Tagged Questions

For questions about estimation and how and when to estimate corectly

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0
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0answers
5 views

Help in maximum likelihood estimation of distance

The model generating the observation is of the form $y_n = A^Tx_n + U_n$ where $x$ is the output of a a linear stationary model and $U$ is a zero mean Gaussian noise of known variance. The set of ...
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0answers
17 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 ...
-2
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1answer
49 views

Estimation of $\int_x^\infty \frac{e^{-t}}{t}$ [on hold]

How can you show $$\int_x^\infty \frac{e^{-t}}{t}\geq \log(1/x)-1 \text{ for }x>0$$ ?
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0answers
33 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 ...
1
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0answers
14 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, ...
0
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0answers
18 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 ...
0
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0answers
17 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
15 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
27 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
29 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 ...
1
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0answers
45 views

Help in finding the optimum value

The model generating the observation is of the form $y_n = A^Tx_n + U_n$ where $x$ is the output of a a linear stationary model and $U$ is a zero mean Gaussian noise of known variance. Now, $error ...
2
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3answers
123 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
20 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
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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
36 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
0
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1answer
28 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
23 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 ...
1
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1answer
75 views

how to apply non-linear least square

I'm trying to implement the example of estimating an angle between a target $\textbf{x}$ and a sensor $x_{p}$. I'm using the example in this book. There are three available measurements of the angle ...
0
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0answers
9 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
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0answers
20 views

How do I determine a sample size for Poisson distributions?

An average ball player scores 20% of his penalties. How many penalty results would I need to receive before I can say that a player scores at a rate of 15% requires coaching on his penalty shots? ...
0
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1answer
45 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
0
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0answers
17 views

Polynomial Expansion Proof Estimation

1-(1-x)^n where x is a value between 0 and 1 and n is a large value. This estimates to around x*n. I am having trouble with the polynomial expansion. According to Pascal's triangle, the first few ...
1
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0answers
28 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
1
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0answers
23 views

Continuous RV - minimizing absolute deviation

We try to find c value minimizing E[|x-c|], "expected value of absolute deviations", for a continuous random variable X. E[|x-c|]=Integral(-inf,inf)[|x-c|]f(x)dx ...
3
votes
1answer
72 views

Estimate large covariance matrix using few samples.

Let $\mathbf{x}$ be a random vector in $\Bbb{R}^n$, such that $\mathbf{x}\sim N(\bar{\mathbf{x}}, \Sigma)$. $N$ observations of $\mathbf{x}$ are available, say $\{\mathbf{x}_i, i=1,\ldots,N\}$. The ...
1
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0answers
21 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
0
votes
1answer
28 views

Is $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$?

I want to show that $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$, where $\log$ is the logarithm to base 2. How can I do that? I tried to make an estimate by first bringing the minus to the other side ...
1
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1answer
28 views

what is the probability that the contractor's estimate will be within 5 weeks of the true mean

A contractor uses sample mean lifetime $x'$ of $250$ compressors as her estimate for population mean lifetime m of all new compressors. If this brand of compressors has a standard deviation of $35$ ...
0
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0answers
16 views

Show that the found value is 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
15 views

Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$. Multiple "samples" or observations for each of these variables are given (not necessarily with the same ...
1
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1answer
38 views

Distribution of the sample variance of n iid exponential variables

I have to check some properties of an estimator, but I can't find its distribution. Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. ...
0
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1answer
44 views

How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
0
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0answers
18 views

density of statistic

Let $X^1$ ... $X^n$ be a random sample coming from a distribution with density : $p(x)=\begin{cases}2\theta^2/x^3, \text{if} \ x\geq 0 \\ \\ 0, \text{if} \ \ x<0 \end{cases}$ Let us denote the ...
1
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0answers
24 views

Statistics and Some Information Challenge

relation between two attribute x,y is $y=\alpha\beta^{-x}$. According to 8 experiments these information were gained. what is the estimation of ( $\alpha, \beta$) using Least Square Error? it's 2010 ...
0
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0answers
26 views

How to use Richardson extrapolation

In comment section in the question "Convergence of $\sum\limits_{k=1}^{\infty} \frac{1}{p_{k^2}}$, where $p_k$ is the $k$th prime" it is suggested that one first calculate $$ f(n) = \sum_{k=1}^{n} ...
0
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0answers
7 views

Prove recursive form of linear least squares estimate.

I have a zero-mean stochastic vector $\mathbf{x}$ which I estimate using observations of a random process $y_i$ where $i=0,1,...$ I denote the linear least squares estimate of $\mathbf{x}$ using all ...
1
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1answer
16 views

Estimating the remainder for Mac Laurin's series

I'm practicing Taylor's series and i found some old task. Calculate value of function $f(x) = e^x + e^{-x}$ at point $x = \frac{1}{\sqrt 2}$ with error not greater than $d=\frac{1}{20}$ So here's ...
0
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0answers
28 views

Estimate on the difference of quotients

The following is supposedly true (found it in a paper), however I fail to see why. Let $L(x)$ be a function that goes to $0$ as $x\rightarrow\infty$, $g(n)$ a sequence which goes to $\infty$ as ...
0
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1answer
35 views

How to estimate the standard deviation in this normal distribution?

There is this simple looking basic statistics question that asks to estimate its mean and standard deviation. I have some doubts and just want to make sure whether my working is correct. For part ...
0
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2answers
43 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
16
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3answers
276 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
0
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0answers
8 views

Using EM versus estimating all parameters directly

My question is really a request for a clarification, and pertains to whether or not there are questions for which it is necessary to use EM or if it's more of a convenience. Let's presume we have n ...
0
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0answers
15 views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
1
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1answer
62 views

Using integral estimation to show that $ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3\ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the ...
0
votes
2answers
76 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
1
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0answers
32 views

Statistics question: Estimating mean when standard deviation is known

I am reading a textbook to learn more about statistics. This section is about estimating the mean of a population when standard deviation of the population is known. My simple question is this: How ...
1
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0answers
31 views

When does l1 regularisation give a sparse solution?

I was maximising a likelihood function, which is convex. I know that the system has a K-sparse solution. I wanted to know the conditions (or some sufficient conditions) on the likelihood function ...
1
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0answers
21 views

Bias of the MLEs for the two-parameter Weibull distribution

Is it possible to obtain a formula for or an equation on the exact bias of the MLE-vector for the two-parameter Weibull distribution (both parameters unknown). I've read papers offering Monte Carlo ...