{"id":"https://openalex.org/W7223027820","doi":"https://doi.org/10.48550/arxiv.2610.24575","title":"The LR1 Lifetime Distribution: Properties, Nonmonotone Hazard, and Statistical Inference","display_name":"The LR1 Lifetime Distribution: Properties, Nonmonotone Hazard, and Statistical Inference","publication_year":2026,"publication_date":"2026-10-17","ids":{"openalex":"https://openalex.org/W7223027820","doi":"https://doi.org/10.48550/arxiv.2610.24575"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.24575","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24575","pdf_url":"https://arxiv.org/pdf/2610.24575","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.24575","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158812663","display_name":"Lalita Singh","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I147727580","display_name":"Dr. Bhimrao Ambedkar University","ror":"https://ror.org/00k4ab982","country_code":"IN","type":"education","lineage":["https://openalex.org/I147727580"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"Singh, Lalita","raw_affiliation_strings":["Agra College , Agra"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Agra College , Agra","institution_ids":["https://openalex.org/I147727580"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5158812663"],"corresponding_institution_ids":["https://openalex.org/I147727580"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10780","display_name":"Reliability and Maintenance Optimization","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2213","display_name":"Safety, Risk, Reliability and Quality"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/gamma-mixture-distribution","display_name":"gamma mixture distribution","score":0.4869999885559082},{"id":"https://openalex.org/keywords/survival-function","display_name":"survival function","score":0.503000020980835},{"id":"https://openalex.org/keywords/order-statistics","display_name":"order statistics","score":0.9390000104904175},{"id":"https://openalex.org/keywords/reliability-analysis","display_name":"reliability analysis","score":0.6060000061988831},{"id":"https://openalex.org/keywords/identifiability","display_name":"identifiability","score":0.8119999766349792},{"id":"https://openalex.org/keywords/maximum-likelihood-estimation","display_name":"maximum likelihood estimation","score":0.7440000176429749},{"id":"https://openalex.org/keywords/expectation-maximization-algorithm","display_name":"expectation-maximization algorithm","score":0.8920000195503235},{"id":"https://openalex.org/keywords/right-censoring","display_name":"right censoring","score":0.6679999828338623},{"id":"https://openalex.org/keywords/asymptotic-confidence-intervals","display_name":"asymptotic confidence intervals","score":0.652999997138977},{"id":"https://openalex.org/keywords/monte-carlo-simulation","display_name":"Monte Carlo simulation","score":0.6819999814033508}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7488999962806702},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.7085999846458435},{"id":"https://openalex.org/C137668524","wikidata":"https://www.wikidata.org/wiki/Q189813","display_name":"Censoring (clinical trials)","level":2,"score":0.6302000284194946},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.5873000025749207},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5584999918937683},{"id":"https://openalex.org/C134261354","wikidata":"https://www.wikidata.org/wiki/Q938438","display_name":"Statistical inference","level":2,"score":0.504800021648407},{"id":"https://openalex.org/C201729545","wikidata":"https://www.wikidata.org/wiki/Q2193887","display_name":"Reliability theory","level":3,"score":0.4643999934196472},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.4602999985218048},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.45410001277923584},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.45210000872612},{"id":"https://openalex.org/C55350006","wikidata":"https://www.wikidata.org/wiki/Q237193","display_name":"Exponential distribution","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.42809998989105225},{"id":"https://openalex.org/C60775368","wikidata":"https://www.wikidata.org/wiki/Q17099337","display_name":"Natural exponential family","level":3,"score":0.4049000144004822},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.37290000915527344},{"id":"https://openalex.org/C149717495","wikidata":"https://www.wikidata.org/wiki/Q117806","display_name":"Gamma distribution","level":2,"score":0.3555000126361847},{"id":"https://openalex.org/C2779793024","wikidata":"https://www.wikidata.org/wiki/Q17299941","display_name":"Indirect Inference","level":3,"score":0.3391000032424927},{"id":"https://openalex.org/C55974624","wikidata":"https://www.wikidata.org/wiki/Q1188504","display_name":"Exponential family","level":2,"score":0.3328999876976013},{"id":"https://openalex.org/C49781872","wikidata":"https://www.wikidata.org/wiki/Q1045555","display_name":"Maximum likelihood","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.328000009059906},{"id":"https://openalex.org/C143791395","wikidata":"https://www.wikidata.org/wiki/Q475208","display_name":"M-estimator","level":3,"score":0.3264000117778778},{"id":"https://openalex.org/C91716921","wikidata":"https://www.wikidata.org/wiki/Q1289366","display_name":"Scale parameter","level":2,"score":0.31929999589920044},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.3176000118255615},{"id":"https://openalex.org/C18294631","wikidata":"https://www.wikidata.org/wiki/Q2621185","display_name":"Location parameter","level":3,"score":0.3174999952316284},{"id":"https://openalex.org/C44082924","wikidata":"https://www.wikidata.org/wiki/Q1767128","display_name":"Order statistic","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C101796028","wikidata":"https://www.wikidata.org/wiki/Q535587","display_name":"Moment-generating function","level":3,"score":0.30559998750686646},{"id":"https://openalex.org/C189508267","wikidata":"https://www.wikidata.org/wiki/Q17088227","display_name":"Density estimation","level":3,"score":0.3046000003814697},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.30169999599456787},{"id":"https://openalex.org/C32645036","wikidata":"https://www.wikidata.org/wiki/Q475164","display_name":"Stochastic ordering","level":2,"score":0.2955999970436096},{"id":"https://openalex.org/C52740198","wikidata":"https://www.wikidata.org/wiki/Q1539564","display_name":"Importance sampling","level":3,"score":0.2953999936580658},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.28610000014305115},{"id":"https://openalex.org/C44249647","wikidata":"https://www.wikidata.org/wiki/Q208498","display_name":"Confidence interval","level":2,"score":0.2840999960899353},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.2761000096797943},{"id":"https://openalex.org/C199435849","wikidata":"https://www.wikidata.org/wiki/Q3179293","display_name":"Shape parameter","level":2,"score":0.2727000117301941},{"id":"https://openalex.org/C43214815","wikidata":"https://www.wikidata.org/wiki/Q7310987","display_name":"Reliability (semiconductor)","level":3,"score":0.27250000834465027},{"id":"https://openalex.org/C48057537","wikidata":"https://www.wikidata.org/wiki/Q3551145","display_name":"Statistical theory","level":2,"score":0.27149999141693115},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.2660999894142151},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.2603999972343445}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.24575","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24575","pdf_url":"https://arxiv.org/pdf/2610.24575","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.24575","doi":"https://doi.org/10.48550/arxiv.2610.24575","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.24575","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.24575","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.24575","pdf_url":"https://arxiv.org/pdf/2610.24575","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223027820.pdf","grobid_xml":"https://content.openalex.org/works/W7223027820.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7222065759","https://openalex.org/W2056519829","https://openalex.org/W4406462050","https://openalex.org/W4400780064","https://openalex.org/W7167247660","https://openalex.org/W2085793343","https://openalex.org/W2086341043","https://openalex.org/W4200532007","https://openalex.org/W2946611877","https://openalex.org/W7165549759"],"abstract_inverted_index":{"The":[0,90,161],"LR1":[1],"distribution":[2],"links":[3],"an":[4,61,71],"exponential":[5,29],"lifetime":[6,11],"and":[7,21,46,59,87,96,123,129,152,155,170,174],"a":[8,13,24,40,103,119,165],"four-stage":[9],"gamma":[10],"through":[12],"single":[14],"positive":[15,109],"parameter":[16,63],"$\u03bb$.":[17],"Their":[18],"rates,":[19],"$2\u03bb$":[20],"$\u03bb/2$,":[22],"have":[23],"fixed":[25,166],"ratio,":[26],"whereas":[27],"the":[28,35,100,115,130,148,156],"mixing":[30],"probability":[31],"$\u03bb^2/(1+\u03bb^2)$":[32],"changes":[33],"with":[34,70],"parameter.":[36],"This":[37],"restriction":[38],"produces":[39],"hazard":[41],"that":[42],"decreases":[43],"near":[44],"zero":[45,93],"eventually":[47],"increases":[48],"toward":[49],"$\u03bb/2$":[50],"for":[51,107],"every":[52,108],"$\u03bb>0$.":[53],"We":[54],"establish":[55],"this":[56],"behavior":[57],"analytically":[58],"obtain":[60],"explicit":[62],"threshold":[64],"separating":[65],"decreasing":[66],"densities":[67,69],"from":[68],"interior":[72],"local":[73],"peak.":[74],"Distributional":[75],"calculations":[76],"provide":[77],"survival":[78],"probabilities,":[79],"moments,":[80,83],"transforms,":[81],"truncated":[82],"quantiles,":[84],"order":[85],"statistics,":[86],"reliability":[88],"summaries.":[89],"density":[91],"at":[92],"establishes":[94],"identifiability,":[95],"strict":[97],"decrease":[98],"of":[99,150,158],"mean":[101],"ensures":[102],"unique":[104],"moment":[105,153],"estimate":[106],"sample":[110],"mean.":[111],"Likelihood":[112],"inference":[113],"includes":[114],"score,":[116],"observed":[117,183],"information,":[118],"constrained":[120],"expectation--maximization":[121],"algorithm,":[122],"asymptotic":[124],"confidence":[125],"limits;":[126],"spacing":[127],"estimation":[128],"likelihood":[131,151],"under":[132],"independent":[133],"right":[134],"censoring":[135],"are":[136],"also":[137],"described.":[138],"Eight":[139],"Monte":[140],"Carlo":[141],"settings,":[142],"comprising":[143],"4,000":[144],"generated":[145],"datasets,":[146],"quantify":[147],"error":[149],"estimators":[154],"coverage":[157],"log-scale":[159],"intervals.":[160],"results":[162],"apply":[163],"in":[164],"working":[167],"time":[168],"unit":[169],"concern":[171],"mathematical":[172],"properties":[173],"correctly":[175],"specified":[176],"simulation,":[177],"rather":[178],"than":[179],"demonstrated":[180],"superiority":[181],"on":[182],"data.":[184]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-21T00:00:00"}