0
$\begingroup$

I have profit (loss) data series with some significant skewness and kurtosis. Is there a way to transform this data to a normal distribution so that I can use the gaussian procedure for the test of hypothesized mean being zero? I am very new to data transformation - tried log with constant addition and cubic root transformations but am not able to achieve normality. Raw data is enclosed below. Any help will be sincerely appreciated as I am hitting a wall here. Have used non -parametric techniques (paired sign and rank test) but am receiving criticism that they suffer from low power of test :(

188.5
0
198.9
847.9
49.5
7
143.2
5595.5
49
57.4
12.9
90
73.1
-47.5
141.2
-9114.9
307.9
241.2
1019.3
368.6
154.3
50.5
65.2
2685.7
413.7
213.1
59.7
316.1
522.9
-303.3
207.9
74.3
32.2
2970.5
-380.5
-1178.9
100.9
208.1
132.5
55.5
-287.2
11.8
4564
4084.9
-26
499
1639
127.7
326.4
6254.7
712.3
1019.5
226
911.7
27.5
-21.2
-70.6
60
0
-365.9
151.7
-155.2
-138.9
303.6
793.6
0
13.3
120.6
287
141.3
34.8
732.8
105
44
55
227.2
-109.5
25.6
140.7
285.2
1449.4
493.7
61.8
-1221.4
-622.5
0
0
132.2
-171.4
-10.1
-66.7
-2417
-727.7
-144.9
63.7
-72.8
248.8
49.7
33.6
267.4
-805.8
584.8
-27.2
-4.1
1165.8
842.7
396.6
34.8
3.5
8.7
107.7
54.4
193.1
-3183.3
1832.6
461.4
4880.8
2468.3
3585.2
53
19.4
515
30.1
2789.6
2702
541
0
14.2
-8.1
0
-140
-2.7
0
36
-134.1
-131.8
-2608.1
-167.4
30
255.1
-5
48.6
46.5
494.2
221.9
-104.2
-750.8
15.9
6.4
56
62.8
250.2
226.8
-4908.9
-80.8
0
0
-986.2
7.1
32
114.2
-1231
-987.2
225.8
52.9
1141.4
7.7
13.5
2927.3
1490.2
258.8
5.5
784.5
654.7
13.8
220
13
-47.8
-106.5
174.2
139.2
269.4
1218.6
7.3
29
260.6
0
0
200.4
4983.3
87.1
117.1
43.3
177.6
-377.4
-273.5
5.5
77.7
16.8
66.5
127.3
15
712.5
277.3
-69
133.8
32.6
777.3
36
268.2
62.7
206.9
1174
-25753.5
-4932.5
-0.7
0.9
11.6
2.5
-34.9
12.4
0
1193.2
-263.9
-84.6
30.1
125.5
54.9
7
-91.7
5442.3
2787.5
44.7
53.7
380.6
2
3200.4
2209.9
27.6
28.1
190.4
99.7
219.3
112.6
49.5
1.5
-256.7
1915.7
-412
18
522.6
510
267.9
62.5
47.5
170
1084.2
2747.4
2032.3
-3.5
22
220.7
337.2
299.5
726.8
326.2
1.5
38.8
107.3
40.6
232
67.2
0
217.3
$\endgroup$
1
$\begingroup$

I wouldn't over-emphasize normal distributions as an ideal here, especially given your outliers:

enter image description here

I used bootstrapping in Stata to get confidence intervals for your mean. The intervals all include zero, whichever way you do it. (Output is slightly edited.)

. set seed 2803

. bootstrap r(mean) , reps(10000) nodots: su profit_loss, meanonly

Bootstrap results                               Number of obs     =        274
                                                Replications      =     10,000

      command:  summarize profit_loss, meanonly
        _bs_1:  r(mean)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   185.2161   124.0647     1.49   0.135    -57.94623    428.3783
------------------------------------------------------------------------------

. estat bootstrap, all

Bootstrap results                               Number of obs     =        274
                                                Replications      =      10000

      command:  summarize profit_loss, meanonly
        _bs_1:  r(mean)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |   185.21605   .7320308   124.06467   -57.94623   428.3783   (N)
             |                                      -81.30128   399.5401   (P)
             |                                      -103.8894   385.3036  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval
$\endgroup$
  • $\begingroup$ Thanks a lot Nick for providing guidance. I have one quick follow up question - just wanted to make sure that I interpret your answer accurately. If I determine CI via bootstrapping, do those confidence intervals hold without any reliance on underlying distribution? I believe that is the case but want to be doubly sure. $\endgroup$ – Ash Dec 18 '17 at 14:09
  • $\begingroup$ That's the main idea here, but do read up on bootstrapping. $\endgroup$ – Nick Cox Dec 18 '17 at 14:18

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