Questions tagged [estimation]

For questions about estimation and how and when to estimate correctly

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28 views

How to estimate a difference?

I have a doubt what is the estimated difference of $91-45$ so I started to think about two methods the first was I will first find the difference of $91-45$ that is 46 than I will just round off 46 ...
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+50

Prove that when $|x|\geq M_1+M_0t, u(x,t)=0$.

For the following Cauchy problem: \begin{cases} \partial^2_tu-a^2(x,t)\partial^2_xu=f(x,t), x\in\mathbb{R},t>0\\u(x,0)=\varphi(x), \partial_t u(x,0)=\psi(x), x\in\mathbb{R},\end{cases} where \begin{...
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50 views

Characterization of Total Error for Estimation of the Parameters of a System

Suppose that we have a system which can receive different kinds of input (the index $k$ signifies the kind of input $I_k$) and performs a calculation on that input based on the internal parameters of ...
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1answer
25 views

"Fusion" of multiple uncertain estimates with covariance matrices

this is my first post here. I tried my best to follow all the guidelines and to state the problem as clear as possible but should something be sloppy or unclear, I will be happy to update the question ...
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19 views

Unscented Kalman Filter Linearly Not Independent Variables

I have an Extended Kalman Filter, where my state vector variables are linearly independent (e.g. conventional position tracking with roll-pitch-yaw angle). Then, when calculating the Jacobians of the ...
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39 views

Computation of whitening matrix for estimation of the covariance matrix for n samples

I am working on data-driven robust optimization, meaning that instead of forming uncertainty sets using conventional approaches such as box-shaped uncertainty sets, I want to determine this set by ...
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2answers
59 views

How big is this number and how would you go about estimating numbers which are too big to compute! [closed]

I have been doing some maths and came across this number, I am just curious to how big it actually is and how you'd go about estimating its size. $$ \frac{\ln \left( \left( \frac{3}{2} + \frac{1}{2^{...
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14 views

Maximum likelihood estimator has the lowest MSE?

I'm reading now Goodfellow's book about deep learning and I found very interesting part about properties of MLE, which is the following: I'm trying to understand it, but I'm not sure what author had ...
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16 views

Expected value of kernel density estimator with sharpened data

My question regards the proof of the bias of the kernel density estimator obtained using "sharpened" data. The method comes from the paper by Choi and Hall (1999). Specifically, assume $X_1, ...
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15 views

Maximum Likelihood Estimator for $r$ with $r^n$ in noise

$x[n]=r^n+w[n]$ for $n=0,1,\ldots,N-1$ where $w[n]$ is WGN with variance $\sigma^2$. I need to estimate $r$. No efficient estimator exists (that I can discover), so I am looking to build a maximum ...
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1answer
31 views

How to infer the actual counts from the observed?

People eat apples, and such events are recorded, so that we can count how many apples were eaten by a specific individual, and how many persons ate exactly $n$ apples. If our observations are ...
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1answer
141 views

Estimating $\tan(\cos(\sin1))$

The precise question through which I came across this particular estimation is as follows: If $x=\alpha$ is the maximum value of $x$ for which $ \left \lfloor{\sin^{-1}(\cos^{-1}(\tan^{-1}x))} \right \...
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26 views

Is this "predictor" equivalent to the typical Kalman filter?

In "Introduction to Stochastic Control" by K. Astrom, page 228 Theorem 4.1, he introduces a state estimator for the discrete-time system: $$ \begin{aligned} x(t+1)&=\Phi x(t)+v(t)\\ y(t)&...
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1answer
43 views

Maximum likelihood estimation intuition for continuous distributions

I was just revisiting the fundamentals and rationale of maximum likelihood estimation when I realised I can't rationalise the continuous case as opposed to the discrete case. For a discrete random ...
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2answers
39 views

Why don't we use the mean of the middle pair to estimate the median in a cumulative frequency curve?

I just learned about cumulative frequency curve. The books says I could use this curve to estimate the median of the data. This is the picture that I cut from my book. As you can see, to estimate the ...
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11 views

Estimation of the interval of a proportion (mark, recapture)

I'm a college studen doing statistics homework and Im looking for help. I have tried to solve this exercise, thinking about it in a number of ways, to no avail. This is the problem: To estimate the ...
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1answer
71 views

Asymptotic formula for a sum involving the exponential function.

Let $n$ be a positive integer, and let $c$ be a positive constant. Consider the asymptotic estimate $$ \sum_{1\le kv < n}\frac{ve^{c\sqrt{n-kv}}}{n-kv} = e^{c\sqrt{n}}\left(1 + \mathcal{O}\!\left(\...
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1answer
26 views

MLE Estimator for paratmer of piece-wise uniform distribution

My question relates to this thread here: How to find MLE of this piecewise pdf? On answer submitted by @StubbornAtom and @Riccardo, the expression used, doesn't that require pdf to be continuous, Is ...
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1answer
27 views

What's the difference between a Bayes Estimator and a Maximum a posteriori (MAP) Estimator?

I have been practicing with two textbooks and one has a chapter on Bayes Estimator (Sheldon Ross) and another has a Chapter on MAP Estimation (Pishro Nik). I seem to get the same answers (to the ...
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1answer
25 views

How to estimate the time taken for a job to complete?

I have a job x which copies file y with data size of 200GB from one disk to another. I would like to estimate an accurate time it takes for the job to complete from current time. This requires basic ...
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19 views

sharp $L^1$ gradient eigenfunction estimate on the unit sphere

Let $u$ be a $L^2$-normalised spherical harmonic of degree $k$ on the unit sphere $\mathbb{S}^{n-1} \subset \mathbb{R}^n$, i.e. $-\Delta_{\mathbb{S}^{n-1}} u = k(k + n - 2)u$ and $\|u\|_{L^2(\mathbb{S}...
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43 views

Modifying a point-estimation to get a $unbiased$ $estimator$ $of$ $variance$

My Exercise Problem: Let x1, x2, ..., x7 be observations of independent random variables X1, X2, ..., X7 with ...
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1answer
31 views

Upper bound for $\sum_{j=0}^k \alpha^j(\beta j)^{k-j}$

I am looking for an upper bound for the sum $$\sum_{j=1}^n \alpha^j(\beta j)^{n-j}$$ for constants $\alpha,\beta <1$ and some $n\in\mathbb{N}$ which is "better" than simply replacing $j$ ...
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2answers
82 views

Estimating the Sum of Square Roots (Calculus Starter)?

I am trying to estimate the value of the Sum below without using the calculator: $$ \sum_{n=1}^{10000} \sqrt{i} $$ I have looked for other ways around and tried to turn it into $n\sqrt{n}\int_{0}^{1} ...
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31 views

Guess the "special number" of an unfair die given the first $N$ tosses

The following problem is something I have come up with recently, and have been trying to solve, but to no avail. I wonder whether the solution is relatively easy, or pretty complicated: Suppose we ...
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1answer
95 views

What is the mistake (A question on uncertainties)?

I have a simple question with a possibly trivial mistake in my answer but I'm unable to figure it out. The question: A sphere fits inside a cube. The length of the cube and the diameter of the sphere ...
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10 views

Plug in estimator for expectation

I trying to understand why the plug in estimator for the expectation is: $\hat{F_n}(x) = \frac{1}{n} \sum_{i = 1}^{n}1\{x_i \leq X\}$ $T(\hat{F}) =\int{\psi}(X)d\hat{F_n}(x) =\frac{1}{n}{\psi}(X_i)$ I ...
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36 views

Sample size for a quantile estimator

Assume that I have a population of size $N$ with an attribute which is a continuous variable. Let's say these are people and the attribute is their height. I want to find the $90\%$ quantile of that ...
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1answer
60 views

Estimator : Mean of ratio of quantities tends to be equal or equal to ratio of mean of quantities

I have the following estimator : $\hat{O}=\dfrac{b_{sp}^{2}\left(\mathcal{D}_{DM}+B^{C}\right)+B_{sp}}{b_{ph}^{2}\left(\mathcal{D}_{DM}+B^{C}\right)+B_{ph}}$ I would like to demonstrate that the mean ...
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30 views

Estimation for the density of a Bessel-3-process

I want to show the following Lemma about Bessel-3-processes: Let $\gamma=\sqrt{\frac{2}{\pi}}$. For every $t>0$ and $x,y\geq 0$ holds \begin{align*} \frac{\gamma z^2}{t^{\frac{3}{2}}}e^{-\frac{z^2}{...
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83 views

estimation integral over minor arcs

I have given the following lemma: Let $\varepsilon>0$ and let $a, q, z$ be such that $$ 1 \leqslant a \leqslant q \leqslant B^{2}, \quad \operatorname{gcd}(a, q)=1, \quad|z| \leqslant \frac{1}{q^{2}...
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1answer
26 views

What effect does uniform convergence of sequences of power series have on the convergence of their coefficients?

Assume we are given a sequence of power series $$ B_n(x) := \sum_{k\ge 1}b_{n,k}x^k, \qquad n\in\mathbb{N} $$ with real-valued coefficients $(b_{n,k})_{n,k\ge 1}$. Further we are given a power series $...
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1answer
23 views

Is there an easy way to algebraically manipulate a large exponential value with a base other than 10 into a base of 10 to be more tangible? IE: 3^3778

Quick preface as with my other posts: I'm not a math major, but I'm enjoying the math portion of my studies, so please keep any explanation on an idiot's level lol - thanks! I came across a problem ...
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2answers
49 views

How to show following estimate with constant depends only on n?

We have $$\frac{k^{1-\frac{2}{n}}}{2k+n-2}\le \frac{k^{1-2/n}}{2k}=\frac{1}{2k^{2/n}}$$ But I am unable to $$\frac{1}{2k^{2/n}}\le c(n) \frac{1}{(k+1)^{2 / n}}$$ I could not able to show original ...
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Variance Covaraince Matrix of Parameters in Logistic Regression

The given below image is taken from book Introduction to Linear Regression Analysis (Douglas C Montgomery) My apologies in advance for not typing , I just want to understand the concept. (1) First ...
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observations on samples are uncertain

We would like to estimate the average height ($\lambda$) of citizens in City-A (the population), we may randomly sample $n$ persons and measure their heights, say $x_1,\cdots,x_n$. Then $E(\lambda)=\...
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19 views

When is the conditional probability formula applicable compared to more rigorous methods?

Imagine that you have a sample of a fruit and some of the fruits of your sample received a certain treatment and others haven't. You want to find out if this treatment has any effect on the acidity of ...
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36 views

Markov inequality to find bound on the probablity over an interval

Is it possible to use Markov inequality, as given in the following, to find bound on the probability over an interval? $$P[X \ge a] \le \frac{E[X]}{a}$$ For example, consider this problem. The number ...
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47 views

Statistical significance in context of financial data?

I understand statistical significance in the general sense: we take a sample from a population and compute some parameter from the sample to infer what is the propulsion parameter to some degree of ...
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8 views

Estimating confidence intervals for the variance of a sampling distribution (from normal), given that the samples are not drawn independently

A random variable $X \sim N(\mu, \sigma^2)$. One day, we draw a sample $X_1, X_2, ..., X_n$ from this population. This gives an estimate of the actual distribution of $X$. What I would like to ...
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1answer
56 views

Mle estimation in hardy Weinberg

There is a population of 3 kinds 1 2 3 occurring in hardy Weinberg proportion $\theta^2, 2\theta(1-\theta), 1-\theta$ If we abserve a sample of 3 individuals and obtain $X_1=1, X_2=2, X_3=1$ To find ...
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1answer
41 views

$r<s$, a bounded sequence of functions $f_{n} \in H^{s}(\mathbb{R})$ converges and is equicontinuous in $H^{s}(\mathbb{R})$, it converges in $H^{s}$.

For $r<s$, there's a bounded sequence of functions in Sobolev space: $f_{n} \in H^{s}(\mathbb{R}) .$ If $f_{n}$ converges in $H^{r}(\mathbb{R})$ and (uniformly) equicontinuous in $H^{s}(\mathbb{R})$...
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66 views

Estimate for differences of exponential integral

Since the function $\exp(-t)/t$ is monotone decreasing for $t>0$ it is immediate to get $$f(x) := \int_x^{2x} \frac{\exp(-t)}{t} dt \le \exp(-x)$$ for $x>0$. Using $$E_1(x) = -\gamma - \log(x) - ...
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Account Resurrection as a function of Days Dead and Time to Resurrect

I am a data scientist working for a company that takes user deposits. I wanted to answer the question of how likely an account that's dropped to $0 on deposit - or dies, in other words - would refund ...
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1answer
91 views

Why, when rounding, do we round 5 up to 10? [duplicate]

If we cut 10 in half, then we get 5. This means two 5s makes 10, meaning, 6, 7, 8, 9, and 10, are the remaining 5 numbers that make it up. So why do we round up 6 numbers? It's such a small thing but ...
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15 views

Confidence Interval for least squares estimator

I had asked this question in another forum and I hope this would not cause a problem. There was a paper by Yasin-Abbasi-Yadkori https://arxiv.org/pdf/1102.2670.pdf titled Online Least Squares ...
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10 views

How to calculate process correlation with sparse unequal observations at different times?

Suppose I have two processes: $dx_t = \sigma_x dW_t$ $dy_t = \sigma_y dV_t$ $dW_t dV_t = \rho dt$ where $W_t,V_t$ are two correlated standard Brownian motions and $\sigma_x,\sigma_y,\rho$ are all ...
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14 views

Quantitative comparison Kalman filter VS Least squares

I've been asked to give a quantitative confirmation of the Kalman filter that I develop. My obvious first idea was to compare the residuals. $$ \chi = {\lvert\lvert C \hat x - y \rvert\rvert}_2^2 $$ ...
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2answers
41 views

Monte carlo variation of $E[1_{\{X>0\}}]$

Let's assume that $X \sim N(0, 1)$. I want to calculate variation of crude monte carlo estimator of parameter $\delta = E[1_{\{X>0\}}]$ My work so far Let's generate $x_1, x_2, x_3,..., x_n \sim N(...
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39 views

Mixing Least squares and Kalman fitler

I'm not sure it is the right department. I try my chance I am wondering if there is a way to make a hybrid formulation of a least-square problem and a Kalman filter. Let me explain what I mean: The (...

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