The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

learn more… | top users | synonyms

0
votes
2answers
36 views

General method to find exact confidence interval?

Say we have a beta random variable with pdf, $f_X (x) = e^{-x/\theta}/\theta$ for positive $x $. Find the exact confidence interval of $\theta$ with 95% confidence. A solution to this says ...
4
votes
1answer
63 views

How to use Laplace method to get the asymptotic expansion of multiple integral

I meet difficulty when I try to get the asymptotic behaviour of multiple integral as x tends to plus infinity. And $-1<$p$<1$ $$\int_x^{+\infty}\int_x^{+\infty}e^{-{\frac{1}{2\sigma^2(1-p^2)}\ \ ...
1
vote
1answer
28 views

Confidence interval and symmetric distribution

Let $X_1,...,X_n$ random sample of $X$~$U[\theta-\frac{1}{2};\theta+\frac{1}{2}]$.Consider $[X_{(1)};X_{(n)}]$ a confidence interval for $\theta$. Find their confidence level and show that ...
1
vote
1answer
66 views

Probability about confidence interval

Let $X_1,...X_n$ be iid $N(\theta,1)$. A 95% confidence interval for $\theta$ is $\overline{X}\pm\frac{1.96}{\sqrt{n}}$.Let p denote the probability that an additional independent ...
0
votes
0answers
9 views

Pitman Asymptotic Relative Efficiency

Let $T_1$ and $T_2$ are two test for testing the same $H_0$ versus $H_1$. Let $t_1$ and $t_2$ are test statistic correspond to $T_1$ and $T_2$, respectively. If $t_1$ and $t_2$ have noncentral chi ...
2
votes
2answers
47 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
1
vote
1answer
59 views

How to write R program to solve the confidence interval?

The problem: let $X_1,\ldots,X_n$ be random variable from $\mathrm{Poisson}(\theta)$. Under $H_0: \theta=\theta_0$, we want to find the $(1-\alpha)100\%$ confidence interval for $\theta$ by using the ...
0
votes
0answers
14 views

Log-likelihood approaches +Inf for a Gaussian process

I am trying to do a standard likelihood maximization for the hyperparameters of a Gaussian Process (details in Chapter 5, Rasmussen & Williams: ...
3
votes
1answer
43 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
0
votes
0answers
18 views

testing significant difference on species richness

I measured the species richness (number species) in three different sites. Now, I want to test whether there is significant difference between each site in terms of species richness. The species ...
0
votes
0answers
22 views

Relation between the Coefficient of Multiple Correlation and Coefficient of Simple Correlation

Consider the regression model $Y=\beta_1 X_1+\beta_2 X_2+\epsilon$, with a sample of size $n$, $Y_i=\beta_1 X_{i1}+\beta_2 X_{i2}+\epsilon_i$, $\epsilon_i \in N(0,\sigma^2)$. Suppossing ...
0
votes
1answer
10 views

Proof that the least square estimators are normally distributed

In my book I have the following proof showing that one of the least square estimators is normally distributed: $\hat\beta_i$ = $\frac {S_{xy}}{S_{xx}}$ = $\frac {1}{S_{xx}}\sum_1^n({x_i}- ...
0
votes
1answer
20 views

Integration to get expected mean and variance

I have the following question: The times in minutes needed to collect the tolls from motorists crossing a toll bridge has the probability density function $$f(x) = 2 exp(−2x), 0 ≤ x < ∞$$ A ...
0
votes
1answer
13 views

Continuum random variable distribution: integral and trapezium-rule methods give different result

Suppose p.d.f. is $\frac{1}{63}x^{2}$. Find the $P(4<x<5)$. I've tried with integration method and trapezium-rule, but they give me different result. With integration, ...
0
votes
1answer
16 views

Ratio laying within the confidence interval still being depicted as having an influence?

I keep seeing this in research papers. The researchers claim that there is a positive correlation between A and B then subsequently show that they odds ratio/sample mean etc. is IN the confidence ...
1
vote
1answer
18 views

Hypothesis-test; test about equal sample means

I'm asked to formulate and do a test of the hypothesis that the sample mean (average grade) of math students and economic students are equal. Data below: I'm not sure how to "attack" this problem. ...
0
votes
1answer
13 views

what is the influence of the specific statistical model selection in a practical project

I hope this is the right place to ask this question. But if it is not, please feel free to migrate. There is a famous quote, which is like "all models are wrong, but a few are useful". So, I was just ...
2
votes
1answer
106 views

What is a easy way to draw from distribution $f(x|u) \propto x^{\alpha-1}I(x<u)$?

We know $u$ and want to draw $x$ from the conditional density $f(x|u) \propto x^{\alpha-1}I(x<u), \alpha>0$. One way is that first draw $r$ from uniform(0,1), and then set $x=u ...
6
votes
1answer
678 views

How to quantify the differencen between 2/4 and 20/40?

Assume I have two methods to do prediction. The first method makes 4 predictions and 2 out of 4 are correct. The second method makes 40 predictions and 20 out of 40 are correct. The prediction ...
1
vote
1answer
65 views

Is there a method to check if two curves (non-linear) are identical

I have two data sets of pollutant concentration on simultaneous days. I have to check whether these two curves follow similar pattern or not ( there might be some time lag between both) on daily ...
1
vote
1answer
99 views

Fisher information matrix of MLE's

I know what it means to compute the fisher information matrix of a vector of parameters. However, how does one compute the fisher information matrix of a vector of MLE's? Specifically, I am working ...
1
vote
1answer
21 views

Topics under Model Based Cluster Analysis

Can anyone recommended topic(s) I could use for my thesis under "Model Based Cluster Analysis"? I initially used "Inference in Model Based Cluster Analysis" as my working topic but appears to be ...
0
votes
1answer
59 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
2
votes
1answer
34 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 ...
0
votes
0answers
9 views

Ellipsoid confedence intervales?

Are Bonforonni, Scheffe, Multivariate t, and Tukey for simultaneous Confidence intervals are ellipsoid? How can I tell from the form of the interval that it is ellipsoid or rectangular?
1
vote
0answers
28 views

Rao-Blackwell theorem and conditional distribution

Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$ i) Find a unbiased estimator of $\lambda$ based ...
1
vote
1answer
13 views

Question regarding the density function of first n prediction

This is an example from Bertsekas' Introduction to Probability 2nd edition example 8.2 Consider now a variation involving the first $n$ dates. Assume that Juliet is late by random amounts $$X_1, ...
0
votes
0answers
16 views

Given a sample determine using Chi-squared test whether these values fit in an EXPONENTIAL distribution

Here I've got such a problem. I was given $n = 20$ values for time of good functioning of a robot between two consecutive defects. 1200, 1432, 1502, 1100, 3286, 4235, 1149, 5236, 2234, ...
2
votes
2answers
28 views

In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
2
votes
2answers
102 views

Calculate the confidence interval of parameter of exponential distribution?

How can I calculate the confidence interval for parameter $\alpha$ of exponential distribution ? I think I can use test-t. Knowing that: $$mean = {1\over\alpha}$$ I found that : $${1\over ...
0
votes
1answer
54 views

Confidence Interval for Nonlinear Regression using F-Test - lmfit

I am trying to understand the implementation for the lmfit confidence interval calculation - in the docs it is stated: "The F-test is used to compare our null model, which is the best fit we have ...
0
votes
1answer
21 views

Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fittet parameters and there after refit ...
0
votes
1answer
29 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 ...
0
votes
1answer
17 views

mean square error comparison

Do you have any idea about how i can solve the question below? $X_1$ and $X_2$ are random variables that satisfy $E[X_1]=E[X_2]=\mu$ and $Var[X_1]=Var[X_2]=1$. Show that when $|\mu - 10| \leq ...
1
vote
1answer
39 views

MLE of Integer Valued Normal Distribution

If Z is a normal random variable on $\mathbb{R}^d$ with parameters $(\mu,\Sigma)$ and we know that $\mu\in \mathbb{Z}^d$ and $\Sigma \in \mathbb{Z}^{d+}$; then how can we solve this MLE problem for ...
1
vote
1answer
25 views

Converting Univariate Time Series to a Multivariate Time Series

$X_{t} =0.9X_{t-1}-0.7X_{t-2} + \epsilon_t$ Its clear that the above process does not exhibit the markov property, i.e the future depending on the present. How would I rewrite the above Time Series ...
1
vote
0answers
44 views

Series of sums of normal variables, likelihood principle

Suppose I have a series of normal variables $Y_i \sim \mathcal N(\theta, 1)$ for $1 \leq i \leq N$. Define: $$S_k = \sum\limits_{i=1}^kY_i$$ Since they're sums of normally distributed variables, ...
0
votes
0answers
2 views

inferring parameters from limting relative frequencies

I refer to my previous question concerning what i call the converse strong law of large numbers (instead o the normal SLLN given the probability=p that with prob1, the limiting relative frequency=p; ...
1
vote
1answer
25 views

Linear Regression quadratic terms

I have a hard time understanding the term 'linear regression'. For what I know, linear means polynomial of degree 1. But then, I found that in one of my lectures, the lecturers are saying that this ...
1
vote
1answer
22 views

What does the notation $\{ \pm 1 \}^X$ in relation to functions and hypothesis classes means in the context of PAC learning over half spaces?

I was reading the following paper (on PAC learning over half-spaces) and encountered the following notation for a hypothesis class (on page 4): $$\mathcal{H} \subset \{ \pm 1 \}^X$$ However, it was ...
0
votes
0answers
20 views

iid sequence of random vectors

If $W_1,...,W_N$ is an iid sequence of random vectors, with $W_i=(X_i,Y_i)^T$, does $W_1,...,W_N$ being an iid sequence imply that $X_i$ will be independent of $Y_i$? Does it imply that $(X_i,Y_i)$ ...
1
vote
0answers
12 views

Statistical Modeling with the combination of two models

I'm having a modeling problem now. Assume we have discrete random variable Y and continuous random variables X and Z. First, we assume a logistic regression between Y and Z.(Assumption One) Also, we ...
0
votes
1answer
22 views

Does scatterplot matrix “work” with quadratic variables?

basically I want to plot a scatterplot matrix using a few variables. For simplicity lets say my model is: $$z=\alpha_0 + \alpha_1w+\alpha_2x+\alpha_3y+\alpha_4y^2 + \epsilon$$ When I plot the matrix, ...
0
votes
0answers
25 views

Uniform most powerful Test for one-sided hypothesis

I am trying to understand this proof above. What I am confusing is (1) The whole theorem correspond to the hypothesis $H_0:θ\leθ_0 \, vs \, H_1:θ\gtθ_0$. But at the beginning of the ...
1
vote
1answer
19 views

Question about law of iterated expectations?

I have this question: Let Y = a + bX + U, where X and U are random variables and a and b are constants.Assume that E(U|X) = 0, and that Var(X) > 0. I need to find E[UX] The answer is zero, found ...
0
votes
0answers
20 views

Is this an instance of the base-rate fallacy?

The following line of probability reasoning is supposedly fallacious, and is an instance of the base-rate fallacy. The argument is that $(1)-(3)$ don't give us enough reason to conclude that $(C)$. ...
-2
votes
1answer
28 views

Distribution of Normal distribution

suppose $X \sim Normal(\mu, \sigma^{2})$. What is the distribution of $Y := N(X)$? where $N$ denotes Standard Normal Cumulative Distribution Function? e.g. in a special case when $\mu = 0$ and $\sigma ...
0
votes
2answers
40 views

A woman has n keys

A woman has n keys, of which one will open her door. After trying one she discards it and tries again if it does not work. What is the expected number of attempts needed? Its straight forward to see ...
0
votes
1answer
15 views

Nonparametric Skew of Data

Recently in my studies of statistics, I have come across the second skewness coefficient to determine the skewness of the set of data. The formula is given by: $$ \frac{3(\mu - \nu)}{\sigma}$$ ...
0
votes
1answer
14 views

function of independent random variables

I have following question: If $X$ and $Y$ are independent, then are $g(X)$ and $g(Y)$ independent as well, for any real function $g$?