## Description

Performs goodness-of-fit test through Khmaladze matringale transformation

## Usage

 `1` ```KhmaladzeTrans(X, Modified = FALSE, strDist, bGraph = FALSE, nNum = 10) ```

## Arguments

 `X` a random sample of n observations `Modified` a logical value which specifies whether or not to use the modeifed version of the test: False calls the original version while True calls the modified version. `strDist` the name of the null distribution for the hypothesis test: Normal, Cauchy, or Logistic. Other distributions such as Gumbel, Weibull and Frechet will be available in later versions. `bGraph` a logical value which specifies whether or not to get the graph of the objective function of the martingale transformation. `nNum` the number of ticks on each segmented interval when drawing the graph of the objective function. The default is 10. Bigger value will result in a smoother graph.

## Value

A list of the following values:

opt.x
• When Modified is False, opt.x is the value of x where the optimum of the objective function - which is also the test statistic - occurs.

• When Modified is True, opt.x is the vector of the value of x's where the infimum and supremum of U_{n} occur.

test.stat
• When Modified is False, test.stat is the test statistic obtained through Khmaladze martingale transformation.

• When Modified is True, test.stat is the vector of the supremum of U_{n}, the infimum of U_{n}, and the difference of them.

graph.data

a data frame which includes the information of the objective function.

graph

a ggplot object which includes the graph of the objective function.

intervals

a list of segmented intervals over which the graph of the objective function is defined.

mu

the point estimate for the location parameter mu

sigma

the point estimate for the scale parameter sigma

## References

 Khmaladze, E.V., Koul, H.L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist., 32. 995-1034

 E.V. Khmaladze, H.L. Koul (2009). Goodness-of-fit problem for errors in nonparametric regression: distribution free approach. Ann. Statist., 37(6A) 3165-3185.

 Kim, Jiwoong (2020). Implementation of a goodness-of-fit test through Khmaladze martingale transformation. Comp. Stat., 35(4): 1993-2017

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66``` ```#################### n = 10 X = rnorm(n, 1,3) # Generate a random sample of n observations from N(1,3) strDist = "Normal" Modified=FALSE lResult = KhmaladzeTrans(X, Modified, strDist, bGraph=TRUE, nNum=10) KMT_OptimalX = lResult\$opt.x KMT_TestStat = lResult\$test.stat KMT_DM = lResult\$graph.data KMT_Graph = lResult\$graph #### Draw the graph of the objective function KMT_Graph KMT_Intervals = lResult\$intervals KMT_Muhat = lResult\$mu KMT_Sigmahat = lResult\$sigma ##################### ##################### n = 10 X = rlogis(n, 1,2) # Generate a random sample of n observations from the logistic distribution strDist = "Logistic" Modified=TRUE lResult = KhmaladzeTrans(X, Modified, strDist, bGraph=TRUE, nNum=10) KMT_Optimal_Positive_X = lResult\$opt.x KMT_Optimal_Negative_X = lResult\$opt.x KMT_Postive_TestStat = lResult\$test.stat KMT_Negative_TestStat = lResult\$test.stat KMT_TestStat = lResult\$test.stat KMT_DM = lResult\$graph.data KMT_Graph = lResult\$graph #### Draw the graph of the objective function KMT_Graph KMT_Intervals = lResult\$intervals KMT_Muhat = lResult\$mu KMT_Sigmahat = lResult\$sigma ##################### ##################### n = 10 X = rcauchy(n, 0,1) # Generate a random sample of n observations from Cauchy distribution strDist = "Cauchy" Modified=FALSE lResult = KhmaladzeTrans(X, Modified, strDist, bGraph=TRUE, nNum=10) KMT_OptimalX = lResult\$opt.x KMT_TestStat = lResult\$test.stat KMT_DM = lResult\$graph.data KMT_Graph = lResult\$graph #### Draw the graph of the objective function KMT_Graph KMT_Intervals = lResult\$intervals KMT_Muhat = lResult\$mu KMT_Sigmahat = lResult\$sigma ##################### ```

GofKmt documentation built on Oct. 24, 2020, 1:07 a.m.