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Demography
The word demography is a combination of two Greek words “demos” and “graphos” meaning “human beings” and “writing/measurement” respectively. A Belgian Statistician “ACHILLE GUILLARD” was the first man to pronounce and designate the term “Demography” in the article “éléments de statistique humaine ou demographique comparee” (elements of human statistics or comparative demography). Definition: Duncan and…
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2D-Random Variables
Let X and Y be two random variables defined on the same sample space S. Then the function (X,Y) that assigns a point in is called a two dimensional random variable. Examples: Joint CDF of (X,Y): The joint cdf of (X,Y) is defined as: Properties of CDF: Proof of Property 1: PROOF OF PROPERTY 2:…
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Censoring Schemes
Many researchers consider survival data analysis to be merely the application of two conventional statistical methods to a special type of problem: parametric if the distribution of survival times is known to be normal and nonparametric if the distribution is unknown. This assumption would be true if the survival times of all the subjects were exact and…
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Normal Distribution
A random variable X is said to follow a Normal Distribution with parameters and if it has a probability density function given by: A random variable X is said to follow a Normal Distribution with parameters and if it has a distribution function given by: First Four Moments: Skewness and Kurtosis Some Interesting Properties of…
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Survival Analysis Notes (1)
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RANDOM NUMBER GENERATORS
What are random numbers? Imagine yourself picking up a card from a well shuffled full deck of cards. What could the card you picked be? The jack of spades, the king of diamonds, or the queen of hearts?? Well it could be the ace of clubs too or it could be any of the other…
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SIMULATION
Simulation is a numerical technique for conducting experiments on a digital computer, which involves certain types of mathematical and logical models that describe the behavior of business or economic system or some component thereof over extended period of time. Simulation deals with both abstract and physical models. Some simulation with physical and abstract models might…
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Monte Carlo Methods
A Monte Carlo method is a computational / numerical method that uses random numbers to compute / estimate a quantity of interest. The quantity of interests may be the mean of a random variable , functions of several means , distributions of random variables or high dimensional integrals. Basically, Monte Carlo methods may be grouped…
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HYPERGEOMETRIC DISTRIBUTION
Suppose an urn contains N coloured balls, out of which M are blue and the rest are red in color. Suppose we draw a sample of ‘n’ number of balls from the urn at random and without replacements. Then if X a random variable denoting the number of blue balls drawn, then the probability that…
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Survival Analysis: Competing Risk Theory
INTRODUCTION A distinctive feature of survival data is the concept of censoring. And an implicit concept in the definition of censoring is that if the study had been prolonged (or if subjects had not dropped out), eventually the outcome of interest would have been observed to occur for all the subjects. Conventional statistical methods for the…
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Ratio Estimation
INTRODUCTION: In Survey Sampling, we often use information on some auxiliary variable, to improve our estimator of the finite population parameter, by giving our estimator protection against selection of bad sample. One such estimator is the ratio estimator introduced as follows: In practice, we are often interested to estimate the ratio of the type: For…
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Linear Algebra
Definition 1 (Vector Space). A vector space over a field F is a quadruple (V, +, ., F )) satisfying the following axioms for all α, β ∈ F and x, y, z ∈ V :1. (V, +) is a commutative group, that is,(a) ’+’ is map from V x V to V. [Closure](b) (x…
