For questions about estimation and how and when to estimate correctly

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24 views

Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$

I was tasked to use the Maclaurin series to calculate $f(0.1)$ of $f(x)=(3+e^{2x})^{0.5}$. I got the Maclaurin expansion of $p_2(x) = \sqrt{3} + 4x +5x^2$ into which I plugged $0.1$ to yield ...
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0answers
17 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
1
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0answers
37 views

equivalent inner-product vector for one

I have a map that projects a $k$ dimensional vector $x$ to an $m$ dimensional vector $\phi(x)$. The vector function (map) $\phi$ can be any linear or non-linear function of $x$, which is not ...
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0answers
12 views

Sample Variance converges in probability

Good evening everyone, I'm sorry that I ask this "stupid" question, but I want to ask you: Let $X_1,...,X_n$ be random variables mit the variance $\sigma^2:=\operatorname{Var}(X_1)$. Let also be an ...
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1answer
10 views

how to use the maximum likelihood function to acquire the maximum likelihood (ML) estimate

I'm reading this book Tracking and Data Association. The author gives an example for estimating an unknown non-random variable $x$ given some observations $z_{j}$ corrupted by Gaussian noise $ w \sim ...
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0answers
35 views

The efficiency property of estimator for second moment. [closed]

Please help to improve the efficiency property of estimation for second moment. Statistical population is normally distributed. Sorry for my bad english, if something is wrong. Thanks in advance.
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2answers
26 views

How to find smallest integer which is greater than N positive primes

I know this can't be computed exactly, but I just need a rough estimate. I know one can compute a rough estimate of the number of primes less than N using the famous formula: ...
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0answers
21 views

Description length in model coding

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with N data points, our estimation error for $\hat ...
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0answers
15 views

When is $E(E(\beta|X)E(\beta'|X))=E(\beta)E(\beta')$?

When is $E(E(\beta|X)E(\beta'|X))=E(\beta)E(\beta')$ true? $E()$ is the expectation, and $\beta$ is vector dependent on matrix $X$, $\beta'$ is the transpose of beta. (could be an OLS estimator). Any ...
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0answers
9 views

Confidence interval for a function of estimators

Let $X_i$ be iid samples and $$I_f = \frac{1}{N}\sum_{i=1}^N f(X_i)$$ be an estimator for the mean of $f(X)$ and $$I_g = \frac{1}{N}\sum_{i=1}^N g(X_i)$$ an estimator for the mean of $g(X)$. How can ...
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101 views

What unbiased statistic should be used to estimate the number of interviewed voters?

Below is a questions in my statistics course and I am struggling on how to start part a. Any suggestions would be helpful. Say a Utah pollster conducted m = 16 polls among people who voted in the ...
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0answers
1 views

Asymptotic distribution of ratio / multiplication of two variables

Suppose $\rightarrow_D $ denotes convergence in distribution. If we know $$ f_1 \rightarrow_D W_1 $$ $$ f_2 \rightarrow_D W_2 $$ Can we say something about the convergence of $$ f_1 f_2 ...
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1answer
14 views

Maximum Likelihood (ML) estimation when 1 estimator is dependent on the other.

Let $\mathcal{L}(\theta_1,\theta_2)$ be the log-likelihood function. If I manage to find an estimator for $\theta_1$, as $\hat{\theta}_1=g(\theta_2,data)$. Then, if I want to find a ML estimator for ...
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0answers
21 views

Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
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0answers
20 views

Total demand for two different prices, where market shares are determinened by logit model

The setting is simple, i.e. formula for demand of service/product is linear $$ d = \alpha - \beta p $$ where $ \alpha $ is maximum demand, $ \beta $ is some coefficient, and $ p $ is price. There ...
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0answers
13 views

Conditional Likelihood estimation

I'm reading a book called "Bayesian Reasoning and Machine Learning" and I have come across a question that gives me some problems. Unfortunately I neither have solutions, nor anyone else who I know ...
2
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2answers
49 views

Estimating integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$

Estimate the definite integral $\int_0^{0.5} \ln(1+\frac{x^2}{4})$ with an error of at most $10^{-4}$, using the Alternating Series Estimation Theorem. My approach is as follows: I found the ...
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1answer
27 views

How to prove Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$?

As the subject states, how can Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$ be proven? Is the proof distribution-dependent or there is a general way to prove it?
2
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0answers
30 views

Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
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0answers
13 views

Simulation Position of a harmonic oscillator system

I am write a simulation for get true Postion of a harmonic oscillator system as Now, I want to write matlab code to get the true postion z of the system. However, ...
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3answers
31 views

Estimations in a ordered field?

My Problem: I am stuck with a proof strategy on the following: So i have got an ordered field $ (K,+,*,<) $ given. I also have $x,y\in K$ and $0\le y < x$ I have to proof that, for every n ...
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0answers
43 views

What exactly is a true value of a parameter?

I am currently studying the properties of the Maximum Likelihood Estimator. One of these properties being the asymptotic normality, I found the following equation: $$\sqrt{n}(\hat{\theta} - ...
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0answers
15 views

Fitted function - Which is better to use?

So I have some data for program running time, that follows a power law relation aN^b. I log-log plotted the data and saw that it became a straight line, so I calculated the slope of this line to get ...
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0answers
7 views

How to calculate the maximal error of a solution of a physical problem found numerically?

Assume we throw a body from a height h with the velocity v0 in some arbitrary direction. Beside the weight ...
0
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0answers
7 views

Sensitivity analysis of paramaters and input variables

I am trying to perform a sensitivity analysis of an optimization problem $f(x,\alpha)= \min_{ Q} {g(x,\alpha , Q)}$ where $x$ is an input variable for our function, and $\alpha $ is a parameter. ...
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1answer
29 views

Question on probability and approximation

Okay I think you are all familiar to YouTube videos and some facts are: to comment, like and dislike on a video you need a Google account. when someone views the video the view count of the video ...
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0answers
22 views

doubled estimation of a series 1/k^2

how can one show, that: $\frac{5}{4}\leq\sum_{k=1}^{\infty}\frac{1}{k^2}\leq\frac{7}{4} $ I can show by the estimation $\frac{1}{n^2}\leq\frac{1}{n(n-1)}$, that $\sum_{k=1}^{\infty}\frac{1}{k^2}\leq2 ...
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0answers
52 views

Maximum Likelihood Estimation with Laplace Distribution

I want to estimate the parameters $a$ and $b$ of the model $y_i = ax_i + b + \varepsilon_i, i=1,...,n $ via Maximum Likelihood. The $\varepsilon_i$ are assumed to be Laplace-distributed with density ...
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0answers
17 views

Least Square estimator: estimating a parameter from a simple signal model

Assume the following signal model: $b_{i,j} = c_i\;d_j \\ b_{j,i} = c_j\;d_i$ where $i,j = [1,2,...,M]$ with $i \neq j$. Both $c_i, d_i \; \forall \; i$ are mutually independent complex random ...
1
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1answer
41 views

Proof about Simple random walk in $\mathbb{R}^{d}$

I have read something about random walks in $\mathbb{R}^{d}$. The random walks is assumed to be started at origin. There is a theorem said that For $d =1$ or $2$, the random walk is recurrent. (i.e. ...
1
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1answer
14 views

Prevalence estimates based on randomized sample of clinical data

This is probably one of the more straight forward questions on here but here it is: I want to use a random number generator to sample X number of charts to look for the # occurrences of Y event. So ...
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0answers
10 views

Efficient estimator with exponential noise

Let us consider the model : $x(n) = \theta + b(n)$ with $\theta$ a constant to estimate and $b(n)$ a noise with an exponential density of probability : $p(b(n)) = \lambda e^{-\lambda b(n)}$ so ...
2
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0answers
17 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
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0answers
31 views

The Hessian Matrix I calculate is twice as much as it should be. Why?

I have a function "fkt." In this example, let it be as simple as $y=a \cdot x+b$. I have a real dataset with values obeying to the model. After regression of the points to the model, I find the ...
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1answer
64 views

Estimation of $\int_x^\infty \frac{e^{-t}}{t}$

How can you show $$\int_x^\infty \frac{e^{-t}}{t} dt\geq \log(1/x)-1 \text{ for }x>0$$? It's quite straigtforward to prove the estimation in the other direction for log(1/x)+1, but I fail ...
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0answers
35 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
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0answers
23 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, ...
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0answers
20 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
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0answers
28 views

Uniform Distribution Min & Max unbiased estimators?

I have $n$ samples ($iid$) from a uniform distribution over $[a,b]$ with unknown $a$ & $b$, let call them $x_i$. Let $Min\{x_i\}=\hat{a}$ and $Max\{x_i\}=\hat{b}$. What are unbiased estimators ...
0
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1answer
15 views

Maximal value of $\vert r^2-n\vert$ with a special condition

let $M,n\in \mathbb{N}$ and $R=\lbrace r\in \mathbb{N} \mid \vert r- \sqrt{n}\vert <M<2\sqrt{n}\rbrace $. I have to show that the maximal value of $\vert r^2-n\vert $ for $r\in R$ is at most ...
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0answers
18 views

estimation question (I should be able to solve it but no, I failed)

Given: $ a = \frac{r+i}{r-i} $ $ b = \frac{r+j}{r-j} $ $ 1 < a < b \le 2a << r $ $ 0 < i < j << r $ How to estimate r given a and b?
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0answers
66 views

Calculating the Z-multiplier and Standard Error in Confidence Intervals

If someone can explain the process of working out the z-multiplier of the z-table. I mean, how do we actually calculate instead of looking up on the table? (my primary question) Also how do you ...
1
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1answer
36 views

asymptotic normality and unbiasedness of mle

Suppose $\hat{\theta}_n$ is the MLE for some parameter $\theta$. Suppose also that the MLE is such that the Cramer regularity conditions are fulfilled, and $\hat{\theta}_n$ is asymptotically normal ...
2
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3answers
131 views

Estimating the series: $\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$

Any idea on how to estimate the following series: $$\sum_{k=0}^{\infty} \frac{k^a b^k}{k!}$$ where $a$ and $b$ are constant values. Greatly appreciate any respond.
2
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1answer
23 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
0
votes
1answer
23 views

linear regression model beta estimate

Suppose we want to estimate $\beta$ by minimizing $L(\beta)=\sum_{i=1}^n(y_i-\beta x_i)^2+\lambda|\beta|$, where $\lambda$ is a fixed positive constant. Calculate the estimate. How would I ...
0
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1answer
37 views

How this integration is solved?

Can anyone explain how this integration has been performed? This is a Bayes estimator for uniform prior assuming quadratic loss function. Thanks in advance
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1answer
49 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
1
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1answer
38 views

$\sum _1 ^n |z_j| \ge 1 \Rightarrow | \sum _1 ^k z_{j_m}| \ge C$

Prove that there exists $C > 0$ such that the following implication holds: If $\{z_1, ..., z_n \} \subset \mathbb{C}$ are such that $\sum _{j=1} ^n |z_j| \ge 1$, then there exists $ \{z_{j_1}, ...
1
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0answers
24 views

Recursive Bayesian Estimation, $p(C_k|x)$ as likelihood

I''ve been struggeling with this problem for the last couple of days. The main goal is to use the probabilistic classification output $p(C_k|x)$, from for example a logistic regression, to enhance ...