Category: Inferential Statistics

Consider the ordinary limits of expected value and variance of the sample mean: When variance goes to zero like this, intuition suggests that the probability mass gets concentrated and some kind of convergence occurs. DEFINITION 9.9 (Convergence in quadratic mean to a number) If E[Xn] = μn and , and the ordinary limits of the sequence of expected values and variances…

The first stochastic convergence concept that we illustrate is not the strongest one, but it can be easier to grasp. DEFINITION 9.7 (Convergence in probability) A sequence of random variables, X1, X2,…, converges in probability to a random variable X if for every  The definition may look intimidating, but it is actually intuitive: Xn tends to X if, for an arbitrarily small ,…

In this section we start considering in some more depth the issues involved in inferential statistics. The aim is to bridge the gap between the elementary treatment that is commonly found in business-oriented textbooks, and higher-level books geared toward mathematical statistics. As we said, most readers can safely skip these sections. Others can just have…

Any Monte Carlo approach relies on the ability of generating variables that look reasonably random. Clearly, no computer algorithm can be truly random, but all we need is a way of generating pseudorandom variables that would trick statistical tests into believing that they are truly random. The starting point of any such strategy is the…

The time we experience in everyday life is continuous. Engineers simulating, e.g., the flight behavior of an aircraft, have to build a continuous-time model accounting for quite complex dynamics. To make the model amenable to numerical simulation, suitable discretization schemes have to be devised; indeed, nothing is continuous in the digital world of computers. The way…

Monte Carlo simulation is a widely used tool in countless branches of physics, engineering, economics, finance, and business in general. Roughly speaking, the aim is to simulate a system on a computer, in order to evaluate its performance under random scenarios. The name was actually invented by physicists and aptly reflects the role of randomness.…

In one-way ANOVA we are testing if observations from different populations have a different mean, which can be considered as the one factor affecting such observations. In two-way ANOVA we consider the possibility that two factors affect observations. As a first step, it is useful to reconsider one-way ANOVA in a slightly different light. What…

Analysis of variance (ANOVA) is the collective name of an array of methods that find wide applications in inferential statistics. In essence, we compare groups of observations in order to check if there are significant differences between them, which may be attributed to the impact of underlying factors. One such case occurs when we compare…

So far, we have been concerned with parameters of probability distributions. We never questioned the fit of the distribution itself against empirical data. For instance, we might assume that a population is normally distributed, and we may estimate and test its expected value and variance. However, normality should not be taken for granted, just like…

We have defined skewness and kurtosis as:17 These definitions are related to higher-order moments of random variables. Just like expected value and variance, these are probabilistic definitions, and we should wonder if and how these measures should be estimated on the basis of sampled data. The “if” should not be a surprise. If we know that the sampled population…