For questions about estimation and how and when to estimate correctly

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2
votes
1answer
57 views

Conditional expectation and rao-blacwell

I am studying on UMVUE, and I'm struggling to find that conditional expectation Let $X_1,\ldots,X_n$ random sample of $X\sim U[0,\theta]$. i) Show that $2X_1$ is a unbiased estimator for $\theta$ and ...
3
votes
1answer
248 views

Kurtosis of sum of Independent Random Variables

Suppose that $X$ and $Y$ are independent random variables with different expected values and variances. Suppose we define kurtosis as $$Kurt(X)=\frac{E[(X- \mu)^4]}{E[(X- \mu)^2]^2}$$ My question is ...
4
votes
1answer
86 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
0
votes
1answer
29 views

Is the event is plausible or not?

An atlete specialized in long jump events jumps an average of $\bar x=7.91m$ in $12$ trials. The standard error of the mean jump distance in these trials is $0.2m$. Is it plausible that when the ...
0
votes
1answer
59 views

Defining bias function for n trial

Let a point estimate for the sample variance be given as $\hat{\sigma}^2 = \frac{1}{n}\sum\limits_{i=1}^n(X_i- \bar{X})^2$ where $n$ is the number of samples. What is the bias in this estimate as a ...
1
vote
1answer
218 views

comparing MSE of estimations of binomial random variables

$X$ is a binomial random variable defined over 12 Bernoulli trials with a success probability of $p$ in each (i.e. $X\sim\operatorname{Bin}(12,p)$. Consider $\hat p=\frac X{10}$ Determine the range ...
-2
votes
2answers
41 views

Integral with logarithm is positive

Given the following integral: $$I(f) = \int_\mathbb{R} f(x) \log \left(f(x) \sqrt{2\pi} e^{\frac{x^2}{2}}\right) dx,$$ where we assume $\int_{\mathbb{R}} f(x)\, dx =1$ and $f\geq 0$ a.e. Assume for ...
0
votes
1answer
28 views

Estimation of the integral

I am trying to compute, or find a good estimate from above the following integral $$ \frac{1}{\pi}\int_{-\infty}^{\infty}|t|^{-1/p}\left|\frac{|t|^{\nu}-1}{t-1}\right|dt, $$where $0<1/p<1$ and ...
3
votes
2answers
51 views

Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = ...
2
votes
2answers
387 views

Simpson's rule to estimate distance traveled given velocity at certain points

Problem: A boat drives a steady course with a variable speed for 4 hours. The speed is registered at regular intervals in meters per second. The registration shows $2.4, 4.4, 7.6, 8.4, 8.6, 7.9, ...
1
vote
1answer
35 views

How to find the estimator using random variables in statistics

I'm doing an assignment for homework in my statistics class. I'm having trouble really understanding what is going on when it comes to estimators, and what the estimator of something is given a random ...
3
votes
0answers
96 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
6
votes
0answers
56 views

Estimation of the order of torsion in $\mathrm{GL}(n,\mathbb Z)$

Let $A \in \mathrm{GL}(n,\mathbb Z)$ be a torsion, I would like to prove that $\mathrm{order}(A)\leq K\exp (cn^{\alpha})$, with $0<\alpha <1$, for $n$ "large enough". I know that if ...
0
votes
0answers
16 views

Simple Hidden Markov Model with Autoregressive Structure - Estimation?

I observe two series over time: $Y_{1:T}=\{ Y_{1}, \dots, Y_{T}\}$ and $X_{1:T}=\{ X_{1}, \dots, X_{T}\}$, where the $X$ series is supposed to be exogenous (I do not define any stochastic process for ...
0
votes
1answer
47 views

Unbiasedness and Minimum Variance of Estimators of Exponential Distribution

We have an exponential random variable $X$ and we take two samples $X_1$ and $X_2$. $f(x,\theta) = (\frac{1}{\theta})\times e^{(\frac {-x}{\theta})}, x\gt0$, $\theta$ being an unknown parameter. We ...
0
votes
2answers
287 views

How do I determine the transfer function of a plant?

I sitting here with a system which I have to determine the transfer function. The unit receives a velocity and position, and move towards that position with the given velocity. What kind of test ...
1
vote
2answers
27 views

Trouble finding an estimator from a discrete RV

Okay, so I am trying to find unbiased and consistent estimators of parameter $a$ from sequence of RVs that represent unfair dice rolls: it rolls 1 with probability of $1+a$, 6 with probability of ...
0
votes
1answer
40 views

Estimating a power series for the order of an entire function

Let $0<s<1$ and consider the power series $$\sum_{n=0}^{\infty}\frac{r^n}{(n!)^{1/s}}.$$ I need to show that for any given $\epsilon>0$, there exists $R>0$ such that for all $r>R$, ...
0
votes
0answers
28 views

There is an estimation of a $ \sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+…a_0}{p} \right) \le (n-1)\sqrt{p}$

Where can I find a proof of the following inequality? ( $n$ is odd) $$ \sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+...a_0}{p} \right) \le (n-1)\sqrt{\vphantom{d}p} $$ I read that ...
0
votes
2answers
92 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
1
vote
0answers
51 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 ...
1
vote
1answer
34 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 ...
1
vote
2answers
47 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 ...
0
votes
1answer
172 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 ...
2
votes
2answers
35 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 ...
0
votes
1answer
43 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 ...
0
votes
1answer
24 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} = ...
0
votes
1answer
54 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 ...
2
votes
1answer
25 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.
0
votes
1answer
14 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 ...
2
votes
0answers
56 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 ...
0
votes
0answers
35 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 ...
0
votes
2answers
101 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 ...
1
vote
2answers
35 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 ...
1
vote
0answers
33 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 ...
1
vote
4answers
59 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 $$ n! > A \cdot B^{2n+1}?$$
1
vote
1answer
155 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 ...
4
votes
0answers
91 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 ...
1
vote
1answer
40 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 ...
2
votes
1answer
51 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 ...
0
votes
1answer
27 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, ...
1
vote
1answer
43 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 ...
0
votes
1answer
60 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 ...
0
votes
1answer
19 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 ...
1
vote
1answer
371 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) ...
0
votes
1answer
279 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 ...
0
votes
1answer
35 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 ...
0
votes
1answer
54 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 ...
0
votes
1answer
103 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 $\delta_i \ll 1$: $$ x_i = X_i(\alpha) = \xi_i \cdot \alpha, \ \ \ \xi_i \sim N(1, \delta_i) $$ ...
0
votes
1answer
106 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 ...