Empirical Distribution Function - Definition

Definition

Let (x1, …, xn) be iid real random variables with the common cdf F(t). Then the empirical distribution function is defined as

\hat F_n(t) = \frac{ \mbox{number of elements in the sample} \leq t}n = \frac{1}{n} \sum_{i=1}^n \mathbf{1}\{x_i \le t\},

where 1{A} is the indicator of event A. For a fixed t, the indicator 1{xi ≤ t} is a Bernoulli random variable with parameter p = F(t), hence is a binomial random variable with mean nF(t) and variance nF(t)(1 − F(t)). This implies that is an unbiased estimator for F(t).

Read more about this topic:  Empirical Distribution Function

Famous quotes containing the word definition:

    “... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
    —Adrienne Rich (b. 1929)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)