over time. For this purpose I run pooled OLS: $$ y_{it} = ...
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1answer
50 views

Coverage probability

I know the definition, but what does it imply if one estimator has greater coverage probability than another estimator? Does it imply that it has converges to the parameter of interest faster?
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30 views

Efficient estimators

let $x_{1},..,x_{n}$ be iid with cauchy distribution. Let $y=median(x_{1},..,x_{n})$ , $z=MLE$. I get that $y$ is not an asymptotically efficient estimator to the location of a cauchy distribution ...
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64 views

Determining the confidence interval of Monte Carlo data

I want to determine the confidence interval for my set of data. I have obtained the data by sampling from several Normal Distributions and running a Monte Carlo Simulation. I was wondering how I could ...
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1answer
87 views

Comparing estimators of location of the Cauchy distribution

I'm comparing the following 4 estimators of location of the Cauchy distribution: Let $x_{1},..x_{n}$ be observations and $l$ be the log likelihood function. $x=median(x_{1},..x_{n})$, ...
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9 views

Estimating variance of prediction error in bootstrapped training sets with clustered data

I have C clusters with m elements each. I split the C clusters into a large training set D and a test set T. Hence, each element in D and T has m related elements, so its a cluster. I want to ...
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2answers
159 views

Bayesian inferencing: how iterative parameter updates work?

I have been struggling with this for a while. A typical optimisation problem can be viewed as optimising some cost function which is a combination of a data term and a penalty term which encourages ...
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0answers
80 views

How many people bought wine?

Rephrased a problem trying to solve for work in terms of people buying wine, also included progress made so far. Set-up: Customers enter a winery with the option of buying bottles of wine. Those who ...
2
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1answer
43 views

Finding estimator for a one-parameter Weibull distribution

I'm doing some practice problems on methods of moments from a textbook. I am stuck on the following question: The pdf of a one-parameter Weibull distribution is given by: $f(x) = \begin{cases} ...
2
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1answer
69 views

estimate a distribution parameters only by data mean and std. dev

I need to estimate a truncated gamma distribution parameters (shape , scale). But, I only know the data mean and std. dev. I do not know the data set. Given the mean and std. dev. of a data set from ...
2
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0answers
34 views

Identifiability in nonlinear mixed-effects models

I am interested in the identifiability of linear mixed effects models. Let's assume $p$ subject are observed at different instants in time. Let $\mathbf{y}_{i}$ $(1 \leq i \leq p)$ the vector ...
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2answers
53 views

Recovering true data from many noisy samples with varying unknown amounts of noise

Input: $k$ vectors $x^1,\ldots,x^k \in \mathbb{R}^n$, where $x^i \sim \mathcal{N}(x,\mathbb{1} \cdot \sigma_i^2)$. Goal: approximate the vector $x$ as well as possible. The quality of approximation ...
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36 views

Standard deviation of a dataset comprised of triplicates (industrial stack sampling data)

I have 4 triplicate data values. each consist of 3 values, for example, this could be 1: Concentration: 45 78 66 These data have a mean of ...
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49 views

Bayesian inference with the wrong distribution

When an observation $x$ is generated by $P(x|\theta)$ for a parameter $\theta$ the Bayesian optimal estimator for the value of $\theta$ is $\hat\theta_{BEST}=\mathbb{E}[\theta|x]=\frac{1}{P(x)}\int ...