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67-415: In addition to its use as a test of fit for distributions, it can be used in parameter estimation as the basis for a form of minimum distance estimation procedure. The test is named after Theodore Wilbur Anderson (1918–2016) and Donald A. Darling (1915–2014), who invented it in 1952. The Anderson–Darling and Cramér–von Mises statistics belong to the class of quadratic EDF statistics (tests based on

134-401: A < X < b  and  c < Y < d ) = ∫ a b ∫ c d f ( x , y ) d y d x ; {\displaystyle \Pr(a<X<b{\text{ and }}c<Y<d)=\int _{a}^{b}\int _{c}^{d}f(x,y)\,dy\,dx;} For two discrete random variables, it is beneficial to generate

201-451: A < b {\displaystyle a<b} , is therefore In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "<"), but the distinction is important for discrete distributions. The proper use of tables of the binomial and Poisson distributions depends upon this convention. Moreover, important formulas like Paul Lévy 's inversion formula for

268-454: A ) = P ⁡ ( a < X ≤ b ) = ∫ a b f X ( x ) d x {\displaystyle F_{X}(b)-F_{X}(a)=\operatorname {P} (a<X\leq b)=\int _{a}^{b}f_{X}(x)\,dx} for all real numbers a {\displaystyle a} and b {\displaystyle b} . The function f X {\displaystyle f_{X}}

335-452: A continuous random variable X {\displaystyle X} can be expressed as the integral of its probability density function f X {\displaystyle f_{X}} as follows: F X ( x ) = ∫ − ∞ x f X ( t ) d t . {\displaystyle F_{X}(x)=\int _{-\infty }^{x}f_{X}(t)\,dt.} In

402-849: A functional returning some measure of "distance" between its two arguments. The functional d {\displaystyle \displaystyle d} is also called the criterion function. If there exists a θ ^ ∈ Θ {\displaystyle {\hat {\theta }}\in \Theta } such that d [ F ( x ; θ ^ ) , F n ( x ) ] = inf { d [ F ( x ; θ ) , F n ( x ) ] ; θ ∈ Θ } {\displaystyle d[F(x;{\hat {\theta }}),F_{n}(x)]=\inf\{d[F(x;\theta ),F_{n}(x)];\theta \in \Theta \}} , then θ ^ {\displaystyle {\hat {\theta }}}

469-485: A continuous random variable can be determined from the cumulative distribution function by differentiating using the Fundamental Theorem of Calculus ; i.e. given F ( x ) {\displaystyle F(x)} , f ( x ) = d F ( x ) d x {\displaystyle f(x)={\frac {dF(x)}{dx}}} as long as the derivative exists. The CDF of

536-627: A cumulative distribution function, in contrast to the lower-case f {\displaystyle f} used for probability density functions and probability mass functions . This applies when discussing general distributions: some specific distributions have their own conventional notation, for example the normal distribution uses Φ {\displaystyle \Phi } and ϕ {\displaystyle \phi } instead of F {\displaystyle F} and f {\displaystyle f} , respectively. The probability density function of

603-430: A real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} , evaluated at x {\displaystyle x} , is the probability that X {\displaystyle X} will take a value less than or equal to x {\displaystyle x} . Every probability distribution supported on

670-404: A shorter notation: F X ( x ) = P ⁡ ( X 1 ≤ x 1 , … , X N ≤ x N ) {\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})} Every multivariate CDF is: Not every function satisfying

737-604: A table of probabilities and address the cumulative probability for each potential range of X and Y , and here is the example: given the joint probability mass function in tabular form, determine the joint cumulative distribution function. Solution: using the given table of probabilities for each potential range of X and Y , the joint cumulative distribution function may be constructed in tabular form: For N {\displaystyle N} random variables X 1 , … , X N {\displaystyle X_{1},\ldots ,X_{N}} ,

SECTION 10

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804-597: Is binomial distributed . Then the CDF of X {\displaystyle X} is given by F ( k ; n , p ) = Pr ( X ≤ k ) = ∑ i = 0 ⌊ k ⌋ ( n i ) p i ( 1 − p ) n − i {\displaystyle F(k;n,p)=\Pr(X\leq k)=\sum _{i=0}^{\lfloor k\rfloor }{n \choose i}p^{i}(1-p)^{n-i}} Here p {\displaystyle p}

871-463: Is exponential distributed . Then the CDF of X {\displaystyle X} is given by F X ( x ; λ ) = { 1 − e − λ x x ≥ 0 , 0 x < 0. {\displaystyle F_{X}(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}} Here λ > 0

938-527: Is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable. If X {\displaystyle X} is a purely discrete random variable , then it attains values x 1 , x 2 , … {\displaystyle x_{1},x_{2},\ldots } with probability p i = p ( x i ) {\displaystyle p_{i}=p(x_{i})} , and

1005-415: Is a conceptual method for fitting a statistical model to data, usually the empirical distribution . Often-used estimators such as ordinary least squares can be thought of as special cases of minimum-distance estimation. While consistent and asymptotically normal , minimum-distance estimators are generally not statistically efficient when compared to maximum likelihood estimators , because they omit

1072-466: Is calculated using An alternative expression in which only a single observation is dealt with at each step of the summation is: A modified statistic can be calculated using If A 2 {\displaystyle A^{2}} or A ∗ 2 {\displaystyle A^{*2}} exceeds a given critical value, then the hypothesis of normality is rejected with some significance level. The critical values are given in

1139-402: Is called the minimum-distance estimate of θ {\displaystyle \displaystyle \theta } . ( Drossos & Philippou 1980 , p. 121) Most theoretical studies of minimum-distance estimation, and most applications, make use of "distance" measures which underlie already-established goodness of fit tests: the test statistic used in one of these tests is used as

1206-411: Is called the survival function and denoted S ( x ) {\displaystyle S(x)} , while the term reliability function is common in engineering . While the plot of a cumulative distribution F {\displaystyle F} often has an S-like shape, an alternative illustration is the folded cumulative distribution or mountain plot , which folds the top half of

1273-724: Is continuous at b {\displaystyle b} , this equals zero and there is no discrete component at b {\displaystyle b} . Every cumulative distribution function F X {\displaystyle F_{X}} is non-decreasing and right-continuous , which makes it a càdlàg function. Furthermore, lim x → − ∞ F X ( x ) = 0 , lim x → + ∞ F X ( x ) = 1. {\displaystyle \lim _{x\to -\infty }F_{X}(x)=0,\quad \lim _{x\to +\infty }F_{X}(x)=1.} Every function with these three properties

1340-396: Is equal to the derivative of F X {\displaystyle F_{X}} almost everywhere , and it is called the probability density function of the distribution of X {\displaystyle X} . If X {\displaystyle X} has finite L1-norm , that is, the expectation of | X | {\displaystyle |X|}

1407-1021: Is finite, then the expectation is given by the Riemann–Stieltjes integral E [ X ] = ∫ − ∞ ∞ t d F X ( t ) {\displaystyle \mathbb {E} [X]=\int _{-\infty }^{\infty }t\,dF_{X}(t)} and for any x ≥ 0 {\displaystyle x\geq 0} , x ( 1 − F X ( x ) ) ≤ ∫ x ∞ t d F X ( t ) {\displaystyle x(1-F_{X}(x))\leq \int _{x}^{\infty }t\,dF_{X}(t)} as well as x F X ( − x ) ≤ ∫ − ∞ − x ( − t ) d F X ( t ) {\displaystyle xF_{X}(-x)\leq \int _{-\infty }^{-x}(-t)\,dF_{X}(t)} as shown in

SECTION 20

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1474-421: Is the function given by where the right-hand side represents the probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} . The probability that X {\displaystyle X} lies in the semi-closed interval ( a , b ] {\displaystyle (a,b]} , where

1541-676: Is the number of elements in the sample, and w ( x ) {\displaystyle w(x)} is a weighting function. When the weighting function is w ( x ) = 1 {\displaystyle w(x)=1} , the statistic is the Cramér–von Mises statistic . The Anderson–Darling (1954) test is based on the distance which is obtained when the weight function is w ( x ) = [ F ( x ) ( 1 − F ( x ) ) ] − 1 {\displaystyle w(x)=[F(x)\;(1-F(x))]^{-1}} . Thus, compared with

1608-746: Is the parameter of the distribution, often called the rate parameter. Suppose X {\displaystyle X} is normal distributed . Then the CDF of X {\displaystyle X} is given by F ( t ; μ , σ ) = 1 σ 2 π ∫ − ∞ t exp ⁡ ( − ( x − μ ) 2 2 σ 2 ) d x . {\displaystyle F(t;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{t}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,dx.} Here

1675-480: Is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n {\displaystyle n} independent experiments, and ⌊ k ⌋ {\displaystyle \lfloor k\rfloor } is the "floor" under k {\displaystyle k} , i.e. the greatest integer less than or equal to k {\displaystyle k} . Sometimes, it

1742-629: Is undefined. Note 2: The above adjustment formula is taken from Shorack & Wellner (1986, p239). Care is required in comparisons across different sources as often the specific adjustment formula is not stated. Note 3: Stephens notes that the test becomes better when the parameters are computed from the data, even if they are known. Note 4: Marsaglia & Marsaglia provide a more accurate result for Case 0 at 85% and 99%. Alternatively, for case 3 above (both mean and variance unknown), D'Agostino (1986) in Table 4.7 on p. 123 and on pages 372–373 gives

1809-623: Is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function ( ccdf ) or simply the tail distribution or exceedance , and is defined as F ¯ X ( x ) = P ⁡ ( X > x ) = 1 − F X ( x ) . {\displaystyle {\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).} This has applications in statistical hypothesis testing , for example, because

1876-500: The Anderson–Darling test are treated simultaneously by treating them as special cases of a more general formulation of a distance measure. Examples of the theoretical results that are available are: consistency of the parameter estimates; the asymptotic covariance matrices of the parameter estimates. Cumulative distribution function In probability theory and statistics , the cumulative distribution function ( CDF ) of

1943-470: The Cramér–von Mises distance , the Anderson–Darling distance places more weight on observations in the tails of the distribution. The Anderson–Darling test assesses whether a sample comes from a specified distribution. It makes use of the fact that, when given a hypothesized underlying distribution and assuming the data does arise from this distribution, the cumulative distribution function (CDF) of

2010-909: The Jacobian usually present in the likelihood function . This, however, substantially reduces the computational complexity of the optimization problem. Let X 1 , … , X n {\displaystyle \displaystyle X_{1},\ldots ,X_{n}} be an independent and identically distributed (iid) random sample from a population with distribution F ( x ; θ ) : θ ∈ Θ {\displaystyle F(x;\theta )\colon \theta \in \Theta } and Θ ⊆ R k ( k ≥ 1 ) {\displaystyle \Theta \subseteq \mathbb {R} ^{k}(k\geq 1)} . Let F n ( x ) {\displaystyle \displaystyle F_{n}(x)} be

2077-542: The Python package Scipy implements this rank test for comparing k samples among several other such rank tests. For k {\displaystyle k} samples the statistic can be computed as follows under the assumption that the distribution function F i {\displaystyle F_{i}} of i {\displaystyle i} -th sample is continuous where Minimum distance estimation Minimum-distance estimation ( MDE )

Anderson–Darling test - Misplaced Pages Continue

2144-406: The characteristic function also rely on the "less than or equal" formulation. If treating several random variables X , Y , … {\displaystyle X,Y,\ldots } etc. the corresponding letters are used as subscripts while, if treating only one, the subscript is usually omitted. It is conventional to use a capital F {\displaystyle F} for

2211-628: The empirical distribution function based on the sample. Let θ ^ {\displaystyle {\hat {\theta }}} be an estimator for θ {\displaystyle \displaystyle \theta } . Then F ( x ; θ ^ ) {\displaystyle F(x;{\hat {\theta }})} is an estimator for F ( x ; θ ) {\displaystyle \displaystyle F(x;\theta )} . Let d [ ⋅ , ⋅ ] {\displaystyle d[\cdot ,\cdot ]} be

2278-470: The empirical distribution function ). If the hypothesized distribution is F {\displaystyle F} , and empirical (sample) cumulative distribution function is F n {\displaystyle F_{n}} , then the quadratic EDF statistics measure the distance between F {\displaystyle F} and F n {\displaystyle F_{n}} by where n {\displaystyle n}

2345-441: The generalized inverse distribution function , which is defined as Some useful properties of the inverse cdf (which are also preserved in the definition of the generalized inverse distribution function) are: The inverse of the cdf can be used to translate results obtained for the uniform distribution to other distributions. The empirical distribution function is an estimate of the cumulative distribution function that generated

2412-423: The inverse distribution function or quantile function . Some distributions do not have a unique inverse (for example if f X ( x ) = 0 {\displaystyle f_{X}(x)=0} for all a < x < b {\displaystyle a<x<b} , causing F X {\displaystyle F_{X}} to be constant). In this case, one may use

2479-467: The normal distribution and the exponential distribution have been published by Pearson & Hartley (1972, Table 54). Details for these distributions, with the addition of the Gumbel distribution , are also given by Shorack & Wellner (1986, p239). Details for the logistic distribution are given by Stephens (1979). A test for the (two parameter) Weibull distribution can be obtained by making use of

2546-487: The (finite) expected value of the real-valued random variable X {\displaystyle X} can be defined on the graph of its cumulative distribution function as illustrated by the drawing in the definition of expected value for arbitrary real-valued random variables . As an example, suppose X {\displaystyle X} is uniformly distributed on the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Then

2613-576: The CDF F X {\displaystyle F_{X}} of a real valued random variable X {\displaystyle X} is continuous , then X {\displaystyle X} is a continuous random variable ; if furthermore F X {\displaystyle F_{X}} is absolutely continuous , then there exists a Lebesgue-integrable function f X ( x ) {\displaystyle f_{X}(x)} such that F X ( b ) − F X (

2680-510: The CDF of X {\displaystyle X} is given by F X ( x ) = { 0 :   x < 0 x :   0 ≤ x ≤ 1 1 :   x > 1 {\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\x&:\ 0\leq x\leq 1\\1&:\ x>1\end{cases}}} Suppose instead that X {\displaystyle X} takes only

2747-621: The CDF of X {\displaystyle X} will be discontinuous at the points x i {\displaystyle x_{i}} : F X ( x ) = P ⁡ ( X ≤ x ) = ∑ x i ≤ x P ⁡ ( X = x i ) = ∑ x i ≤ x p ( x i ) . {\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)=\sum _{x_{i}\leq x}\operatorname {P} (X=x_{i})=\sum _{x_{i}\leq x}p(x_{i}).} If

Anderson–Darling test - Misplaced Pages Continue

2814-514: The above four properties is a multivariate CDF, unlike in the single dimension case. For example, let F ( x , y ) = 0 {\displaystyle F(x,y)=0} for x < 0 {\displaystyle x<0} or x + y < 1 {\displaystyle x+y<1} or y < 0 {\displaystyle y<0} and let F ( x , y ) = 1 {\displaystyle F(x,y)=1} otherwise. It

2881-478: The adjusted statistic: and normality is rejected if A ∗ 2 {\displaystyle A^{*2}} exceeds 0.631, 0.754, 0.884, 1.047, or 1.159 at 10%, 5%, 2.5%, 1%, and 0.5% significance levels, respectively; the procedure is valid for sample size at least n=8. The formulas for computing the p -values for other values of A ∗ 2 {\displaystyle A^{*2}} are given in Table 4.9 on p. 127 in

2948-627: The best empirical distribution function statistics for detecting most departures from normality. The computation differs based on what is known about the distribution: The n observations, X i {\displaystyle X_{i}} , for i = 1 , … n {\displaystyle i=1,\ldots n} , of the variable X {\displaystyle X} must be sorted such that X 1 ≤ X 2 ≤ . . . ≤ X n {\displaystyle X_{1}\leq X_{2}\leq ...\leq X_{n}} and

3015-534: The case of a random variable X {\displaystyle X} which has distribution having a discrete component at a value b {\displaystyle b} , P ⁡ ( X = b ) = F X ( b ) − lim x → b − F X ( x ) . {\displaystyle \operatorname {P} (X=b)=F_{X}(b)-\lim _{x\to b^{-}}F_{X}(x).} If F X {\displaystyle F_{X}}

3082-403: The case of a scalar continuous distribution , it gives the area under the probability density function from negative infinity to x {\displaystyle x} . Cumulative distribution functions are also used to specify the distribution of multivariate random variables . The cumulative distribution function of a real-valued random variable X {\displaystyle X}

3149-411: The data can be transformed to what should follow a uniform distribution . The data can be then tested for uniformity with a distance test (Shapiro 1980). The formula for the test statistic A {\displaystyle A} to assess if data { Y 1 < ⋯ < Y n } {\displaystyle \{Y_{1}<\cdots <Y_{n}\}} (note that

3216-401: The data must be put in order) comes from a CDF F {\displaystyle F} is where The test statistic can then be compared against the critical values of the theoretical distribution. In this case, no parameters are estimated in relation to the cumulative distribution function F {\displaystyle F} . Essentially the same test statistic can be used in

3283-577: The diagram (consider the areas of the two red rectangles and their extensions to the right or left up to the graph of F X {\displaystyle F_{X}} ). In particular, we have lim x → − ∞ x F X ( x ) = 0 , lim x → + ∞ x ( 1 − F X ( x ) ) = 0. {\displaystyle \lim _{x\to -\infty }xF_{X}(x)=0,\quad \lim _{x\to +\infty }x(1-F_{X}(x))=0.} In addition,

3350-554: The discrete values 0 and 1, with equal probability. Then the CDF of X {\displaystyle X} is given by F X ( x ) = { 0 :   x < 0 1 / 2 :   0 ≤ x < 1 1 :   x ≥ 1 {\displaystyle F_{X}(x)={\begin{cases}0&:\ x<0\\1/2&:\ 0\leq x<1\\1&:\ x\geq 1\end{cases}}} Suppose X {\displaystyle X}

3417-421: The distance measure to be minimised. Below are some examples of statistical tests that have been used for minimum-distance estimation. The chi-square test uses as its criterion the sum, over predefined groups, of the squared difference between the increases of the empirical distribution and the estimated distribution, weighted by the increase in the estimate for that group. The Cramér–von Mises criterion uses

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3484-437: The distribution or of the empirical results. If the CDF F is strictly increasing and continuous then F − 1 ( p ) , p ∈ [ 0 , 1 ] , {\displaystyle F^{-1}(p),p\in [0,1],} is the unique real number x {\displaystyle x} such that F ( x ) = p {\displaystyle F(x)=p} . This defines

3551-402: The fact that the logarithm of a Weibull variate has a Gumbel distribution . Fritz Scholz and Michael A. Stephens (1987) discuss a test, based on the Anderson–Darling measure of agreement between distributions, for whether a number of random samples with possibly different sample sizes may have arisen from the same distribution, where this distribution is unspecified. The R package kSamples and

3618-413: The graph over, that is where 1 { A } {\displaystyle 1_{\{A\}}} denotes the indicator function and the second summand is the survivor function , thus using two scales, one for the upslope and another for the downslope. This form of illustration emphasises the median , dispersion (specifically, the mean absolute deviation from the median ) and skewness of

3685-487: The integral is of a weighted version of the squared difference, where the weighting relates the variance of the empirical distribution function ( Parr & Schucany 1980 , p. 616). The theory of minimum-distance estimation is related to that for the asymptotic distribution of the corresponding statistical goodness of fit tests. Often the cases of the Cramér–von Mises criterion , the Kolmogorov–Smirnov test and

3752-477: The integral of the squared difference between the empirical and the estimated distribution functions ( Parr & Schucany 1980 , p. 616). The Kolmogorov–Smirnov test uses the supremum of the absolute difference between the empirical and the estimated distribution functions ( Parr & Schucany 1980 , p. 616). The Anderson–Darling test is similar to the Cramér–von Mises criterion except that

3819-444: The joint CDF F X 1 , … , X N {\displaystyle F_{X_{1},\ldots ,X_{N}}} is given by Interpreting the N {\displaystyle N} random variables as a random vector X = ( X 1 , … , X N ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}} yields

3886-568: The joint CDF F X Y {\displaystyle F_{XY}} is given by where the right-hand side represents the probability that the random variable X {\displaystyle X} takes on a value less than or equal to x {\displaystyle x} and that Y {\displaystyle Y} takes on a value less than or equal to y {\displaystyle y} . Example of joint cumulative distribution function: For two continuous variables X and Y : Pr (

3953-423: The notation in the following assumes that X i represent the ordered observations. Let The values X i {\displaystyle X_{i}} are standardized to create new values Y i {\displaystyle Y_{i}} , given by With the standard normal CDF Φ {\displaystyle \Phi } , A 2 {\displaystyle A^{2}}

4020-749: The one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic , T , has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value t {\displaystyle t} of the test statistic p = P ⁡ ( T ≥ t ) = P ⁡ ( T > t ) = 1 − F T ( t ) . {\displaystyle p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).} In survival analysis , F ¯ X ( x ) {\displaystyle {\bar {F}}_{X}(x)}

4087-449: The parameter μ {\displaystyle \mu } is the mean or expectation of the distribution; and σ {\displaystyle \sigma } is its standard deviation. A table of the CDF of the standard normal distribution is often used in statistical applications, where it is named the standard normal table , the unit normal table , or the Z table . Suppose X {\displaystyle X}

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4154-478: The points in the sample. It converges with probability 1 to that underlying distribution. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function. When dealing simultaneously with more than one random variable the joint cumulative distribution function can also be defined. For example, for a pair of random variables X , Y {\displaystyle X,Y} ,

4221-639: The real numbers, discrete or "mixed" as well as continuous , is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) F : R → [ 0 , 1 ] {\displaystyle F\colon \mathbb {R} \rightarrow [0,1]} satisfying lim x → − ∞ F ( x ) = 0 {\displaystyle \lim _{x\rightarrow -\infty }F(x)=0} and lim x → ∞ F ( x ) = 1 {\displaystyle \lim _{x\rightarrow \infty }F(x)=1} . In

4288-418: The same book. Above, it was assumed that the variable X i {\displaystyle X_{i}} was being tested for normal distribution. Any other family of distributions can be tested but the test for each family is implemented by using a different modification of the basic test statistic and this is referred to critical values specific to that family of distributions. The modifications of

4355-422: The statistic and tables of critical values are given by Stephens (1986) for the exponential, extreme-value, Weibull, gamma, logistic, Cauchy, and von Mises distributions. Tests for the (two-parameter) log-normal distribution can be implemented by transforming the data using a logarithm and using the above test for normality. Details for the required modifications to the test statistic and for the critical values for

4422-425: The table below for values of A ∗ 2 {\displaystyle A^{*2}} . Note 1: If σ ^ {\displaystyle {\hat {\sigma }}} = 0 or any Φ ( Y i ) = {\displaystyle \Phi (Y_{i})=} (0 or 1) then A 2 {\displaystyle A^{2}} cannot be calculated and

4489-510: The test of fit of a family of distributions, but then it must be compared against the critical values appropriate to that family of theoretical distributions and dependent also on the method used for parameter estimation. Empirical testing has found that the Anderson–Darling test is not quite as good as the Shapiro–Wilk test , but is better than other tests. Stephens found A 2 {\displaystyle A^{2}} to be one of

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