For questions about estimation and how and when to estimate correctly

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41 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
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1answer
20 views

Estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with 2D Taylor

I need to estimate $\ln\left( 1.04^{0.25} + 0.98^{0.2} -1 \right)$ with a Taylor approximation of a two variable function (i.e. x and y). Eventually I managed to pull the (presumably) correct ...
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2answers
34 views

How to estimate the axis of symmetry for an even function with error?

I have a situation here, where, for an unknown $t$, and an unknown but nice* real function $f$, for which $x\rightarrow f(x-t)$ is even, I measure $f(x) + \epsilon_x$, where $\epsilon_x$ is some kind ...
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1answer
94 views

least mean squares(conditional expectation) problem

The lifetime of a type-A bulb is exponentially distributed with parameter $2$. The lifetime of a type-B bulb is exponentially distributed with parameter $3$. You have a box full of lightbulbs of the ...
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20 views

Maximum likelihood estimation (MLE) calculation [closed]

Find Maximum likelihood estimation (MLE) of theta. Find the MLE's probabiliy distribution. this is what i tried, but i dont know how to simplify it?
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2answers
29 views

Statistics - Estimation problem

I am struggling with a statistics problem that seems quite easy but don't know what to do. In a factory a product is given to two experts - X and Y. They have to independently test the product and ...
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1answer
37 views

chose the best path for estimation

I have a Cartesian grid (100x100) in which some of the points are known (30 out of 10,000) and the rest are unknown. I want to use the known points and estimate the other cells. Is there any ...
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1answer
16 views

standard error for the parameters of a linear regression model

Given a linear model $\mathbf{y} = \beta \mathbf{X} + \epsilon$, it is well known that the estimate for $\beta$ that gives the minimum residual sum of squares (RSS) is given by $\hat{\beta} = ...
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1answer
37 views

interval learning unbiased estimator

Suppose that I draw $n$ points uniformly at random from $\mathcal{U}(-a,a)$ for some $a\in\mathbb{R}_+$. Denote this set of points $\{X_i\}_{i=1}^n$. Now for some $-a < x_1 < x_2 < a$, let a ...
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0answers
18 views

How to estimate $\zeta(s)/\zeta(ks)$ in zero-free regions?

Riemann zeta function $\zeta(s)$ $(s=\sigma+i\tau)$, the best zero-free region to known to date, namely $$\sigma \ge 1-c(\log \tau)^{-2/3}(\log\log \tau)^{-1/3} \quad (\tau \ge 3).$$ and we have ...
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1answer
22 views

Show $\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C\, 2^{-2n}\log(n)$

I'd like to prove $$\sum_{n\leq k\leq 2n}2^{-2k}\log(k)\leq C \, 2^{-2n}\log(n),$$ where $C>0$ is a constant. Can someone give me a hint.
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1answer
11 views

Density Estimation and Analysis

This is an excerpt from BW Silverman's 'Density Estimation for Statistics and Data Analysis.' The oldest and most widely used density estimator is the histogram. Given an origin $x_0$ and a bin ...
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0answers
20 views

Maximum likelihood estimator(MLE)

Consider a sample from a distribution with PDF $$f(x) = \begin{cases} \frac{1}{2}(1+\theta x), & -1 \leq x \leq 1\\ 0, & otherwise \end{cases} $$ find the maximum likelihood estimator of ...
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0answers
24 views

estimating the probability density function of a random variable $X$

I have a random variable $X$ that is a sum of two non-independent random variables $X_1$ and $X_2$. Since $X_1$ and $X_2$ are non-independent, then convolution theorem cannot be used to find the pdf ...
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2answers
85 views

parameter estimation for propotional equations of three variables

I am modeling a system that should estimate a parameter $\beta \in [0,1]$ which is directly proportional to two other variables $P \in [0,1]$ and $NV \in \{0,1,2,\cdots,N\}$, and inversely ...
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15 views

Phasor estimation

Given the following system: $s_k$ ==> [ C ] ==> $r_k$ Where $\left\{ s_k, k=1\ldots n\right\}$ is a set of complex scalars and $r_k$ is given by: $r_k=s_k e^{j\Theta} e^{j\delta k}$ Assuming that ...
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2answers
28 views

Problems with this max likelihood estimation

I have the following density function: $f(x;\omega) = \omega*x^{(\omega-1)}*I_{(0,1)}(x)$ for $\omega > 0$ First I want to have the Likelihoodfuntion, which is $\prod_{i=1}^n f(x_i;\omega)$ I ...
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0answers
14 views

Squared error consistent is asymptotically unbiased lim$_{n→ \infty} E[(\hat\theta_n - \theta)^2] = 0$

An estimator $\hat\theta_n$ is said to be squared error consistent for $\theta$ if lim$_{n→ \infty} E[(\hat\theta_n - \theta)^2] = 0$ a) Show that any squared error consistent $\theta_n$ is ...
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4answers
50 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
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1answer
28 views

Expectation - Sample Covariance

I am trying to derive the expectation $\mathbb E$ of the sample covariance $$\overline{cov}_{X,Y} := \frac{1}{n-1}\cdot \sum_{i=1}^n (X_i-\overline X)(Y_i - \overline Y)$$ where $\overline X = \frac1n ...
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0answers
19 views

Can the limit of the MSE of an estimator be infinity?

Is it ever possible for the limit of the MSE of an estimator be infinity? I was doing an exercise and it turns out that the estimator is consistent but the limit of the MSE is infinity, so I am ...
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1answer
17 views

Given f(x) and two correlated random variables x & y, how do I estimate the correlation of f(x) & f(y)

I have a smooth continuous well-behaved function f(x), where f(x) is positive and mononically increasing with x, and x is positive real continuous variable. Given the mean, variance, and correlation ...
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1answer
45 views

Help show a statistic converges in probability given another statistic that converges in probability

Let $Y = (Y_1,\dots,Y_n)$ be a random sample from $N(\mu,1)$ and $\bar{Y}=\sum\limits_{i=1}^nY_i/n$ I am given that $\bar{Y}^2$ converges in probability to $\mu^2$ and now need to show that ...
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1answer
23 views

Determine an appropriate size of sample

Let's say I have a pool of $N$ balls, which can be of $n$ colors $A_1, \cdots, A_n$. $N$ is much bigger than $n$. What number of balls must I draw if I want to have a good estimate of $R_1, \cdots, ...
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0answers
13 views

Density - Excess Bunching - Bunching Estimator

Saez defines excess bunching at the kink as the area under the density in the dominated region: $$ B = \int^{z^*+d z}_{z^*} h(z)dz \approx h(z^*)dz^* $$ where income $z$ is distributed according to a ...
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1answer
37 views

How to figure out the respective sufficient statistic for a given vector of parameters?

Let $Y$ be a random sample from $N(\mu,\sigma^2)$ where both $\mu$ and $\sigma^2$ are unknown. Let $\theta$ be the vector of parameters of interest $\theta=(\mu,\sigma^2)$. I need to find the ...
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0answers
24 views

Show the linear combiantion is not sufficient for $p$

Let $X_1, X_2, X_3$ be a set of three independent Bernoulli random variables with unknown parameter $p = P(X_i = 1)$. Where it is given that $ \hat p = X_1 + X_2 + X_3$ is sufficient for $p$. Show ...
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1answer
22 views

Mean Square Error & Bias

The random variable $Y$ is related to the angle at which muon particles decay. Y has density function: $$f(y)=\frac{1+\alpha y}{2} \; \; -1 \leq y \leq 1$$ where $\alpha$ is a parameter satisfying ...
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14 views

What is the distribution for this function?

Suppose we let X1, X2,...,Xn be an IID random sample of observations on the random variable X. So what my question is: Assuming that X ~ (μ,σ^2), find the distribution of √n(μ̂-μ)/σ. This is a ...
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1answer
14 views

On the estimation of the average of a deteriministic, scalar, real-valued function of two variables.

Suppose we have a scalar, real-valued function of two variables $F(x,y)$ where ($x$,$y$) belongs to a discrete domain $D$ of $F$, which has finite number of elements. Let $\mid D \mid$ be the number ...
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1answer
73 views

Exponential decay estimate for $(x,t) \in U_T$

Suppose $u$ is a smooth solution of $$\begin{cases}u_t - \Delta u + cu = 0 & \text{in }U \times (0,\infty) \\ \qquad \qquad \quad \, \,u=0 & \text{on } \partial U \times [0,\infty) ...
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1answer
63 views

Exponential decay estimate

Assume $u$ is a smooth solution of $$\begin{cases} u_t - \Delta u = 0 & \text{in }U \times (0,\infty) \\ \qquad \quad u=0 & \text{on }\partial U \times [0,\infty) \\ \qquad \quad u = g ...
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1answer
22 views

Sample Size Estimation with unknown variance

After spending a lot of time I'm still nowhere. What I have: Let d denote the desired margin of error so that the interval for µ is of width 2d. Let (1 − α) denote the probability associated with ...
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1answer
33 views

probability, unbiased median?

I am trying to work a problem from the book. Problem: As an alternative to imposing unbiasedness, an estimator's distribution can be "centered" by requiring that its medium be equal to the unknown ...
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1answer
97 views

The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$

We have $n$ sensors $X_i$ which estimate the scalar value $\alpha$ with different relative accuracies $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
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1answer
21 views

Two chips drawn from urn, calculate $P(|\hat\theta - 3| > 1.0)$.

Two chips are drawn without replacements from an urn containing 5 chips, numbered 1-5. The average of the two drawn is to be used as an estimator, $\hat\theta$, for the true average of all the chips ...
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1answer
31 views

Likelihood and maximum likelihood

what is the likelihood, log-likelihood and MLE of; $$θ(θ+1)x^{θ−1}(1−x)$$ any help greatly appreciated
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1answer
36 views

Is it possible to estimate $e$ based on $N$?

Consider a sequence of random numbers $u_1,\dots,u_n$ obtained from a continuous distribution $F$. Let $N$ be the first one that is greater than its immediate predecessor. In othe words, ...
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0answers
8 views

Estimating mortality rates when $n_t=0$

I am using a series of observations to calculate mortality rates: $m=1-(n_t/n_0)^{1/t}$ $n_0$ number of individuals in the first observation $n_t$ number of individuals at time $t$ I am comparing ...
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1answer
35 views

Find the MLE estimator for θ

Let $Y_1 ,Y_2 ,...,Y_n$ be a random sample from a distribution with pdf $f(y) = e^{-(y -θ) }$ for $y \geq 0 $ and $0$ else a) Find the Method of Moments estimator for θ b) Find the MLE estimator ...
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23 views

Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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40 views

Estimate uncertainty of a function

I have a question about estimate the uncertainty of a function. I have a variable $x$. I assume that I can predict the variable $x$ is followed by a function $f(x)$ such as $$f(x)=\exp(-x^2)$$ Now, ...
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1answer
28 views

Compound distribution with unknown distribution of its hyperparameter

Suppose $X\sim \mathcal{N}(0,\sigma)$, and $\sigma$ is another random variable in a sense that we only know that it is some constant random variable with finite support, i-e $\sigma \in [\sigma_\max, ...
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3answers
60 views

Estimate the number of typos there are in a book, based on two editors' finds

This is one question from an interview I have just taken: Suppose there is a book full of typos. Tom and Jerry found $x$ and $y$ typos throughout the book, respectively. There are $z$ typos that ...
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34 views

Why is my approximation of this alternating series incorrect?

I've been working on some calc problems and I'm stuck on the second part of a problem consisting of estimating the value of a series with a given error. I tried calculating it by hand.. wolfram, ...
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34 views

Estimating $\int_0^1\sin(x^2)dx$ using the Taylor expansion of $\sin(x)$

Problem: a) Find the Taylor polynomial $T_6(x)$ for $f(x) = \sin(x)$ about $x=0$. I found this to be $x-\frac{x^3}{6} + \frac{x^5}{120} + O(x^6)$. b) Use this to estimate $\int_0^1\sin(x^2)dx$ with ...
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11 views

Optimum delay of complex signals for LS minimization

Say I have a complex periodic signal $s(t)$ with period $T$. Note that $s(t)$ is not necessarily a complex sinusoide. Now I define $h(t)$ as \begin{equation} h(t) = \sum_{i}^{N} s(t - \tau_{i} ) ...
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8 views

How to do density estimation restricted to a linear model (multidimensional plane)?

I would like to fit a plane to sampling data. So I would like to do a density estimation but restricted to linear models on the main domain. Is there any standard method? This is also a problem in ...
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1answer
57 views

Deduce the following estimate

Let $u$ be a smooth solution of the uniformly elliptic equation $Lu=-\sum_{i,j=1}^n a^{ij}(x)u_{x_i x_j}$ in $U$. Assume that the coefficients have bounded derivatives. Set $v:=|Du|^2+\lambda ...
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16 views

Decomposing change in estimates

In this scenario, an estimate of a variable is given by the observed data multiplied by a representative weight. Let the observed data at the current time be $d_t$ and the weight $w_t$. The difference ...