For questions about estimation and how and when to estimate correctly

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7 views

Existance of a UMVUE

$ \{X_{i}: 1\leq i \leq n \} $ is a random sample, i.i.d $ N(\mu, 1) $ with $ \mu $ unknown. For a fixed $ x_{0} $, does there exist a UMVUE for $ \phi(x_{0}-\mu) $, where $ \phi $ denotes standard ...
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16 views

MLE and unbiased estimator of $P\{X_{i}=1\}$ given poisson distribution

$\{X_{i}: 1\leq i \leq n\}$ is an i.i.d. Poisson random sample with unknown mean $\lambda$. Find the MLE of $P\{X_{i}=1\}$. Is the MLE unbiased? Does there exist an unbiased estimator of $P\{X_{i}=...
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0answers
24 views

Estimator bias and consistency

Let $x_1, x_2, \ldots,x_n$ be a simple random sample from a random variable $X$ with support $\{0,1,2,3,4\}$ and probability function $p(0)=\frac{5}{12}(1-\lambda)^2$, $p(1)=\lambda$, $p(2)=\lambda(1-\...
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0answers
16 views

Check if the MLE is unbiased and/or consistent

Let $X_1, X_2,..., X_n$ be iid random variables with probability density function $$f(k|\theta) = \begin{cases} \theta, & \text{if $k=0$} \\ \theta(1-\theta), & \text{if $k=1$}\ \ \ \ \ \ \ \ ...
2
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1answer
25 views

Definition of Bias for Baysian Estimation

I know for parameter estimation the estimator $ \hat{\theta}(X)$ of $\theta$ based on observation $\theta$ is said to be unbiased if \begin{align} E[ \hat{\theta}(X)]=\theta, \ \forall \theta. \end{...
3
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2answers
41 views

Estimate growth of a recurrence convolution

Consider the following recurrence relation $$ a_{m+1} = (4 m + 1) \sum_{k=1}^m a_k a_{m-k+1}, \qquad a_1 = 1. $$ The first several values are $$ a_1 = 1,\; a_2 = 5,\; a_3 = 90, \; a_4 = 2665, \; a_5 = ...
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1answer
24 views

Two methods for estimate $s_{\bar{X}}$?

When reading in my book Mathematical statistics and Data Analysis (Ross), I figured there where two different methods to compute the estimate $s_{\bar{X}}$, with $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$...
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1answer
28 views

Can we estimate $P(x)$ using $P(1)$?

Given a polynomial $P(x)$, is it possible to estimate/lower bound/upper bound the value of $P(k)$ for some $k \in \mathbb{N}$ if we know $P(1)$? We can also assume $P(x)$ has only natural ...
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37 views

How to combine correlated estimates to test variable is > 0?

Let X1 and X2 be two unbiased but correlated Gaussian estimators of a true value x. 1. What is the proper way to combine two observations of X1 and X2 to test whether x > 0? 2. How does the answer ...
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1answer
36 views

Estimate counts with different sample sizes

Given an arbitrary time period, lets say one week, but it could be five days, one month etc.., I have a sample from a population. My sample consists of shoppers at a store. For week one my sample is ...
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2answers
31 views

How to know when to use t-value or z-value?

I'm doing 2 statistics exercises: The 1st: An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with a standard deviation of 40 hours. If ...
2
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1answer
37 views

Asymptotic lower bound of this function

Suppose that $n$ is an even number. Let $$f(n)=\frac{\sum_{j=1}^{n/2}\binom{n}{2j}\log(2j)}{2^{n-1}}.$$ Can we find some function $g(n)$ (e.g. $\log(n)$ or $n^\alpha$) such that $f(n)=\Omega(g(n))$? ...
3
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0answers
37 views

unbounded variation of $\sin(x)/x$

How can I show that the variation of $sin(x)/x$ is unbounded? Could you please help me. I know that I have to use but how can I rough estimate that this is bigger than infinity?
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1answer
65 views

Estimation of fraction of integrals

(edited for more clarity) For a given function $f$, which is continuous, and $a < b$ real numbers, I need to make an estimation of the type $ \Bigg| \frac{\int_a^b f(t) (-t)dt}{\int_a^b f(t)dt} \...
2
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1answer
24 views

Variance of Least Squares Estimator

Suppose a fit a line using the method of least squares to $n$ points, all the standard statistical assumptions hold, and I want to estimate that line at a new point, $x_0$. Denoting that value by $\...
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1answer
59 views

Integral of conditional probability density function

As far as I understand, when we fix the condition for the conditional density, we get probability distribution and the integral over all the space is $1$ $P(X|Y=y_0)$: $$\int_{\mathbb{R}}f_{X \mid Y}(...
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1answer
72 views

On characterization of MRE estimators

I have some trouble understanding the second equality in the proof of theorem 6;
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1answer
73 views

On randomized estimators [closed]

I been reading the following text on randomized estimators, I cant manage to understand how the randomisation is incoparated into the randomized estimator. How does the random mechanism fit in, ...
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0answers
24 views

Probability if variable has $15\%$ CV

I have a relatively simple question, but I am not sure if I understand it right. I have estimated through my calculations the value $X$. $X$ depends on many things, but one of them is $Y$ and I know ...
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3answers
78 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
3
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0answers
24 views

How to handle Finite-state-machine with correlated inputs?

My system can be represented by the following state-diagram. where each arch represents Input/Output when a transition is made from one state to the other. The inputs to this FSM are correlated. ...
3
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1answer
51 views

Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then $$\lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ \varphi\left(\frac{\sqrt{...
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1answer
18 views

Estimate line in [theta, rho]-space given 2 points

Given 2 points (x1,y1), (x2,y2) I wish to estimate a line defined by [cos(θ) sin(θ) -r], where r is the distance from origin to the line along a vector perpendicular to the line, and the angle theta ...
2
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1answer
41 views

Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
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1answer
10 views

Analogue Tape how long do I have to record?

If I have 1200ft (feet) of tape. How long will I be able to record for at 7.5ips (inches per second) Thank you
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1answer
39 views

Find zeros of a function or at least say things about their location?

Let $a>0$ be a fixed parameter. I would like to find the (I think there are only two) $x\in \mathbb{R}$ such that $$(x-a)e^{-\frac{1}{2}(x-a)^2} = (x+a)e^{-\frac{1}{2}(x+a)^2}.$$ I know this might ...
1
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0answers
34 views

Bayesian Estimation Derivation

I am trying to understand Bayesian estimation and I come across this line in my lecture notes: θ(Bayesian) = E_θ|x[θ] = E[π(θ|x)] So it's meant to reader that ...
1
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2answers
110 views

Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: $$D:=\left|\,\varphi\left(\frac{x-\alpha^2-\varepsilon}{\alpha}\right)-\varphi\left(\frac{x-\alpha^2+\varepsilon}{\alpha}\...
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0answers
14 views

Check inequality

Let $x\in \mathbb{R}^d$ and $e_j$ be a basis vector with 1 at the $j$-position (otherwise $0$). Is it true that $\frac{1}{\mid x+e_j\mid^{d-2}}-\frac{1}{\mid x\mid^{d-2}}=O(\mid x\mid^{-d+1})$? Does ...
0
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1answer
17 views

variance of a Maximum Liklihood estimator

I am reading a book on Bayesian Estimation and Sensor Fusion and I want to know where the formula below come from. In fact, what is the relation between the variance and the second derivative of the ...
3
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3answers
53 views

Question for the estimation of $\sum_{i=1}^x \frac{1}{w+i}$ as $x \to \infty$.

I have a question of the estimation of this summation: $$ \frac{1}{w+1}+\frac{1}{w+2}+\cdots+\frac{1}{w+x}$$ Which is: $$\sum_{i=1}^x \frac{1}{w+i}$$ What I have tried: applying limit to the ...
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0answers
10 views

If a family of densities is not complete then is it necessary that there isn't any MVUE?

The question is about the truth of this statement: "If the family $\{f(x;\theta):\theta\in\Omega\}$ is not complete, then there doesn't exist any MVUE" MVUE is an abbreviation for "Minimum Variance ...
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0answers
14 views

Computing an expression with limes and limit superior and floor-function

Let $2\leq e\leq r$. I am trying to compute or estimate (from above) $$ \lim_{k\to\infty}\limsup_{n\to\infty}\frac{1}{n}\log\left[(e+r+1)^{2(n+k)-1}-((e+r)\cdot 2r+e+r+(e+r+1)\cdot r^2)^{c_{n,k}}\cdot ...
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0answers
12 views

How to improve a poisson based estimator using variance reduction techniques

Given a random number $X \sim Pois(\mu)$ for some random, i.i.d $\mu$, I'm trying to estimate $P(X \ge x)$ by simulation. The approach is to use a raw/classic estimator, where I generate a bunch of $X$...
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2answers
31 views

What's the best guess for the parameter of an exponentially distributed sample?

I have a sample of size $N$ values. I know the values are exponentially distributed, i.e. they are distributed according to this probability density function: $$ f(x;\lambda) = \begin{cases} \lambda ...
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0answers
11 views

Estimate average change over time with matlab

I'm trying to use matlab to estimate the average difference in location and height for tectonic plates for a university assignment. Not alot of guidelines or pushes in any direction as how to solve ...
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33 views

Estimating distribution from two distributions

I have been doing a survey on Family Incomes in India. The income of male and females are denoted by x and y. x and y are strictly positive. Per chance, individual values of y were deleted. I only ...
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1answer
23 views

Stability (wrt parameters) of elliptic partial differential equation

consider the equation $$\mathcal Lu=f \quad \text{in } \Omega $$ With some appropriate boundary condition, $\Omega$ regoular as you like, $ \mathcal L$ to be defined by $$\mathcal Lu=-div(A(x)\...
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0answers
9 views

Estimate Results From a Database

I have some data in an Excel document, from differents projects i've made. Now I would like to use this data in order to estimate the results for new projects with different input values. Each ...
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1answer
40 views

Energy estimate in Evans PDE book

Before Theorem 6 in Chapter 7.4 in Evans' PDE book Evans claims that there exists $\beta > 0$ such that $$ \beta\|u\|_{H^1(\Omega)}^2 \leq B[u,u]\,, \quad \forall u \in H_0^1(\Omega)\,. $$ From how ...
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2answers
28 views

Optimal estimation of the fusion of two measurements

Suppose I have a sensor measuring a quantity $\text R$. For example the sensor could be a radar estimating the range of a target. We can write: $$R(t)=r(t)+\nu_0(t)$$ where $r(t)$ is the real range ...
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1answer
17 views

An issue related to the expectation maximization algorithm for a coin toss experiment

I just read a very nicely written introduction paper for the expectation maximisation algorithm published in Nature biotechnology by Do and Batzoglou (http://www.nature.com/nbt/journal/v26/n8/full/...
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14 views

Error propagation in complex formula

I'm currently trying to use error propagation formulae to calculate an estimate for the error in the following molecular dynamics formula: $C_v^* = \frac{3}{2}\bigg[1-\frac{2}{3NT^{*2}}\big\langle (\...
0
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1answer
19 views

Predicting values based on given data

num value 8 ? 10 ? 12 24.33 16 22.5 20 22.29 24 22.41 28 22.55 32 ? 36 ? let's say i have the following ...
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1answer
24 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
0
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1answer
16 views

Extimate error, payment rate, continous compounded

Give an estimate of the error, when the payment rate $x_m = r P_0 \frac{(1+r/m)^{mT}}{(1+r/m)^{mT}-1}$ (compounding and repayment m times per year) ist approximated bei $x_{\infty}=\frac{r P_0* e^{rt}}...
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1answer
24 views

Can a prediction interval be interpreted as a probability?

Suppose I find a 90% prediction interval for some data distribution. This implies that if I sample large enough data from this distribution, then 90% of such data will lie inside the prediction ...
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0answers
12 views

Representation of the optimal filter measure as the measure of a diffusion process

In "Mitter SK, Newton NJ. A Variational Approach to Nonlinear Estimation. SIAM J Control Optim. 2003 Jan;42(5):1813–33", it is shown that the path estimation measure $P_{X|Y}(\cdot,y)$ for the ...
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1answer
18 views

Given the estimator find wheter

X has an uniform distribution on interval $(0,\theta]$ where $\theta$ is a positive parameter Given the estimator: $$T(X_1,X2, \ldots, X_n)=\frac{2}{n} \sum_{i=1}^n X_i$$ Find whether this estimator ...
0
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1answer
37 views

Bias and variance of estimator

I have the following estimator, $E = 1/\bar{X}$ of $E = 1/\lambda$ where X is exponentially distributed with parameter $\lambda$. I'm trying to find the bias and variance of this estimator. For the ...