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Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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Least Square Estimation

In the following example question (from Bertsekas, edition 1), i have two questions: Why the value of fX|Y(x|y) is 1/2? Can anyone explain in more detail about the graph on the right (especially the ...
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Proving independence between Beta estimated and Delta in OLS

I know that in ordinary least squares $b$(beta estimated) and $\delta^2$(variance estimated) are independent, but how do I prove that?
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Proving a gradient estimate for a continuous function in an open ball

I need to prove the estimate $|\partial_{x_i}u(x_0)| \leq \frac{N}{R}u(x_0)$ where $u \in C(B(x_0,R))$ and $u$ is nonnegative and harmonic in $B(x_0,R)$, and $i=1, \ldots,N$ I found this ...
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50 views

What adaptive controller can be used in embedded system with low RAM?

This is not a question for data science, hardware or programming languages. This is a more practical question about adaptive control for embedded systems, but still a math question. I have tried to ...
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Find the time gap between two vehicle based on km/h of the vehicle

I am currently researching on VANET for identifying the road capacity based on vehicle moving speed. Below is my question, If a car moving on 8km/h and distance to the next car is 15 meters, what ...
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29 views

How to find equation for $\theta_\lambda$

Suppose we have data $D=\{x_i,y_i\}_{i=1}^n$ where $x=(x_{i,1},x_{i,2},1)^T \in \mathbb{R}^3$. An estimator for these data, $$y=f(x;\theta)=(\theta_1 x_1+\theta_2 x_2+\theta_3) ~~~(\theta \in \mathbb{...
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Estimate for the fundamental solution of Laplace equation from Evans

After the definition of the fundamental solution of Laplace equation (page 22) $\Phi(x) = \begin{cases} -\frac{1}{2\pi} \, \log(|x|), \, & n=2, \\ \frac{1}{n \, (n-2) \, \omega_n} \, \frac{1}{|x|...
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Which of $s^2$ and $S^2$ is a better estimator of $\sigma^2$ in the sense of the mean squared error?

Let's recall that: $$s^2=\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar X)^2\quad\&\quad S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar X)^2$$ We actually know that $S^2$ is an unbiased estimator of the ...
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What fraction of children would have no parents?

This question has pop-culture origins but is mathematical for sure. In the movie "Avengers: Infinity War" (SPOILERS AHEAD) the main villain eliminates half of all life on the Earth (technically the ...
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Likelihood ratio test for the exponential distribution

Let $Y$ be a random exponentially distributed variable, with mean $\lambda$. That is, its probability density function is $$f(y) = \left\{ \begin{array}{ c c } \frac{1}{\lambda}e^{\frac{...
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How is “continuous dependence on initial conditions” defined? And how to prove it?

EDIT: I tried to prove the continuous dependence of the problem by somehow use a weak Maximum principle and posted a modified Version of this post on mathoverflow: https://mathoverflow.net/questions/...
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25 views

Find the Maximum Likelihood Estimator of a binomial distribution

X ~ Binomial (5,p) , where X is taken with replacement from a simple random sample $X_1 =x_1 , ... X_n = x_n$ (I guess it is until $X_5$ since n=5) Here is the full question in case the context is ...
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Text book recommendations for statistical estimation theory -specifically MLEs and confidence intervals

I am looking for a textbook on statistical estimation theory. In particular I am interested in a book that explains MLEs and confidence intervals. Preferably accompanied by exercises. The book should ...
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Solution of $\int_x^1y^{a-1}\left(1-y\right)^{b-1}dy = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$

When is $f=g$ on $(0,1)$ for $f = \int_x^1y^{a-1}\left(1-y\right)^{b-1}dy$ $g = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$ Let me show their graphs. They are small, so I multiplied ...
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Mean squared error for vectors

I know that when we compare estimators $\hat{b_1}$ and $\hat{b_2}$ to an unknown parameter $\beta$, in classical statistics an estimator $\hat{b_1}$ is said to be "better" than $\hat{b_2}$ if: $$ ...
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About De Moivre-Laplace theorem proof

The theorem relates the binomial distribution and the normal distribution. For $npq\gg 1$ $$\binom{n}{k} p^kq^{n-k} \simeq \frac{1}{\sqrt{2\pi npq}} e^{- \dfrac{{(k-np)}^2}{2npq}} \label{Eq_1}\tag*{...
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estimating $\prod_{i=1}^k a_i\leq \max_i a_i^{p_i}$, where $\sum_i 1/p_i=1$

Let $1<p_1,\dots,p_k<\infty$ such that $\sum_{i=1}^k\frac{1}{p_i}=1$. Moreover, let $a_1,\dots,a_k\geq 0$. I have to show that $$\prod_{i=1}^k a_i\leq\max_i a_i^{p_i}$$ I want to use induction ...
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40 views

Where and Why do we use measurable functions in modern probability theory?

I am a newbie to the measure theoretic framework for probability. While studying MMSE Estimation I came across the term "borel function" which upon further googling made me understand that it's a "...
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How can I write $\rho$ in function of $\Sigma$?

Be $\Sigma = (1-\rho) I_p + \rho 1_p 1_p^\top$. I need to write $\rho$ in function of $\Sigma$ to calculate the maximum likelihood estimator of $\rho$ and I know that the maximum likelihood estimator ...
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40 views

$1435\binom{20000}{10000}*0.515^{10000}*0.485^{10000} \leq 0.01$

This inequality is part of a larger probability question and must be shown without explicitly calculating it. The only hint I have is to use Stirling's Approximation so $1 \leq \frac{n!}{\sqrt{2\pi n}...
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Find maximum of $\log(1+x)(1-I_x(a,b-a))$

I am struggling to find (or at least set some bounds on) $$\arg \max_x \log(1+x)(1-I_x(a,b-a)),$$ where $I_x(a,b-a)$ is the regularized incomplete beta function, i.e $$I_x(a,b-a) = \frac {\int_0^x ...
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Time series forecasting with ARIMA

I use Statgraphics for time series forecasting with ARIMA. But i get nonsense future values and i dont know why. My forecast values will be in ascending or descending order. Can be my data not ...
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40 views

Show that the only unbiased estimator for the zero-truncated Poisson distribution is absurd

Consider the zero-truncated Poisson distribution on the striclty positive integers, i.e. \begin{align} \mathbb{P}_{\theta}(X=k) = \frac{\theta^k}{k!(e^{\theta}-1)}\, \, \, , \, \, k=1, 2, ... \end{...
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42 views

Find the best unbiased estimator for $\mu^T \mu + 1^T \mu$.

Let $X_1, X_2, \dots, X_n$ independent n-dimensional vectors with the same distribution $N(\mu, I)$. Find the best unbiased estimator for $$ \mu^T \mu + 1^T\mu $$ where $1^T = (1, 1, \dots, 1)$. ...
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47 views

Localization based on Distance Measurements via Least Squares

Estimating a vector $x \in \mathbb{R}^2$ knowing its distance to four beacons $v_1, \dots, v_4\in \mathbb{R}^2$ via least-squares means finding the least-squares solution to $A x = y$, where $y\in\...
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Moment estimator $\hat{\theta}$ of $\mathrm{Beta}(\theta,1)$ and bias of $\hat{\theta}$

I'm trying to find the moment estimator for the density function $$f(y)=\theta y^{\theta-1}$$ and check whether this is biased. I know this is a $Beta(\theta,1)$ distribution and it looks like I only ...
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How to find the estimator of a signal with additive white noise

We have to derivate the MAP estimation of a signal $x(m)$ observed in AWGN $n(m)$, resulting $y(m) = x(m) + n(m)$, supposing no zero mean in any process. I have done this: To find the MAP estimator, ...
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53 views

Expectation, Variance and Moment estimator of Beta Distribution

I'm given a beta distributed random variable: $X \sim \text{Beta}_{(\theta, 1)} =: \mathbb{P}_\theta$. Where $\theta \geq q$ and $$\mathbb{f}_\theta(x) = \theta \cdot x^{\theta-1} \cdot \mathbb{1}_{[...
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estimate value of $\sqrt[30]{0.05}$

Yesterday I got an exam in which there was a problem and its solution results in $$\sqrt[30]{0.05}$$ I didn't go further calculation. Still I can't. My lecturer said, even I'm still not sure if he ...
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Estimate exponential equation from graph

Output y of a system is a non increasing function of an input x Based on my studies I have found that it can be estimated as ...
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25 views

Extended Kalman Filter for Orientation without Control Input

I'd like to implement an Extended Kalman Filter to estimate the state$$s=[w,x,y,z,av_x,av_y,av_z]$$with $[w,x,y,z]^T$ respresenting the current orientation as a quaternion and $[av_x,av_y,av_z]^T$ ...
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45 views

Finding an upper bound for the following expression

Consider the following function $$f(t) := \begin{cases} |t|,~~ &\text{if $|t| > 1$} \\ 1,~~ &\text{if $|t| \leq 1$}. \end{cases}$$ I'm trying to find a reasonably accurate constant $M > ...
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47 views

Sufficient statistic for Double Exponential

Let $X_1,X_2,...X_n$ be a random sample from $f(x,\theta)=\frac{1}{2 \theta}e^{\frac{-|x|}{\theta}}$.We know by Factorisation theorem that $\frac{\sum |X_i|}{n}$ is sufficient for $\theta$. But can ...
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Compute the bias of $\frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}}$

I found the maximum likelihood estimator of $x$. Now, how to compute the bias of: $$ \hat{x}_{ML} = \frac{\sum_{i=1}^N\frac{z_i}{\sigma_i^2}}{\sum_{i=1}^N\frac{1}{\sigma_i^2}} $$ Where $\hat{x}_{ML}$ ...
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A sum of polyharmonic series

Could anyone suggest how to obtain good estimates from above $\vee$ below for the following finite series: $$F(N,k):=\sum_{n_1 + \dots + n_k < N, \; n_i\ge 1}\frac{1}{n_1(n_1+n_2)\dots(n_1 + \dots ...
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Smallest $n$ such that $\;\binom{n}{k}\bigl(1-\frac{1}{2^k}\bigr)^{(n-k)}<1$

As part of a problem, I'm trying to find an estimation for the smallest $n$ (as a function of $k$) such that: $$\binom{n}{k}\biggl(1-\frac{1}{2^k}\biggr)^{(n-k)}<1$$ It's probably not possible to ...
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error covariance of MMSE estimator relation to other error covariance estimators

I'm trying to prove the following: let $ \Lambda_{e}$ be the error covariance of an estimator $\,\hat{\theta}(y)$ of $\,\theta$ based on $\,y$. I want to show that the error covariance of MMSE ...
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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 ...
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Estimating an integral involving the n-th power of the cdf of the normal distribution.

I am trying to solve integrals that look like the following: $\int_{-\infty}^{\infty} \phi(cx+d)^n e^{-(x-a)^2}dx$ where $\phi$ is the cdf of the standard normal distribution. I have no idea how to ...
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Let $X_{x,1}$ and $X_{x,2} $be two unbiased estimators for a real parameter $X$ [closed]

Let $X_{x,1}$ and $X_{x,2} $be two unbiased estimators for a real parameter $X$. Let us define $X_{x}=\alpha X_{x,1}+\beta X_{x,2}$ with $\alpha,\beta\in \mathbb R$. For which values of $\alpha$ and ...
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Bayesian inference considering known correlation

Let's suppose there are many states that we want to observe: $x_i, i\in \{1...N\}$, and suppose that for each state $x_i$ observation equation $p(y|x_i)$ is given. In this case, given an observation $...
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What is the Variance of the estimator $ \hatμ_n = \frac{1}{n}(5X_1+ \sum_{i=2}^{n-1}X_i-3X_n)$

I am given the random sample X1, . . . , Xn, which is identically and independently distributed over f(X), for a population density f(X) with finite mean μ and variance $σ_2$. I am given the estimator ...
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Estimation of a geometric distribution's parameter by the reciprocal of the sample mean

I came accross this exercise when studying statistics, but I can't get to what's the solution. The exercise simply asks to show whether the reciprocal of the sample mean is an unbiased estimation for ...
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Skewness of residual estimator in time series model

I've started to read the book "Introductory Time Series with R", and in page 20 the author mentions a time series model: $\log(x_t) = m_t + s_t + z_t$ where the relevant part is the $z_t$ which are ...
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Estimation / Calibration of Transformation of 2DOF laser pointing system in 3D space

Im creating a system where a Laser pointer should be able to point to various objects to direct a certain workflow. This laser pointer has two degrees of freedom, rotations about the local X and Y ...
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Big $O$ of integral and estimate

Suppose $f: (0,+\infty) \to \mathbb{R}$ is s.t. \begin{equation}\tag{1} f(s) = \frac{1}{4\pi s} + \mathcal{O} \biggl (\frac{1}{\sqrt{s}} \biggr ) \quad \quad\text{ as } s \to 0^+ \end{equation} and $f ...
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Converse of ML inequality for contour integrals

If the ML inequality estimates a region of an integral in the complex plane to be zero then that's the actual value of the integral, and I've been using this for evaluating some integrals along the ...
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29 views

Consistent estimator problem

The Problem Check if estimator $Y_n=max(X_1,X_2,...,X_n)$ where $X_1,X_2,...,X_n$ ~ $U[0,X]$ with parameter $X$ is consistent or unbiased. We also assume that $X_1,X_2,...,X_n$ are independent. What ...
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108 views

Conditional probability with MLE of Poisson variable

I'm having some trouble with this study question and would appreciate any help. This may be a duplicate but I have not been able to find any others. Question: Leaves of plants are examined for bugs. ...
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Regressions with two independent variables where one is part of another.

I am doing simple regression analysis where I use one-way fixed effect model to estimate the effects of two variables on dependent variable. The question I am asking is how to interpret these two ...