For questions about estimation and how and when to estimate correctly

learn more… | top users | synonyms

0
votes
0answers
19 views

Estimation and combinatorics

I have a single user A, who needs to be matched to another user B. The criteria for matching A and B is 1 characteristic only. B is present in either one of the groups described below or present in ...
0
votes
0answers
46 views

expectation and variance of an implicit estimator

Suppose the following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+...
2
votes
1answer
38 views

Bounding the absolute error of the linear approximation by $|E|\le\frac{n^2M}{2}\|\mathbf h\|^2$

Let $f: D\subseteq \Bbb R^n \to \Bbb R$ be a $C^2$ function. I'm trying to show that the absolute value of the error of the first order Taylor approximation of $f(\mathbf x+\mathbf h)$ is bounded ...
0
votes
0answers
40 views

Expectation and Variance of an Estimator

Imagene following equation holds \begin{align*} p_2=\int\limits_{-\infty}^{\Phi^{-1}(p)}\int\limits_{-\infty}^{\Phi^{-1}(p)} \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\bigg({-\frac{1}{2}\frac{x^2-\rho xy+y^2}{...
1
vote
1answer
18 views

Proving OLS estimator of variance

According to Gujarati, author notes that in a simple linear equation form $Y_i=\alpha +\beta X_i + \epsilon_i$ where regression model is defined as $\hat Y_i =\hat \alpha + \hat \beta X_i$ OLS method ...
1
vote
0answers
21 views

Knowing the rates of gain and loss, estimate the average amount.

This question was presented to me as such: Say that people grow 100 hairs a day and lose 100 hairs a day and hairs have a lifespan of 100 days. Estimate how much hair an average person would have. ...
0
votes
1answer
14 views

Show that an estimator is unbiased and it's variance goes towards 0? Poisson distribution

Let X be Poisson Distributed with expectation λt. Show that the estimator is unbiased and has a variance that goes towards 0 when the times goes to infinity. λ^=X/t I know that for these questions ...
1
vote
1answer
21 views

Linear estimation of an exponential distribution

QUESTION We have $Y \sim \mathrm{Exp}(1/6)$. We define $T = e^{−4Y}$ Calculate the best linear estimator of $T$ according to $Y$ ANSWER Ok it sounds pretty simple at first I got my $f_Y(y)=\frac{...
0
votes
1answer
34 views

How to check if an estimator is expectation right?

I am sorry if the terminology is a little bit off, and anyone that knows the correct terminology please correct me. Let us assume we have 3 independent measures, X1, X2, and X3 from the same ...
0
votes
0answers
5 views

How to compute the probability and CI of replicating multiple previously observed statistically significant p-value?

The FDA often requires a sponsor to conduct multiple clinical trials prior to approval. Given the following observations in a ph2 and ph3 trial, how would you go about predicting the probability of ...
2
votes
1answer
15 views

Lest number of terms for a partial sum to estimate sum within error

So here is my question. I used the Alternating series to prove that the sum does converge absolutely, so it converges in general. I then tried to say that $\frac{1}{n^4}<10^{-8}$ and solve for n ...
1
vote
1answer
22 views

It's possible to generalize the Ml inequality (also call Estimation Lemma)?

The ML inequality property in complex integral says $|\int_{c}f(z)dz| \leq ML$. If I have two function in the integral, I can write the inequality: $|\int_{c}f(z)g(z)dz| \leq ML|\int_{c}g(z)dz| $ ?. ...
1
vote
1answer
23 views

How to adjust estimation of probability according to new information

Suppose $\{a_1,a_2,\dots,a_n\}$ is a permutation of $\{1,2,\dots,n\}$. The probability of $a_i=j$ is estimated to be $p_{ij}$. The probability matrix might look like this $$ P=\left( \begin{matrix} ...
1
vote
0answers
48 views

a consequence of Prime Number Theorem

By Prime Number Theorem we have $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$, so $\frac{p_{n+1}}{p_n}=1+a_n$ where $a_n\to 0$. How fast does $(a_n)$ converge to $0$ ? Does for example $a_n\ln n$ or $...
0
votes
0answers
27 views

Forecasting Query on probabilities

many Thanks if you can help with this query. Player A and Player B are due to play a darts match. Player A has a record of scoring between 80 and 140 in each of his last 35 rounds of throws. Best ...
0
votes
0answers
9 views

Question about the process of inducement of “MAP, MLE, MDE, and MCE”

Question about "Maximum A Posteriori (MAP), Maximum Likelihood Estimation (MLE), Minimum Distance Estimation (MDE), Maximum Correlation Estimation (MCE)" Let the space of $\mathbf{x}_k$'s $$\...
0
votes
1answer
26 views

Calculate average life time in system with inputs and observed items inside system in a given time

In a given time (t) i observe the inputs that enter a system and the total items that are the sistem, i.e ...
0
votes
0answers
12 views

Estimation of copulas

For estimation of a parameter of bivariate copula, we really need bivarite data? or from two different one dimensional data we can estimate copula parameter?
1
vote
0answers
27 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}=...
2
votes
0answers
28 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-\...
1
vote
0answers
17 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 $k=1$}\ \ \ \ \ \ \ \ ...
3
votes
2answers
43 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 = ...
0
votes
1answer
25 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 $\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
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 ...
1
vote
2answers
38 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
votes
1answer
38 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
votes
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?
1
vote
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
votes
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 $\...
0
votes
1answer
71 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}(...
1
vote
1answer
73 views

On characterization of MRE estimators

I have some trouble understanding the second equality in the proof of theorem 6;
-1
votes
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, ...
1
vote
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 ...
1
vote
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
votes
0answers
31 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
votes
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{...
0
votes
1answer
24 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
votes
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 ...
-1
votes
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
1
vote
1answer
40 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
vote
0answers
36 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
vote
2answers
111 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}\...
0
votes
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
votes
1answer
18 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
votes
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 ...
0
votes
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 ...
0
votes
0answers
15 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 ...
0
votes
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 $X$...