In pondering this question, I used Mariana's function to generate the following data.
data = {356, 403, 49, 677, 109, 566, 111, 233, 189, 395, 72, 103, 394,
108, 255, 201, 197, 101, 112, 144, 262, 231, 171, 349, 522, 262,
189, 128, 97, 188, 285, 459, 182, 220, 301, 154, 243, 250, 199,
293, 141, 302, 64, 196, 106, 560, 115, 172, 54, 236, 183, 133, 218,
614, 111, 161, 310, 224, 134, 427, 130, 200, 380, 87, 430, 183,
800, 368, 210, 221, 105, 104, 78, 213, 103, 586, 395, 312, 384,
203, 141, 224, 107, 106, 172, 304, 141, 298, 250, 226, 268, 288,
108, 116, 347, 123, 622, 135, 223, 229, 79, 74, 144, 88, 130, 284,
272, 500, 310, 325, 247, 149, 612, 41, 100, 257, 229, 400, 486,
142, 140, 136, 56, 411, 489, 83, 142, 59, 108, 264, 108, 160, 347,
129, 137, 120, 100, 247, 117, 188, 121, 132, 316, 280, 336, 227,
197, 156, 397, 144, 101, 317, 624, 171, 189, 72, 276, 261, 102, 92,
131, 384, 256, 87, 109, 390, 97, 62, 172, 311, 188, 506, 239, 269,
403, 356, 268, 397, 214, 202, 321, 148, 120, 169, 74, 75, 235,
129, 90, 423, 514, 63, 233, 61, 82, 104, 167, 251, 198, 203, 316,
309, 310, 305, 743, 334, 95, 169, 185, 1074, 126, 278, 343, 857,
119, 80, 102, 92, 223, 151, 309, 127, 253, 346, 286, 240, 251, 413,
101, 158, 462, 77, 138, 333, 275, 223, 224, 123, 129, 251, 72,
225, 174, 237, 530, 110, 295, 153, 136, 183, 137, 79, 182, 187,
177, 152, 293, 165, 124, 118, 163, 154, 222, 111, 110, 67, 96, 269,
255, 190, 297, 72, 216, 129, 166, 83, 52, 252, 168, 82, 491, 208,
427, 470, 462, 110, 365, 465, 135, 131, 165, 166, 420, 190, 511,
928, 246, 349, 274, 184, 291, 145, 298, 470, 232, 302, 212, 182,
209, 730, 106, 105, 761, 91, 124, 244, 351, 119, 462, 101, 262,
233, 146, 512, 156, 138, 155, 76, 385, 168, 146, 430, 172, 208,
121, 170, 271, 206, 120, 233, 210, 953, 353, 186, 199, 221, 272,
494, 136, 292, 107, 265, 162, 235, 185, 214, 90, 167, 315, 238,
109, 102, 425, 713, 149, 438, 41, 247, 233, 145, 268, 580, 174,
115, 132, 99, 136, 140, 223, 149, 371, 520, 300, 301, 117, 69, 403,
143, 941, 107, 126, 234, 212, 139, 197, 558, 133, 45, 82, 91, 118,
554, 457, 340, 239, 600, 222, 136, 211, 182, 359, 171, 96, 161,
68, 181, 118, 171, 226, 121, 309, 222, 149, 95, 304, 177, 204, 194,
123, 129, 126, 160, 353, 108, 249, 170, 326, 620, 83, 252, 104,
134, 246, 154, 268, 152, 303, 143, 168, 422, 298, 186, 128, 97, 92,
316, 100, 182, 230, 198, 140, 217, 823, 371, 457, 122, 257, 207,
53, 260, 112, 190, 66, 244, 267, 98, 210, 276, 189, 61, 107, 123,
180, 93, 213, 207, 233, 155, 541, 339, 95, 314, 77, 314, 219, 609,
354, 121, 208, 272, 244, 201, 134, 428, 45, 214, 254, 115, 223,
145, 155, 287, 60, 138, 382, 132, 124, 218, 256, 255, 221, 142,
246, 116, 184, 275, 147, 161, 378, 156, 149, 492, 91, 143, 325,
181, 301, 275, 255, 240, 249, 578, 136, 177, 160, 107, 395, 151,
233, 149, 386, 38, 214, 243, 188, 582, 513, 176, 234, 87, 70, 130,
321, 123, 450, 125, 145, 594, 164, 600, 54, 335, 124, 310, 262,
470, 442, 338, 219, 73, 951, 158, 229, 139, 129, 364, 257, 231,
392, 468, 136, 157, 222, 108, 351, 306, 78, 121, 137, 347, 128,
239, 219, 92, 259, 213, 98, 151, 170, 202, 446, 336, 293, 174, 183,
100, 345, 203, 194, 280, 330, 251, 335, 202, 198, 371, 399, 241,
588, 527, 305, 621, 101, 124, 516, 311, 192, 228, 281, 127, 351,
116, 468, 126, 155, 237, 282, 470, 427, 150, 80, 438, 232, 180,
128, 482, 169, 224, 105, 362, 136, 135, 94, 137, 172, 292, 186, 91,
109, 144, 304, 184, 239, 285, 232, 89, 131, 376, 153, 298, 60, 97,
83, 583, 187, 338, 196, 75, 125, 161, 294, 115, 182, 51, 328, 232,
68, 339, 322, 171, 57, 331, 235, 113, 127, 176, 165, 240, 213,
310, 96, 250, 171, 221, 140, 115, 145, 186, 343, 188, 146, 226,
559, 103, 348, 272, 157, 156, 296, 218, 143, 306, 435, 150, 380,
121, 163, 213, 283, 155, 290, 156, 372, 212, 172, 120, 336, 280,
152, 101, 202, 325, 160, 98, 91, 259, 135, 209, 385, 210, 147, 214,
644, 102, 76, 576, 133, 52, 424, 187, 628, 421, 147, 211, 276,
468, 592, 99, 391, 302, 191, 441, 164, 136, 223, 212, 101, 122,
274, 198, 161, 648, 243, 210, 346, 330, 311, 123, 484, 183, 215,
450, 255, 680, 532, 569, 102, 97, 151, 321, 151, 164, 198, 289,
171, 103, 118, 172, 101, 340, 176, 206, 70, 233, 170, 190, 448,
339, 387, 33, 239, 295, 200, 131, 322, 111, 516, 313, 365, 203, 85,
134, 134, 191, 228, 270, 125, 80, 145, 272, 229, 106, 151, 117,
289, 120, 644, 140, 247, 133, 525, 232, 109, 243, 74, 152, 516,
311, 179, 247, 191, 308, 355, 102, 598, 382, 153, 108, 77, 197,
210, 200, 83, 86, 315, 304, 243, 329, 397, 282, 140, 578, 129, 211,
293, 219, 113, 471, 260, 160, 179, 341, 622, 311, 187, 175, 403,
140, 239, 258, 193, 358, 83, 241, 320, 457, 111, 206, 96, 179, 152,
158, 574, 199, 217, 189, 663, 336, 388, 258, 351, 362, 369, 155,
66, 230, 501, 247, 330, 383, 202, 567, 349, 117, 161, 524, 349,
197, 162, 121, 1005, 343, 325, 255, 59, 303, 79, 203, 505, 337,
607, 272, 170, 190, 129, 503, 780, 304, 243, 272, 146, 135, 689,
105, 287, 406, 119, 58, 466, 90, 194, 111, 69, 113, 262, 145, 95,
79, 93, 154, 272, 245, 238, 135, 65, 90, 209, 154, 455, 77};
plot1 = Histogram[data, Automatic, "PDF", Frame -> True,
PlotLabel -> "Histogram PDF"]
Now I specify a candidate symbolic Gamma distribution and use DistributionFitTest
candidatedist = GammaDistribution[a, b, c, d];
fittestdatatable =
DistributionFitTest[data, candidatedist, "TestDataTable",
Method -> Automatic]
DistributionFitTest[data, candidatedist, "TestConclusion"]
dist2 = DistributionFitTest[data, candidatedist, "FittedDistribution"]
pdf2 = PDF[dist2, x]
plot2 = Plot[pdf2, {x, 0, 1500}, PlotStyle -> Thick, Frame -> True,
PlotLabel -> "dist2 PDF", MaxRecursion -> 2, PlotPoints -> 1000,
PlotRange -> All];
plot3 = Show[plot1, plot2,
PlotLabel -> "dist2 vs Histogram Comparison"]
The resulting output tells me that it failed to converge within 100 iterations and that the fit is rejected at the 5% level presumably due to the low P-value of 0.0409 via Pearson Chi Square. Yet the PDF/Histogram plot and the ProbabilityPlot both look good.
Q1. Is there a way to use something like MaxIterations to go beyond 100 ?
Q2. Did it truly "not converge" ?
Next, I use the very parameters found (a,b,c and d) by DistributionFitTest to define a specific numerical Gamma distribution and repeat the analysis.
a = 7.22667435871025;
b = 4.0245321518298836;
c = 0.5170201479394287;
d = 31.933431399237044;
dist3 = GammaDistribution[a, b, c, d]
DistributionFitTest[data, dist3, "TestDataTable", Method -> Automatic]
DistributionFitTest[data, dist3, "TestConclusion"]
pdf3 = PDF[dist3, x]
plot5 = Plot[pdf3, {x, 0, 1500}, PlotStyle -> Thick, Frame -> True,
PlotLabel -> "dist3 PDF", MaxRecursion -> 2, PlotPoints -> 1000,
PlotRange -> All];
plot6 = Show[plot1, plot5,
PlotLabel -> "dist3 vs Histogram Comparison"]
Now Mathematica appears quite happy with the numerically defined Gamma distribution and produces a P-value of 0.9625 via Cramer-von Mises and reports that the distribution is not rejected at the 5% level. And as before (and as expected) the PDF/Histogram and ProbabilityPlots look good and identical to the previous plots.
Q3. Why is the same distribution first rejected at the 5% level and then subsequently not rejected at the 5% level? Which is the truth ?
Thanks for any insight.
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