# Identifying the best distribution to this data?

I'm trying to fit an appropriate distribution to a data with 216 values and estimate parameters. From Cullen and Frey graph, it looks like lognormal could be a good fit. From q-q plot, Weibull seems to be way off(not shown here), gamma and lognormal both have a decent fit. However, from the plots, I noticed that values near the tails for gamma and lognormal deviate from theoretical quantiles. Per KS goodness of fit test, both gamma and lognormal should be rejected. Can anyone help me find the best distribution to fit this data? Maybe generalized gamma or some other distribution?
I also noticed that when using fitdist() function for gamma and lognormal, I get warning messages saying "produced NaNs". I know this has something to do with starting values. Is it a huge concern as I still get results? There is no reason to believe that this data necessarily comes from a parametric distribution but it'd help me a great deal for my application even if it's an approximate fit.

data:

sample_dat <- c(1 = 24190.6519978227, 2 = 17577.6779580857, 3 = 17783.6137418753,
4 = 18502.2588058986, 5 = 20987.7098031672, 6 = 19082.1828830684,
7 = 19867.7051305278, 8 = 19805.6604578152, 9 = 21073.7410563297,
10 = 20235.1985438263, 11 = 20280.7721982665, 12 = 20618.4731913037,
13 = 20015.0345952459, 14 = 20178.8937222835, 15 = 21165.010526476,
16 = 19869.912957752, 17 = 19936.3647075856, 18 = 19732.6497421531,
19 = 19769.7478650236, 20 = 20275.7268959984, 21 = 19728.5633481974,
22 = 19550.0486566074, 23 = 19986.08929126, 24 = 19518.044670615,
25 = 19880.6560083398, 26 = 20038.4361737349, 27 = 19727.7660781244,
28 = 20575.5693632931, 29 = 19893.3495137753, 30 = 19819.6912196116,
31 = 20037.9126248003, 32 = 19762.9865189421, 33 = 20163.420948739,
34 = 20655.8319892201, 35 = 20157.5385415838, 36 = 20494.4752591327,
37 = 20342.3076327613, 38 = 19921.9235301919, 39 = 20168.4958393548,
40 = 19767.7258943828, 41 = 19980.6964110633, 42 = 19906.6818024689,
43 = 19769.8473448909, 44 = 20026.813630786, 45 = 19777.3262456911,
46 = 19492.905169283, 47 = 19959.1501418138, 48 = 19670.974502509,
49 = 19796.871772798, 50 = 20639.4272084155, 51 = 20279.0230729833,
52 = 20250.3425534929, 53 = 19817.1066831987, 54 = 19346.2985293944,
55 = 20015.3672895762, 56 = 19581.2802602078, 57 = 19507.1241669358,
58 = 19838.9956176621, 59 = 19371.4990351715, 60 = 19656.6360612319,
61 = 20173.4070999522, 62 = 19604.065377713, 63 = 20229.0647995393,
64 = 19395.0701456043, 65 = 19235.7844177276, 66 = 20408.4168183013,
67 = 20212.0917841551, 68 = 19940.5398807866, 69 = 20069.1312811576,
70 = 19388.361897448, 71 = 20218.6934536489, 72 = 20288.8044431891,
73 = 20175.4766982743, 74 = 21102.8588369018, 75 = 20367.3406272503,
76 = 20197.9482860951, 77 = 20605.3123087575, 78 = 23190.1151473022,
79 = 20421.8632368977, 80 = 19838.9569703404, 81 = 19903.3910024236,
82 = 20529.2543352289, 83 = 24423.8018209449, 84 = 19789.4554741556,
85 = 20409.2450942893, 86 = 19187.6856447278, 87 = 19854.0293926443,
88 = 19820.6523140919, 89 = 19647.3078820941, 90 = 19856.0398360484,
91 = 18668.2689169183, 92 = 18806.1743770072, 93 = 19316.0908116071,
94 = 19201.275058431, 95 = 19934.6394421779, 96 = 19320.3075192736,
97 = 18532.5267692405, 98 = 19400.0720216917, 99 = 21950.2266577088,
100 = 18570.1177147663, 101 = 18936.2431216396, 102 = 18192.0699954351,
103 = 19097.0451612151, 104 = 19359.9731844248, 105 = 18836.556092507,
106 = 19735.0745856312, 107 = 19722.3443123634, 108 = 19173.0339230909,
109 = 20365.6006661866, 110 = 20123.9044385451, 111 = 19825.54937388,
112 = 19885.0111644767, 113 = 19414.0353346761, 114 = 20210.774891742,
115 = 20660.8666708364, 116 = 20596.7855769297, 117 = 20663.6530884631,
118 = 20196.0492755493, 119 = 20226.1463575532, 120 = 20700.2336558365,
121 = 20287.0968525683, 122 = 21250.5489155952, 123 = 21268.4163649977,
124 = 21246.6291697841, 125 = 24939.3311103555, 126 = 20597.7926248392,
127 = 20490.695199506, 128 = 20756.9106472625, 129 = 20006.7323761338,
130 = 20185.2391558478, 131 = 20496.0140834608, 132 = 19893.8369321575,
133 = 20142.9907377186, 134 = 19569.9811926461, 135 = 18962.784022162,
136 = 19537.9950479324, 137 = 19530.6810590182, 138 = 19673.8046574291,
139 = 20381.2464924931, 140 = 19161.8961947673, 141 = 19399.6581385452,
142 = 19628.3405896661, 143 = 19328.3622864074, 144 = 19485.182297409,
145 = 18996.538950381, 146 = 18585.1310826446, 147 = 18743.3310031845,
148 = 19093.3003591165, 149 = 19133.6434814352, 150 = 19133.1047974024,
151 = 19094.0654002703, 152 = 19245.7200898926, 153 = 19663.5836236574,
154 = 20065.1241427255, 155 = 20245.0576735299, 156 = 20133.2636199266,
157 = 20316.8734153867, 158 = 20600.7727873761, 159 = 21675.8632677,
160 = 22756.7527085958, 161 = 20876.9359622793, 162 = 20053.3238209932,
163 = 20233.9246955674, 164 = 20038.3857493922, 165 = 20095.1027652358,
166 = 20129.8451905491, 167 = 20362.1187876726, 168 = 20167.0602340497,
169 = 19577.0523241914, 170 = 19544.2179732263, 171 = 19743.3815680761,
172 = 19469.9349326817, 173 = 19744.759325937, 174 = 20133.4187073207,
175 = 20076.4050336208, 176 = 19686.557716286, 177 = 19851.9233283836,
178 = 19499.7859626472, 179 = 19767.7692779779, 180 = 20236.7830376002,
181 = 19916.5202836501, 182 = 19852.2658957894, 183 = 22708.7299098016,
184 = 19535.2342988054, 185 = 19881.1852072082, 186 = 21978.69950247,
187 = 19719.4917230461, 188 = 19845.0343713612, 189 = 19483.6957534967,
190 = 19710.6439591109, 191 = 19475.1964533578, 192 = 19487.602916967,
193 = 19961.0092562518, 194 = 21665.1446008963, 195 = 19469.3228093585,
196 = 19595.5475973323, 197 = 19822.4656998112, 198 = 19638.9709378275,
199 = 19808.01019673, 200 = 19164.8759623877, 201 = 19424.6565379976,
202 = 19538.3303853448, 203 = 19253.1915738142, 204 = 19805.8673118959,
205 = 19778.0830250914, 206 = 19961.3991008866, 207 = 20514.8743721598,
208 = 20744.9688513231, 209 = 20364.2239731278, 210 = 21234.1232167435,
211 = 20251.8204798059, 212 = 20432.7852139617, 213 = 20233.9181878248,
214 = 20340.3176742371, 215 = 19693.2478441763, 216 = 18786.5621200915
)


Have you seen the poweRlaw package? They have a bootstrap estimation of the p-value for lognormal fit to the data.