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21 views

Does random sampling from a dataset produce the same distribution as the original space?

Let's say we have a dataset $D$ with $N$ rows and $M$ columns. Each column is a feature. And for each feature $X_1, X_2,..., X_N $~ iid $F_p$ where $F_p$ is the distribution for feature p. Now let's ...
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0answers
41 views

Population vs Sampling Frame vs Sample

Could someone please explain how the sampling frame is different from population and sample? I understand that the population is all the sampling unit that match our criteria for the study. And the ...
0
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1answer
32 views

Why is the infinite-population perspective usually taken in sampling? What underlying bias/convergence implications are there?

In sampling literature and causal inference literature, there usually is a distinction made about how to view observed data. The first is usually to view some observed data as having come from a ...
1
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1answer
43 views

Prove property of a confidence interval

How does one go about proving the characteristic of a confidence interval that: A 95% confidence interval means if you were to randomly sample the same way 1000 times and create 1000 confidence ...
1
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1answer
32 views

Generate random sample of X1 and X2

If X2 is dependent on X1, how to generate the random sample of (X1,X2)? One scenario is that we know the prior distribution of X1 and functional relationship between X1 and X2, how to generate the ...
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0answers
18 views

Why the ratio of (distance from expectation)^2 / expectation in a goodness fit test follow a chi-square distribution?

I know that the sum of square of normal random variables follow a chi-square distribution. But when I learn how to do a goodness-fit test I don't know why the ratio of (O-E)^2/E follows a chi-square. ...
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0answers
10 views

Renormalizing a distribution to reduce variance

I have a predictive model $M$ that generates an empirical predictive distribution $P_M$ via a set of samples. I cannot change the predictive model. I can evaluate the predictive performance using ...
1
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0answers
28 views

Does a data-dependent sampling rule induce correlation?

[This question is cross-posted on math SE here ] Suppose I have two iid streams of data that are independent of each other: $X = (X_1, X_2, \ldots)$ and $Y = (Y_1, Y_2, \ldots)$. I want to estimate ...
1
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0answers
56 views

Sampling without replacement - Normal sampling distribution [duplicate]

Most of the introductory stats textbooks, treat the sampling distribution of the mean as a normal distribution when sampling is done without replacement and n/N > 0.1. They just use of the finite ...
0
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1answer
148 views

Why do we use rejection sampling even we know the distribution?

I already read this post and I have the exact same questions. Below I pulled the first question and the answer from the post. Therefore we still use the distribution of p for the randomly ...
4
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2answers
2k views

Difference between “Sampling” and “Subsampling”?

I just got this question in my mind, because I have seen so many times in the literature that these two words are being used alternatively, Sampling and ...
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2answers
43 views

Should we identify outliers from population prior to taking sample?

I am revising undergrad statistics course via this course, where i am learning technique to pull out sample from population. While ensuring that sample is decent representative of population, i am ...
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0answers
30 views

Expected payoff from a weighted random sampling without replacement?

In an evolutionary game theoretic context, I am interested in calculating expected payoffs of different strategies in a 2x2 game, given a weighted random sampling without replacement from a population....
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0answers
31 views

Where do the values for t tables come from?

For a given normal distribution, figuring out what percentage of scores fall between two bounds is straight forward. Calculate the z score and look it up on a z table. Or one can also evaluate the ...
3
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0answers
169 views

minimum number of rolls necessary to determine how many sides a die has

For example, say you have a black box that has a number of n-sided dice in it. You have 4-sided dice, 6-sided, 8, 10, 12, 20 and so on. The die sides are all A except for one side that says B, e.g. ...
13
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3answers
4k views

Why is bootstrapping useful?

If all you are doing is re-sampling from the empirical distribution, why not just study the empirical distribution? For example instead of studying the variability by repeated sampling, why not just ...
4
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0answers
90 views

Maximizing a computationally expensive function

Let $f:[0,1]^{80} \rightarrow [0,1]$ be some function, and say I have a computationally expensive way to calculate $f(x)$ for each $x \in [0,1]^{80}$ (expensive = 40s per query). The goal is to ...
4
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3answers
170 views

Statistics can't be a function of a parameter - but isn't the sample a function of the parameter?

I have a question that relates to this post: Can a statistic depend on a parameter? But on it, the discussion focuses much on the t-statistic given as an example by the question asker. My doubt in a ...
4
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2answers
2k views

What does it mean to sample a data point from or according to a distribution? [duplicate]

I know this is an extremely basic question, but I have never had a course on statistics or applied probability. The only probability I had was in a measure theory course. Now I am doing machine ...
1
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1answer
29 views

Need help with a PMF example

Was reading through my lecture slides and I saw this question I am not sure what or how the prof approached this problem after step 6. It'd be greatly appreciated if anyone can tell me what's going ...
10
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2answers
442 views

Easy proof of $\sum_{i=1}^n \left(Z_i - \bar{Z}\right)^2 \sim \chi^2_{n-1}$?

Let $Z_1,\cdots,Z_n$ be independent standard normal random variables. There are many (lengthy) proofs out there, showing that $$ \sum_{i=1}^n \left(Z_i - \frac{1}{n}\sum_{j=1}^n Z_j \right)^2 \sim \...
4
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3answers
897 views

How are estimators like the Horvitz-Thompson Estimator derived?

The Horvitz-Thompson Estimator is usually given by: $$ \hat{Y}_{HT} = \sum_{i=1}^n \pi_i ^{-1} Y_i $$ The proof that it is unbiased is trivial to do. In additional, there exists other estimators out ...
1
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1answer
292 views

Using binomial vs central limit theorem in hypothesis testing

This is the question that I was working on: "An airline claims that the proportion of luggage that is lost is less than or equal to 0.06. A random sample of size 200 is taken. Out of the 200 ...
4
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3answers
239 views

Expected value of SRSWOR sample maximum

If I draw a sample of size n without replacement from the set {1,2,3,...,N}- what is the expected value of the sample maximum? (n < N). Possible to get a closed form solution?
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0answers
76 views

Ratio of a matrix's sample standard deviation to the sd of the matrix of its row averages approximately = the square root of the sample size?

Given a large matrix, why is the ratio of a matrix's sample standard deviation and the standard deviation of the matrix of its row averages approximately equal to the square root of the sample size? ...
0
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1answer
44 views

Which sampling distributions of the normal correspond to which sample statistics? [closed]

Question: Which sampling distributions of a normally distributed, i.e. $\mathcal{N}(\mu, \sigma^2)$, population correspond to which sample statistics? In particular, which sample statistics (if any), ...
1
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0answers
48 views

What is the purpose of a t-distribution? [duplicate]

As far as I know, we use a t-distribution if the sample size is small and the population is normally distributed. But if the population is normally distributed, then the sample mean is also normally ...
5
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1answer
1k views

Rao-Blackwell unbiased estimator binomial distribution

I have the iids $\ X_1,X_2, ... , X_n$ with pmf $\ P(X_i = x_i) = {{m}\choose{x_i}}\theta^{x_i}(1-\theta)^{m-x_i}, 0 \leq x_i \leq m$ I have the unbiased estimator $\ X_1/m$, the sufficient statistic ...
2
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3answers
164 views

explanation of proof that sample mean is unbiased

William Cochran's book on sampling gives the following proof that a sample mean is unbiased: Since every unit appears in the same number of samples, it is clear that $E[Y_1 + \cdots + Y_n]$ must be ...
0
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1answer
183 views

Covariance of sampling mean and sampling variance

Let a simple random sampling design in a finite univers U (i.e. all possible samples of size n are equally likely to occur). How to calculate the covariance of sampling mean $\bar{y}$ and ...
1
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2answers
71 views

Questions on population and sample

As someone new to statistics, I had a debate with my supervisor today on the definition of population and sample in the following case. Suppose we track all new users (1,000 in total) that used an ...
5
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1answer
537 views

Proof of the Horvitz-Thompson result

I'm trying to find an elementary derivation (proof was the wrong word) of the Horvitz-Thompson estimator: $$ \hat{Y}=\sum_{i\in s}\frac{y_{i}}{\pi_{i}} $$ where $i \in s$ if and only if unit $y_{i}$ a ...
1
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0answers
78 views

Sampling from the conditional distribution of continuous random variables?

Suppose I have a prior $\pi(\theta)$ and likelihood $f(x|\theta)$, where $\theta$ takes values in $\{0,1\}$. I implement the following procedure For $i = 1, \dots, T$ Simulate $\theta_i \sim \pi(\...
0
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1answer
2k views

How to calculate confidence level for a given sample size and population size?

It's been a while since I had statistics in uni, and so I'm a little rusty. I need some help with a fairly straight forward calculation of the confidence level of a sample size. I've been trying to ...
4
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1answer
3k views

The effect of temperature in temperature sampling

I was reading this while I found: The high temperature sample displays greater linguistic variety, but the low temperature sample is more grammatically correct. Such is the world of temperature ...
0
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1answer
62 views

How to obtain the minimum number of individuals needed to estimate a character frequency

I have 150 individuals coming from the same population. I want to know what is the minimum number of individuals I need to estimate the frequency for ~1600 different characters. Each character can ...
2
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1answer
550 views

Sample space and outcome of birthday problem

Suppose, calculating the probability of having at least two peoples same birthday from 25 people. What is the sample space and outcome of the experiment? As far as I pondered, S = 365^25 and outcome ...
0
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1answer
36 views

What is the difference between simulation design and simulation data? [closed]

I read that there is a simulation design such as GARCH(1,1) and this is called simulation design. However, when I have a model (for example a simple linear model) then I can simulate as many data ...
0
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0answers
85 views

How to draw samples from a multivariate Gaussian distribution without having access to a function that does the job? [duplicate]

I am using the programming language Lua which does not have any built-in function for drawing samples from a multivariate Gaussian distribution. So I wonder, how can one implement a function that does ...
1
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0answers
33 views

Procedure for calculating a sampling distribution

I'm still trying to understand the basics of understanding the intuition of sampling distributions and calculating the sampling distributions of common estimators. For example, I understand the ...
3
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1answer
303 views

“Let random variables $X_1,\dots, X_n$ be a iid random sample from $f(x)$” - what does it mean?

In books it is often written, Let random variables $X_1,\dots, X_n$ be a iid random sample from $f(x)$. What does it mean? Are $X_1,X_2,\dots,X_n$ different values of one random variable $X$ which ...
0
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1answer
257 views

What are those variables in Cochran's formula

I cannot find any information about the Cochran's formula below: $$\text{Sample Size} = \frac{n}{1 + (n/\text{population})}$$ in which $n$ is equal to $Z * Z [P (1-P)/(D*D)]$ So I assume Z is the z-...
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0answers
33 views

Pandey and Dubey estimator.

I am studying sampling theory Pandey and Dubey(1988) proposed the following product estimator. $\bar y_{PD} = \bar y \left( \frac{\bar x + C_x}{\bar X +C_x}\right)$ And its Mean square error is ...
0
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0answers
769 views

Central Limit Theorem, Skewed Distribution and outlier detection

I'm trying to see how the CLT can help when trying to detect outliers in a distribution. Say I'm pricing houses in a location and that the resulting distribution of my sample of prices is rightly ...
2
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1answer
60 views

Basic question on sampling stream of data

I am reading up an article and came across sampling scenario, but I am not able to come up with intuition behind the numbers presented. Scenario: User issuing search queries to a search engine. "...
1
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1answer
361 views

what is the intuition behind the SRSWOR formula?

I earlier asked about Slovin's Formula (https://math.stackexchange.com/questions/1410492/what-is-the-intuition-behind-slovins-formula), and learned shortly thereafter that it was derived from this ...
2
votes
1answer
556 views

Mean and variance for unequal samples

I have a sampling of variable sized plots. Each plot contains the number of trees present on the plot. Given: $n=$ the number of plots $s_i=$ the size of the $i^{th}$ plot $y_i=$ the number of trees ...
0
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1answer
495 views

How do I split a normal distributed sample into groups of percentiles but with an additional random noise component for uncertainty?

I have a sample of students that I want to divide into smaller groups based on a their IQ but with a certain random noise component - how can I do that? I need to cluster the best, the average and ...
1
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1answer
37 views

What is the standard way to distinguish the errors associated with sampling and measurement in statistics?

This is probably a very basic, yet not easy, question in applied science. I was just wondering, what are usually the standard ways to deal with it? Any pointers to further references are greatly ...
1
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1answer
616 views

Proof for the sampling variance of the Neyman Estimator

I'm going through Imbens and Rubin's new book and I just for the life of me can't figure out 1 minor detail in their proof for the sampling variance of the Neyman estimator $\bar{Y}^{obs}_{t} - \bar{Y}...