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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14 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
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0answers
16 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 $n$ ...
1
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0answers
25 views

Help in maximum likelihood estimation of a parameter not included in distribution [on hold]

There is a metric $H$ defined as = $\sum_{i=1}^{N} \min |u_i - u_j| * ..*|u_{i+d-1} - u_{j+d-1}|$ where $u$ is a multi dimensional vector of dimension $d$ and $u_i,u_j$ $\in \mathcal{R}^d$ are the ...
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1answer
10 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 ...
0
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1answer
23 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 ...
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0answers
40 views
+50

Help in maximum likelihood estimation derivation

The problem is estimating a metric $Vol(D)$ for the following situation : Given noisy observations the observed data are a random sample $Y_1,\ldots,Y_n$ where $Y_i \in \mathcal{R}^d$. The model for ...
2
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2answers
135 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 ...
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1answer
17 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
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1answer
23 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 ...
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0answers
11 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 $N$ ...
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0answers
23 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 ...
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1answer
59 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 ...
4
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0answers
23 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 ...
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0answers
52 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 ...
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0answers
57 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
1answer
35 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
84 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
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2answers
57 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 ...
2
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0answers
34 views

Finding $Var(S^2), E(S^4),$ and unbiased estimator for $\sigma^4$ from random, normal samp

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution $N(\mu,\sigma^2)$ and let $S^2$ be the sample variance. (a) Find $V(S^2)$ and derive $E(S^4)$. (b) find an unbiased ...
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2answers
48 views

$Var(\bar{X})$ for a random sample from Bernoulli Distribution

Let $X_1,...,X_n$ be a random sample of size $n$ from a Bernoulli distribution with parameter $p$ where $0< p< 1$ is unkown. (a) Find $\theta^2=Var(\bar{X}).$ (b) Find the value of $c$ so that ...
3
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2answers
62 views

Correlation estimation on Half-Normal Distribution

Let $$(X,Y)\sim N\left(\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}1&\rho\\\rho&1\end{pmatrix}\right)$$ and let we are observing $(|X_1|,|Y_1|),\dots,(|X_n|,|Y_n|)$ independently. I wish ...
0
votes
1answer
44 views

Estimating age distribution from first names

Is it possible to estimate an age distribution of a specific population (e.g. actors, customers, etc.) from a list of their first names? Given that data of the number of individuals grouped by birth ...
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0answers
18 views

How to calculate the volume, area and other measurements of a human body from picture? [closed]

Picture 1 shows that there are different sizes, shoulders (if body-builder), etc. Picture 2 illustrates this in real life animation and side view as this would be key to work out the ...
2
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0answers
21 views

Truncated Gamma distribution parameter estimation

I need to estimate the parameter of a Gamma distribution form the data, but I only have samples from 0.05 to 3 (most of the samples are concentrated here). I tried MLE but due to the truncation is ...
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0answers
16 views

adaptive surveying for a maximal income that depends on a parameter

I have a product that I would like to price for the highest income. The income $I$ from this product will depend on the asking price $c$: $$ I(c) = N \cdot E(c)$$ where $E(c)$ is the expected ...
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1answer
38 views

R - MLE of modified Champernowne density

I've come across an article (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=704903), in which author wrote about maximum likelihood estimates of parameters in the so called modified Champernowne ...
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0answers
27 views

Help in estimation : MLE formulation

The intrinsic dimension (ID) of a data set generated from dynamical system is the minimum number of free variables needed to generate the data. Given a time series expressed in $m$ dimension (ID) we ...
1
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2answers
67 views

Seasonal adjustment for a series that has already been adjusted

A dataset I am working with (from the OECD), for harmonised unemployment seems to be seasonally adjusted: The unemployment rates shown here are calculated as the number of unemployed persons as a ...
1
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0answers
45 views

When to use minimum mean square error (MMSE) estimate instead of maximum a posteriori (MAP) estimate

What are the advantages and disadvantages of MMSE estimates and MAP estimates?
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0answers
42 views

what's the difference between MMSE estimation and MAP estimation

can anyone answer the question regarding difference between minimize estimation error and minimize wrong estimation? Thank you
0
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1answer
27 views

Use of infinity norm instead of SSE for machine learning accuracy?

Are there any examples or arguments in favor of using an infinity norm (or equivalent) over sum of squared errors or root mean squared error for evaluating machine learning algorithms?
2
votes
1answer
24 views

Dirichlet Process Hyperparameter Estimation with Sampling

I have a dirichlet process for which I need to learn the concentration (strength) hyperparameter (with gamma prior on it). One way of doing is via maximizing the Likelihood. Another way of doing this ...
0
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0answers
16 views

Online moving median [duplicate]

So I can use "EWMA" (1) to update an estimate of the mean as each new measurement is received. If I know the window size of the smooth($\eta$), the previous estimate($ \bar{x}_t$), and the new ...
1
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1answer
39 views

Finding the unbiased variance estimator in high dimensional spaces

The problem comes from linear regression. Assume the regression function is linear, i.e. $$ f(X) = \beta_0+\sum_{j=1}^pX_j\beta_j $$ .Given a set of training data $(x_1, y_1),\ldots,(x_N,y_N)$,we try ...
0
votes
1answer
43 views

R Bayesian prediction of a Gaussian process

I have a Gaussian model with mean zero, variance is arbitrary constant, and correlation function $e^{-\theta(x-x')^2}$ where $\theta$ is again an arbitrary constant. I've plotted some realizations of ...
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0answers
7 views

Question on the estimation of a complex matrix

I would like to find a complex matrix C from (huge) set of the data { y }, where every complex-scalar y can be described by a following model which includes the matrix C (whose size is say M x M), y ...
6
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0answers
59 views

Estimating number of balls by successively selecting a ball and marking it

Lets say I have N balls in a bag. On my first draw, I mark the ball and replace it in the bag. On my second draw, if I pick up a marked ball I return it to the bag. If, however I pick up a ...
5
votes
1answer
203 views

Why don't we use the unbiased sample variance to calculate the standard error?

The standard error is an approximation of the standard deviation of the sampling distribution of the sample means. The real standard deviation of the sampling distribution, $\sigma _{\bar x}$ is: ...
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0answers
17 views

multidimensional time series nonlinear parameter estimation

I am trying to fit time series data for performing parameter estimation of a nonlinear multidimensional dynamical model (grey-box). At the moment I'm successfully using MATLAB's ...
5
votes
0answers
236 views

How to find the functional form of pdf from time series using Kernel density estimate

I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. I have read about Kernel Density Estimate for the case when we don't have ...
0
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0answers
19 views

Determine a distribution of a gaussian stochastic with different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
1
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0answers
18 views

Marginal pseudo-likelihood and consistency?

For a given set of random variables, $X_1,...,X_n$ we know that in many cases finding the maximum of the pseudo likelihood: $$PL(x_1,\ldots,x_n) = \prod_{i=1}^n p(x_i | ...
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0answers
34 views

Estimate the covariance matrix of a normal distribution if the mean vectors is given by a linear rule

Let $X=(x_1,\ldots,x_n)^\top\in\Bbb{R}^n$ be a random vector that follows a multivariate Gaussian distribution with known mean vector $\mu=(\mu_1,\ldots,\mu_n)^\top\in\Bbb{R}^n$. The covariance matrix ...
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0answers
27 views

demostrate Jensen's inequality (i think) [duplicate]

Please any help is very good. I was trying to start with the definition of expected value but I don't know how to finish.
2
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0answers
19 views

Error of bounded mean estimation with quantization

I have a distribution whose mean $\theta \in [0,1]$. Given $n$ observations, I compute the sample mean (say $\hat{\theta}$) and quantize it to accuracy $\frac{1}{n}$ (basically I can use only ...
2
votes
1answer
37 views

Sufficient and complete statistic

Let $ X_1, ... , X_n $ be i.i.d random variables with pdf given by $$f(x;\theta) = \exp(-(x-\theta))I_{(\theta, \infty)}(x)$$ It is asked to find a sufficient statistics for $ \theta $ and to verify ...
0
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1answer
92 views

Unbiased estimator based on geometric mean for random sample on an exponential distribution

Let $X_1, \ldots, X_n$ be a random sample on an exponential distribution with mean $\theta$. Obtain an unbiased estimator for $\theta$ based on $G$, where $G$ is the geometric mean of the ...
3
votes
2answers
75 views

Show that the value is, indeed, the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
1
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0answers
29 views

Distribution of sample variance for non-normal random variable [duplicate]

For a sample of size $n$ of non-normal random variables $x_1,...,x_n$. Is it possible to know which distribution the sample variance estimator follows? Details: The distribution of $x_i$'s is not ...
2
votes
1answer
23 views

Estimation probability in related binomial distribution

I have two binomial experiments with an unknown probability - But i do know that the ratio between the probability in the first experiment and the probability in the second experiment - For example I ...