asymptotic – Consistency of jackknife and bootstrap estimates for simple statistics

For both jackknife and simple bootstrap, I should prove the consistency of the variance and bias estimators. (Just for the simple case that our statistics are a function of the sample mean.)

I already have the evidence for the jackknife estimators, but it is difficult to find some for the bootstrap (especially for the bias estimator).

My question is: We know that the jackknife variance and bias estimators are linear approximations of the respective bootstrap estimates. Can not the results for the jackknife estimators be simply transferred to the bootstrap estimator so that the evidence for the jackknife is not required?

(Also: I have already proved that the jackknife bias correction removes the first-order term in the bias (again for statistics as functions of the sample mean).) Since the jackknife bias estimator is a linear approximation of the bootstrap estimator, it can we do not just transmit the result for the bootstrap bias correction?

Thank you for your help!

Statistics – What is the difference between the common distribution function and the common probability of a random variable?

What is the difference between the common distribution function and the common probability of a random variable?

Book is "A First Course in Probability 9th Edition"
Chapter 6.1 is the common distribution
and 6.7 is the common probability of the function of the random variables

Statistics – How do I find a sufficient and unbiased estimator for this distribution?

(Xi Yi) ~ MN distribution means (u1, u2) and corvariance matrix (σ1 ^ 2 pσ1σ2 pσ1σ2 σ2 ^ 2) i = 1 2 3 … n

1.p σ1 σ2 are known to find a sufficient and unbiased estimator for u1-u2.

2.p σ1 σ2 are unknown, find a sufficient and unbiased estimator for u1-u2 σ1 σ2 and p.

I really have no idea how to deal with these two problems.
Any kind of help will be super helpful!

Probability or Statistics – Bootstrap Resampling: Why are the bootstrap resamples independent?

For the simple bootstrap method we consider n i.i.d. Random variables with common distribution function F estimated by ^ F (parametric or non-parametric). For the procedure we then generate the so-called bootstrap samples by sampling with replacement from the distribution ^ F.

My question is: how can the generated bootstrap samples become independent again? (Already for the simple case that ^ F is the empirical distribution function of the data x1, …, xn does not make sense: Let's assume that a random variable is drawn more than once for the same resample: Xi can only be independent, if it is almost constant, I have no answer for the general parametric case either.

I can not find that in any book, but it's written everywhere without explanation.

Thank you for your help!

SQL Server – Does the STATISTICS IO output contain Reads for the Version Store?

STATISTICS IO contains no reads for the version store, at least for the version store in tempdb.

Here is a demo to prove:

--setup script
USE master
GO

CREATE DATABASE TestDB
GO

ALTER DATABASE TestDB
SET ALLOW_SNAPSHOT_ISOLATION ON
GO

USE TestDB
GO

TROPICAL TABLE WHEN EXISTS dbo.Test
GO

CREATE TABLE dbo.Test (ID int Identity PRIMARY KEY, Junk int)

INSERT dbo.Test
SELECT TOP (100000) 1
FROM master.dbo.spt_values ​​a
CROSS JOIN master.dbo.spt_values ​​b

Start a 30s update loop in an SSMS tab

--Update loop
SET NOCOUNT
DECLARE @stop datetime = DATEADD (SECOND, 30, GETDATE ())

WHILE GETDATE () <@stop
BEGIN
START TRAN

UPDATE dbo.Test
SET junk + = 1

UNDERTAKE
THE END

UPDATE dbo.Test
SET junk = 1

Run two identical queries during the loop SNAPSHOT With STATISTICS IO ON, separated by 15s, so that versions can accumulate.

USE TestDB
SET STATISTICS IO ON
GO

SET TRANSACTION ISOLATION LEVEL SNAPSHOT

START TRAN

SELECT MAX (Junk)
FROM dbo.Test

Wait for delay & # 39; 00: 00: 15 & # 39;

SELECT MAX (Junk)
FROM dbo.Test

UNDERTAKE

The IO statistics show identical read operations:
Stats IO

However, the actual execution plan shows that searching for the second query takes much longer due to reading the version store.
Current plans

Probability or Statistics – Average of samples in a given period

In an experiment, I take samples that yield low (around 0) and high (around 60) values.

In many tests there are higher values ​​at the end of the experiment than at the beginning.
The time is represented by the X values ​​and the results are the Y values.

The representation of the data result of the experiment is as follows:

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Enter the image description here

I would like to calculate the mean of the experiment but taking into account the time.

Probably the solution is to find the area of ​​the plot and divide it by the time span, but I do not know if there is an easy way to solve it.
I want to calculate the mean without integration.

greetings

Statistics – Which algorithm should be counted for a certain period of time?

We tried to develop a real-time statistics engine to measure how hot / trend content is. We know that reaching high-end stuff like youtube / FB / twitter is difficult, our goal is simpler:

  • Count click / read / how / etc .. during the last hour / day / week / etc ..

  • Calculate some base indexes like "LikeCount / ClickCount", "ReadCount / TurnOverCount" etc.

  • Use the data in Step 2 above for reporting or entering other data

We used mysql to log every user action for a content. After one month, the spreadsheet had 10 billion rows when our site had about 2,000 items and 1 million visitors a day. After 2 months, the query slowly became unbearable:

Select count (*) from log_table, where content_id = xxx and timestamp> (now - yyy)

Of course, we have an index for content_id and timestamps and the database size has become very large.

I think our solution is pretty childish. Is there a better algorithm to solve this? (How do Youtube / Twitter solve this problem in general?)

P / S: I recently read about HyperLogLog, somehow understood, can it help in this case?

statistics – Why can not we use x / average for standardization instead of z-scores?

When I checked the data standardization and the Z-score theory, I had this intuition.
For example, suppose you have the results of people who performed two different tests:

Test A (mean = 70%, std.dev = 6%)
+ -------------- + ------- + --------- + ------- +
| Participant # | Score | z-score | x / avg |
+ -------------- + ------- + --------- + ------- +
| 1 | 60 | -1,66 | 0.85 |
| 2 | 65 | -0.83 | 0.92 |
| 3 | 80 | 1.66 | 1,14 |
| 4 | 90 | 3.33 | 1.28 |
| 5 | 40 | -5.00 | 0.57 |
| ... | | | |
+ -------------- + ------- + --------- + ------- +

Test B (mean = 75%; std.dev = 7%)
+ -------------- + ------- + --------- + ------- +
| Participant # | Score | z-score | x / avg |
+ -------------- + ------- + --------- + ------- +
| 1 | 60 | -2,14 | 0.8 |
| 2 | 70 | -0.71 | 0.93 |
| 3 | 80 | 0.71 | 1.06 |
| 4 | 90 | 2.14 | 1,2 |
| ... | | | |
+ -------------- + ------- + --------- + ------- +

We can see that Participant # 3 in Test A has a higher z-score than Participant # 3 in Test B and that he is relatively better than his counterpart.

I can not find any information about the name of the measurement x / avg, but I have the intuition that it could be used as a proxy for standardized data.

I'm certainly wrong, as it is nowhere mentioned, but why?