Comparison of experiments for a class of positively dependent random variables

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

Let {X1,...,XN} be a set of N independent random variables, and let S n be a sum of n random variables chosen without replacement from the set {X1,...,XN} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S n under mild conditions.

A Monte Carlo method for yield estimation in charge-redistribution A/D converters is described, where the capacitances are random variables with finite correlation among themselves. The symmetric property of the resultant covariance matrix was exploited to decorrelate the random variables into another set of independent random variables.

On the expectations of maxima of sets of independent random variables

A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively (negatively) correlated when the tail of the distribution is heavier (lighter) than exponential. Here we use these correlations to detect heavy-tailed behavior in small sets of independent random variables. The method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations.

The paper studies a model of K-distributed clutter in a radar detection problem. The clutter is modelled with a multidimensional Gaussian probability density function where the variance values are represented by a set of independent random variables distributed with a function. The detection of an a prior known target against the considered K-clutter is studied using the likelihood ratio test. To evaluate the probabilities of false alarm (Pfa) and detection (Pd), we propose, as an alternative to the usual Monte Carlo simulation, a procedure that, resorting to analytical and numerical methods, approximates Pfa and Pd by a power series expansion in terms of a suitable parameter of the model. The numerical work is greatly reduced in the region where the expansion holds. In fact, the high-dimensional integrals involved in the power series expansion are reduced to products of low-dimensional ones. A numerical comparison with the standard Monte Carlo computation has been performed on a test case. The approach used here can be adapted to K-models with correlated Gamma variables.

Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least $0.669\dots$ times as great as that of the prophet. In fact, there exists a threshold stopping rule which guarantees a gambler-to-prophet ratio of at least $1-\frac1e=0.632\dots$. In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is $1/2$. In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed---namely, the forward and reverse orderings---the gambler-to-prophet ratio improves to $φ^{-1}=0.618\dots$, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from $0.5$ to $φ^{-1}$, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed $φ^{-1}+o(1)$ until the number of allowed permutations grows to $O(\log n)$. The ratio reaches $1-\frac1e-\varepsilon$ for a suitably chosen set of $O(\text{poly}(\varepsilon^{-1})\cdot\log n)$ permutations and does not exceed $1-\frac1e$ even when the full set of $n!$ permutations is allowed.

Recent papers have reported that 54Mn, which decays by electron capture (a weak nuclear interaction) with half-life ∼312 days, is influenced by solar activity. Should this actually occur, new physics would be needed to explain it. This paper reports results of an analysis of 54Mn activity measured over a time interval of ∼3.6 half-lives. If standard nuclear physics applies, the logarithmic residuals of 54Mn activities should form a stationary set of independent random variables whose statistics are determined solely by a constant decay rate λ and initial mean count μ. Analysis of the time-variation, autocorrelation, and power spectra of the 54Mn logarithmic residuals agrees exquisitely with standard nuclear physics. Computer-simulated activities exhibiting periodic decay of amplitude show that anomalies would be detectable by these statistical tests for values of α as low as ∼1 part in 104. This limit is about 10 times lower than reported deviations from exponential decay.

Partial least squares (PLS) regression (a.k.a. projection on latent structures) is a recent technique that combines features from and generalizes principal component analysis (PCA) and multiple linear regression. Its goal is to predict a set of dependent variables from a set of independent variables or predictors. This prediction is achieved by extracting from the predictors a set of orthogonal factors called latent variables which have the best predictive power. These latent variables can be used to create displays akin to PCA displays. The quality of the prediction obtained from a PLS regression model is evaluated with cross‐validation techniques such as the bootstrap and jackknife. There are two main variants of PLS regression: The most common one separates the roles of dependent and independent variables; the second one—used mostly to analyze brain imaging data—gives the same roles to dependent and independent variables. Copyright © 2010 John Wiley & Sons, Inc.This article is categorized under: Statistical Models > Linear Models Algorithms and Computational Methods > Least Squares

We would like to know whether year-to-year changes, or month-tomonth changes in crop yields or prospects are consistent with observed weather data. Generally, historical weather records extend back farther than records of crop yields. We wish to make use of weather data for the entire period of record even though yield data may be available for a much shorter period. This paper reports on an exploratory inverted matrix approach used in one phase of a crop-weather study. The application of multiple regression methods in the study of relationships between crop yields and weather factors is, of course, not new, but the large amount of computational labor involved has discouraged many workers and our people from attempting correlations studies on a very extensive scale. As pointed out by R. A. Fisher, the use of the inverse matrix of a set of normal equations greatly reduces the amount of computations when the same set of independent variables is used repeatedly; in addition, it serves to simplify the calculation of sampling errors of the regression coefficients. However, a large amount of computational work is still required when the various dependent variables are available for only relatively few years, and these periods vary from crop to crop because of the fact that the data or series were started at different points in time. We would like some way of utilizing all the weather and crop yield data available. Therefore, we would like to devise what might be called inverse matrix solution for a given State or area which could be used whenever the given set of weather factors were appropriate. However, the sampling errors of the regression coefficients cannot be computed using the elements of this generalized where the dependent variables are used for only a subperiod. The inverse matrix is obtained for a given set of independent variables (i.e., weather factors) for the entire period of the weather

Traditional radar-based rainfall estimation is typically done by known functional relationships between the rainfall intensity (R) and radar measurables, such as R–Zh, R–(Zh, ZDR), etc. One of the biggest advantages of machine learning algorithms is the applicability to a non-linear relationship between a dependent variable and independent variables without any predefined relationships. We explored the potential use of two supervised machine learning methods (regression tree and random forest) in rainfall estimation using dual-polarization radar variables. The regression tree does not require normalization and scaling of data; however, this method is quite unstable since each split depends on the parent split. Since the random forest is an ensemble method of regression trees, it has less variability in prediction compared with regression trees, but consumes more computer resources. We considered several different configurations for machine learning algorithms with different sets of dependent and independent variables. The random forest model was appropriately tuned. In the test of variable importance, the specific differential phase (differential reflectivity) was the most important variable to predict the rainfall rate (residual that is the difference between the true rainfall rate and the one estimated from the R–Z relationship). The models were evaluated by 10-fold cross-validation. The best model was the random forest model using a residual with the non-classified training set. The results indicated that the machine learning algorithms outperformed the traditional R–Z relationship. Then, we applied the best machine learning model to an S-band dual-polarization radar (Mt. Myeonbong) and validated the result with ground rain gauges. The results of the application to radar data showed that the estimates of the residuals had spatial variability. The stratiform and weak rain areas had positive residuals while convective areas had negative residuals, indicating that the spatial error structure driven by the R–Z relationship was well captured by the model. The rainfall rates of all pixels over the study area were adjusted with the estimated residuals. The rainfall rates adjusted by residual showed excellent agreement with the rain gauge, especially at high rainfall rates.

Enjoyment Mediates Associations of the Physical and Family Environment With Adolescent Physical Activity: A Structural Equation Modeling Approach

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

The stress-strength models have been intensively investigated in the literature in regards of estimating the reliability θ = P (X > Y) using parametric and nonparametric approaches under different sampling schemes when X and Y are independent random variables. In this paper, we consider the problem of estimating θ when (X, Y) are dependent random variables with a bivariate underlying distribution. The empirical and kernel estimates of θ = P (X > Y), based on bivariate ranked set sampling (BVRSS) are considered, when (X, Y) are paired dependent continuous random variables. The estimators obtained are compared to their counterpart, bivariate simple random sampling (BVSRS), via the bias and mean square error (MSE). We demonstrate that the suggested estimators based on BVRSS are more efficient than those based on BVSRS. A simulation study is conducted to gain insight into the performance of the proposed estimators. A real data example is provided to illustrate the process.

Application of hierarchical linear modelling to the study of trajectories of behavioural development