The transmuted family of distributions has been receiving increased attention over the last few years. For a baseline<i> G</i> distribution, we derive a simple representation for the transmuted-<i>G</i> family density function as a linear mixture of the<i> G</i> and exponentiated-<i>G</i> densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, R&#xe9;nyi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set.

Marcelo Bourguignon

Indranil Ghosh

Gauss M. Cordeiro

General Results for the Transmuted Family of Distributions and New Models

Nakagami distribution is considered. The classical maximum likelihood estimator has been obtained. Bayesian method of estimation is employed in order to estimate the scale parameter of Nakagami distribution by using Jeffreys&#x2019;, Extension of Jeffreys&#x2019;, and Quasi priors under three different loss functions. Also the simulation study is conducted in R software.

Kaisar Ahmad

S. P. Ahmad

A. Ahmed

Classical and Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution

In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M1" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>, expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.11981pt" id="M2" height="9.46863pt" version="1.1" viewBox="-0.0498162 -8.34882 26.7929 9.46863" width="26.7929pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g><g transform="matrix(.013,0,0,-0.013,9.12,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,20.383,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg> occurrences of success runs of length <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M3" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg> with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M4" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 6.16729 8.8423" width="6.16729pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-127" d="M442 569C442 628 407 660 362 660C328 660 292 648 255 616C200 568 165 508 123 334L106 264C80 242 52 222 23 202L30 176C54 191 76 205 100 222L87 166C73 103 79 43 95 19C108 0 139 -12 169 -12C242 -12 318 38 369 106L354 129C322 93 260 49 215 49C173 49 147 81 170 187L189 274C293 344 442 452 442 569ZM380 565C380 478 267 377 196 322L216 406C254 556 282 623 332 623C364 623 380 596 380 565Z"/></g></svg>-overlapping. The case <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6370001pt" id="M5" height="9.2729pt" version="1.1" viewBox="-0.0498162 -8.6359 27.354 9.2729" width="27.354pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-127"/></g><g transform="matrix(.013,0,0,-0.013,9.679,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g><g transform="matrix(.013,0,0,-0.013,20.942,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g></svg>, where there must be a gap of at least <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.15067pt" id="M6" height="11.439pt" version="1.1" viewBox="-0.0498162 -9.28833 13.0309 11.439" width="13.0309pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><use xlink:href="#g113-127"/></g><g transform="matrix(.013,0,0,-0.013,9.466,0)"><use xlink:href="#g113-9"/></g></svg> trials between success runs, is also studied. Next we treat the distribution of the waiting time for the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455406pt" id="M7" height="9.48169pt" version="1.1" viewBox="-0.0498162 -9.23615 16.6324 9.48169" width="16.6324pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-115"/></g><g transform="matrix(.013,0,0,-0.013,5.488,0)"><path id="g121-114" d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z"/></g><g transform="matrix(.013,0,0,-0.013,9.531,0)"><path id="g121-102" d="M513 0V26C455 31 447 37 447 103V278C447 398 390 449 309 449C255 449 202 412 166 376V712C127 700 67 684 19 677V653C85 647 87 643 87 580V103C87 37 79 31 19 26L18 0H231V26C171 32 166 39 166 103V341C194 373 232 392 270 392C337 392 368 351 368 269V103C368 38 360 32 302 26V0H513Z"/></g></svg> nonoverlapping appearance of a pair of successes separated by at most <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3499298pt" id="M8" height="9.63826pt" version="1.1" viewBox="-0.0498162 -9.28833 26.3971 9.63826" width="26.3971pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g><g transform="matrix(.013,0,0,-0.013,9.445,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,19.981,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg> failures (<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.11977pt" id="M9" height="10.4081pt" version="1.1" viewBox="-0.0498162 -9.28833 27.8499 10.4081" width="27.8499pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g><g transform="matrix(.013,0,0,-0.013,10.171,0)"><use xlink:href="#g117-94"/></g><g transform="matrix(.013,0,0,-0.013,21.434,0)"><use xlink:href="#g113-51"/></g></svg>).

S. J. Dilworth

S. R. Mane

Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials

Scan statistics are mostly used to detect spatial areas or time intervals in which the mean level of a given variable is more important. Sometimes, when this variable is continuous, there is an interest in looking for clusters in which its variability is more important. In this paper, two scan statistics are introduced for identifying clusters of values exhibiting higher variance. Like many classical scan statistics, the first one relies on a generalized likelihood ratio test whereas the second one is based on ratios of empirical variances. These methods are useful to identify spatial areas or time intervals in which the variability of a given variable is more important. In an application of the new methods, I look for geographical clusters of high-variability income in France and then for residuals exhibiting higher variance in a linear regression context.

Lionel Cucala

Scan Statistics for Detecting High-Variance Clusters

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Journal of Probability and Statistics