The term consistent estimator is short for “consistent sequence of estimators,” an idea found in convergence in probability. The basic idea is that you repeat the estimator’s results over and over again, with steadily increasing sample sizes. Eventually — assuming that your estimator is consistent — the sequence will converge on the true population parameter. This convergence is called a limit, which is a fundamental building block of calculus.
Levinsohn, J. & MacKie-Mason, J. (1989). A simple, cons. est. for disturbance components in financial models. National Bureau of Economic Research. Technical working paper No. 80. Retrieved January 7, 2017 from http://www.nber.org/papers/t0080.pdf.
When you’re talking about a construct in relation to testing and construct validity, it has nothing to do with the way a test is designed or constructed. A construct is something that happens in the brain, like a skill, level of emotion, ability or proficiency. For example, proficiency in any language is a construct.
Construct validity is one way to test the validity of a test; it’s used in education, the social sciences, and psychology. It demonstrates that the test is actually measuring the construct it claims it’s measuring. For example, you might try to find out if an educational program increases emotional maturity in elementary school age children. Construct validity would measure if your research is actually measuring emotional maturity.
It isn’t that easy to measure construct validity–several measures are usually required to demonstrate it, including pilot studies and clinical trials. One of the reasons it’s so hard to measure is one of the very reasons it exists: in the social sciences, there’s a lot of subjectivity and most constructs have no real unit of measurement. Even those constructs that do have an acceptable measurement scale (like IQ) are open to debate.
After World War II, many efforts were made to apply statistics to construct validity, but the solutions were so complicated they couldn’t be used in real life. Experience and judgment of the researcher are the acceptable norms to testing construct validity. In some circumstances, such as in clinical trials, statistical tests like a Student’s t-test can be used to determine if there is a significant difference between pre- and post tests.
A contour integral is what we get when we generalize what we’ve learned about taking integrals of real functions along a real line to integrals of complex functions along a contour in a two-dimensional complex plane.
It’s not quite as difficult as it sounds. To directly calculate the values of a contour integral around a given contour, all we need to do is sum the values of the “complex residues“, inside of the contour. A residue in this case is what remains when you integrate around the origin. We can also apply the Cauchy integral formula, or use an application of the residue theorem.
What is a contour in the complex plane? Think about it as a finite (fixed) number of smooth curves. We can define it more exactly as a directed curve, that is made up of a finite sequence of directed smooth curves. Each of these curves must be matched to give just one direction.
Integrating over a contour might sound intimidating, so let’s start with something a bit simpler. Suppose we want to integrate the function f(x) over the curve Γ, and suppose M ∈ â„‚1[I] defines a curve such that Γ = M(I).
That’s all well and good. But what if we want to integrate over a contour which is defined by M1,…Ml ∈ C1[I]? We could describe our contour this way:
Convergence of random variables (sometimes called stochastic convergence) is where a set of numbers settle on a particular number. It works the same way as convergence in everyday life; For example, cars on a 5-line highway might converge to one specific lane if there’s an accident closing down four of the other lanes. In the same way, a sequence of numbers (which could represent cars or anything else) can converge (mathematically, this time) on a single, specific number. Certain processes, distributions and events can result in convergence— which basically mean the values will get closer and closer together.
When Random variables converge on a single number, they may not settle exactly that number, but they come very, very close. In notation, x (xn → x) tells us that a sequence of random variables (xn) converges to the value x. This is only true if the https://www.statisticshowto.com/absolute-value-function/#absolute of the differences approaches zero as n becomes infinitely larger. In notation, that’s:
Each of these definitions is quite different from the others. However, for an infinite series of independent random variables: convergence in probability, convergence in distribution, and almost sure convergence are equivalent (Fristedt & Gray, 2013, p.272).
If you toss a coin n times, you would expect heads around 50% of the time. However, let’s say you toss the coin 10 times. You might get 7 tails and 3 heads (70%), 2 tails and 8 heads (20%), or a wide variety of other possible combinations. Eventually though, if you toss the coin enough times (say, 1,000), you’ll probably end up with about 50% tails. In other words, the percentage of heads will converge to the expected probability.
The concept of a limit is important here; in the limiting process, elements of a sequence become closer to each other as n increases. In simple terms, you can say that they converge to a single number.
Convergence in distribution (sometimes called convergence in law) is based on the distribution of random variables, rather than the individual variables themselves. It is the convergence of a sequence of cumulative distribution functions (CDF). As it’s the CDFs, and not the individual variables that converge, the variables can have different probability spaces.
In more formal terms, a sequence of random variables converges in distribution if the CDFs for that sequence converge into a single CDF. Let’s say you had a series of random variables, Xn. Each of these variables X1, X2,…Xn has a CDF FXn(x), which gives us a series of CDFs {FXn(x)}. Convergence in distribution implies that the CDFs converge to a single CDF, Fx(x) (Kapadia et. al, 2017).
Several methods are available for proving convergence in distribution. For example, Slutsky’s Theorem and the Delta Method can both help to establish convergence. Convergence of moment generating functions can prove convergence in distribution, but the converse isn’t true: lack of converging MGFs does not indicate lack of convergence in distribution. Scheffe’s Theorem is another alternative, which is stated as follows (Knight, 1999, p.126):
Statistics MagicLet’s say that a sequence of random variables Xn has probability mass function (PMF) fn and each random variable X has a PMF f. If it’s true that fn(x) → f(x) (for all x), then this implies convergence in distribution. Similarly, suppose that Xn has cumulative distribution function (CDF) fn (n ≥ 1) and X has CDF f. If it’s true that fn(x) → f(x) (for all but a countable number of X), that also implies convergence in distribution.
Almost sure convergence (also called convergence in probability one) answers the question: given a random variable X, do the outcomes of the sequence Xn converge to the outcomes of X with a probability of 1? (Mittelhammer, 2013).
As an example of this type of convergence of random variables, let’s say an entomologist is studying feeding habits for wild house mice and records the amount of food consumed per day. The amount of food consumed will vary wildly, but we can be almost sure (quite certain) that amount will eventually become zero when the animal dies. It will almost certainly stay zero after that point. We’re “almost certain” because the animal could be revived, or appear dead for a while, or a scientist could discover the secret for eternal mouse life. In life — as in probability and statistics — nothing is certain.
The difference between almost sure convergence (called strong consistency for b) and convergence in probability (called weak consistency for b) is subtle. It’s what Cameron and Trivedi (2005 p. 947) call “…conceptually more difficult” to grasp. The main difference is that convergence in probability allows for more erratic behavior of random variables. You can think of it as a stronger type of convergence, almost like a stronger magnet, pulling the random variables in together. If a sequence shows almost sure convergence (which is strong), that implies convergence in probability (which is weaker). The converse is not true — convergence in probability does not imply almost sure convergence, as the latter requires a stronger sense of convergence.
Convergence in mean is stronger than convergence in probability (this can be proved by using Markov’s Inequality). Although convergence in mean implies convergence in probability, the reverse is not true.
The overlap between the two functions can be evaluated by a convolution integral, which is a “generalized product” of two functions when one of the functions is reversed and shifted.
Other names for the convolution integral include faltung (German for folding), composition product, and superposition integral (Arkshay et al., 2014). These integrals have many applications anywhere solutions for differential equations arise, like engineering, physics, and statistics.
If the Laplace transforms (F(s) = L{f(t)} and G(s) = L{g(t)}) both exist for s > a ≥ 0, then H(s) = F(s)G(s) = L{h(t)} for s > a, where:
Note that the integral is commutative, i.e. f * g = g * f: it doesn’t matter which function is evaluated first. This useful fact means that you can try placing either of the functions first or second, to see which resulting integral is easier to evaluate.
In general, integrating convolution integrals can be challenging except in the simplest of cases. It can be easier to work first in the Laplace Domain, then transform back into the whatever domain you’re working with, like the time domain (Cheever, 2020).
For someone recovering from complex mental health issues, this amount of detail is about the right amount for initial first introduction to this topic of study . Thank you..
A correlogram (also called Auto Correlation Function ACF Plot or Autocorrelation plot) is a visual way to show serial correlation in data that changes over time (i.e. time series data). Serial correlation (also called autocorrelation) is where an error at one point in time travels to a subsequent point in time. For example, you might overestimate the value of your stock market investments for the first quarter, leading to an overestimate of values for following quarters.
Correlograms can give you a good idea of whether or not pairs of data show autocorrelation. They cannot be used for measuring how large that autocorrelation is (for a mathematical way to test for serial correlation, try the Durbin Watson test).
A correlogram gives a summary of correlation at different periods of time. The plot shows the correlation coefficient for the series lagged (in distance) by one delay at a time. For example, at x=1 you might be comparing January to February or February to March. The horizontal scale is the time lag and the vertical axis is the autocorrelation coefficient (ACF). The plot is often combined with a measure of autocorrelation like Moran’s I; Moran’s values close to +1 indicate clustering while values close to -1 indicate dispersion.
The above image shows relatively small Moran’s I (between about -0.2 and 0.35). In addition, there is no pattern in the autocorrelations (i.e. no consistent upward or downward pattern as you travel across the x-axis). This set of data likely has no significant autocorrelation.
On the other hand, this next image shows fairly high Moran’s I values and an upward trend. This indicates that autocorrelation is highly likely for your set of data.